6081:
5364:
7483:
6076:{\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{\text{pwr}})&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {y_{i}}{p_{i}}}-{\hat {Y}}_{pwr}\right)^{2}\\&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {n}{n}}{\frac {y_{i}}{p_{i}}}-{\frac {n}{n}}\sum _{i=1}^{n}w_{i}y_{i}\right)^{2}={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(n{\frac {y_{i}}{\pi _{i}}}-n{\frac {\sum _{i=1}^{n}w_{i}y_{i}}{n}}\right)^{2}\\&={\frac {n^{2}}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\\&={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\end{aligned}}}
14318:
6955:
13675:
16434:
8639:
7478:{\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{{\text{pwr (known }}N{\text{)}}})&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left({\check {\Delta }}_{ii}{\check {y}}_{i}{\check {y}}_{i}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}}{\pi _{i}}}{\frac {y_{i}}{\pi _{i}}}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}\end{aligned}}}
17580:
14313:{\displaystyle {\begin{aligned}\operatorname {E} &={\frac {\sum \limits _{i=1}^{N}\operatorname {E} }{N}}\\&=\operatorname {E} )^{2}]-{\frac {1}{N}}\operatorname {E} )^{2}]\\&=\left({\frac {N-1}{N}}\right)\sigma _{\text{actual}}^{2}\\\operatorname {E} &={\frac {\sum \limits _{i=1}^{N}w_{i}\operatorname {E} }{V_{1}}}\\&=\operatorname {E} )^{2}]-{\frac {V_{2}}{V_{1}^{2}}}\operatorname {E} )^{2}]\\&=\left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)\sigma _{\text{actual}}^{2}\end{aligned}}}
15995:
17298:
8162:
12807:
4746:
16429:{\displaystyle {\begin{aligned}\mathbf {C} &={\frac {\sum _{i=1}^{N}w_{i}}{\left(\sum _{i=1}^{N}w_{i}\right)^{2}-\sum _{i=1}^{N}w_{i}^{2}}}\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)\\&={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{V_{1}-(V_{2}/V_{1})}}.\end{aligned}}}
10062:
8157:
11128:
10662:
9623:
14953:
17575:{\displaystyle {\begin{aligned}{\bar {\mathbf {x} }}&=\left(\mathbf {C} _{1}^{-1}+\mathbf {C} _{2}^{-1}\right)^{-1}\left(\mathbf {C} _{1}^{-1}\mathbf {x} _{1}+\mathbf {C} _{2}^{-1}\mathbf {x} _{2}\right)\\&={\begin{bmatrix}0.9901&0\\0&0.9901\end{bmatrix}}{\begin{bmatrix}1\\1\end{bmatrix}}={\begin{bmatrix}0.9901\\0.9901\end{bmatrix}}\end{aligned}}}
8634:{\displaystyle {\hat {R}}={\hat {\bar {Y}}}={\frac {\sum _{i=1}^{N}I_{i}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}I_{i}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}'_{i}}{\sum _{i=1}^{N}{\check {1}}'_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y'_{i}}{\sum _{i=1}^{N}w_{i}1'_{i}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}={\bar {y}}_{w}}
1746:
5110:
6290:
12548:
2226:
9632:
10800:
10374:
4464:
4479:
7859:
9363:
19124:
Weighted means are typically used to find the weighted mean of historical data, rather than theoretically generated data. In this case, there will be some error in the variance of each data point. Typically experimental errors may be underestimated due to the experimenter not taking into account all
11772:
Gatz et al. mention that the above formulation was published by
Endlich et al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size, based on the formulation published by Cochran (1977), as an approximation to the ratio
11136:
We have (at least) two versions of variance for the weighted mean: one with known and one with unknown population size estimation. There is no uniformly better approach, but the literature presents several arguments to prefer using the population estimation version (even when the population size is
14700:
8922:
first-order linearization, asymptotics, and bootstrap/jackknife. The Taylor linearization method could lead to under-estimation of the variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be
17287:
17168:
11144:
For the trivial case in which all the weights are equal to 1, the above formula is just like the regular formula for the variance of the mean (but notice that it uses the maximum likelihood estimator for the variance instead of the unbiased variance. I.e.: dividing it by n instead of (n-1)).
6898:
16636:
15627:
6098:
1507:
124:
The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade
15421:
11141:), the version with unknown population mean is considered more stable. Lastly, if the proportion of sampling is negatively correlated with the values (i.e.: smaller chance to sample an observation that is large), then the un-known population size version slightly compensates for that.
3411:
is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities. I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as
9121:
4270:
16940:
7827:
5268:
1891:
13490:
12802:{\displaystyle {\begin{aligned}{\hat {\sigma }}^{2}\ &={\frac {\sum \limits _{i=1}^{N}\left(x_{i}-\mu \right)^{2}}{N}}\\{\hat {\sigma }}_{\mathrm {w} }^{2}&={\frac {\sum \limits _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{*}\right)^{2}}{\sum _{i=1}^{N}w_{i}}}\end{aligned}}}
4858:
11767:
11773:
mean. However, Endlich et al. didn't seem to publish this derivation in their paper (even though they mention they used it), and
Cochran's book includes a slightly different formulation. Still, it's almost identical to the formulations described in previous sections.
18389:
In the scenario described in the previous section, most frequently the decrease in interaction strength obeys a negative exponential law. If the observations are sampled at equidistant times, then exponential decrease is equivalent to decrease by a constant fraction
16705:
The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a
2778:
890:
Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights may not be negative in order for the equation to work. Some may be zero, but not all of them (since division by zero is not allowed).
2007:
13133:
19425:
8917:
depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily
10057:{\displaystyle {\widehat {V({\hat {R}})}}={\frac {1}{{\hat {Z}}^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\frac {y_{i}-{\hat {R}}z_{i}}{\pi _{i}}}{\frac {y_{j}-{\hat {R}}z_{j}}{\pi _{j}}}\right)={\frac {1}{{\hat {Z}}^{2}}}\left}
4741:{\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}={\frac {{\frac {1}{n}}\sum _{i=1}^{n}{\frac {y'_{i}}{p_{i}}}}{N}}\approx {\frac {\sum _{i=1}^{n}{\frac {y'_{i}}{\pi _{i}}}}{N}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}.}
8152:{\displaystyle R={\bar {Y}}={\frac {\sum _{i=1}^{N}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}_{i}}{\sum _{i=1}^{N}{\check {1}}_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y_{i}}{\sum _{i=1}^{N}w_{i}}}}
11123:{\displaystyle {\widehat {V({\bar {y}}_{w})}}={\frac {1}{{\hat {N}}^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}-{\bar {y}}_{w}}{\pi _{i}}}\right)^{2}={\frac {1}{(\sum _{i=1}^{n}w_{i})^{2}}}\sum _{i=1}^{n}w_{i}^{2}(y_{i}-{\bar {y}}_{w})^{2}.}
4293:
15263:
10657:{\displaystyle {\widehat {V({\hat {R}})}}={\widehat {V({\bar {y}}_{w})}}={\frac {1}{{\hat {N}}^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\frac {y_{i}-{\bar {y}}_{w}}{\pi _{i}}}{\frac {y_{j}-{\bar {y}}_{w}}{\pi _{j}}}\right).}
8897:
885:
11137:
known). For example: if all y values are constant, the estimator with unknown population size will give the correct result, while the one with known population size will have some variability. Also, when the sample size itself is random (e.g.: in
17045:
722:
11541:
18807:
must correspond to the actual decrease of interaction strength. If this cannot be determined from theoretical considerations, then the following properties of exponentially decreasing weights are useful in making a suitable choice: at step
2339:
2901:
2394:, is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one).
18025:
4034:
6679:
8758:
17934:
12130:
1352:
19241:
11930:
3382:
14948:{\displaystyle {\begin{aligned}s_{\mathrm {w} }^{2}\ &={\frac {{\hat {\sigma }}_{\mathrm {w} }^{2}}{1-(V_{2}/V_{1}^{2})}}\\&={\frac {\sum \limits _{i=1}^{N}w_{i}(x_{i}-\mu ^{*})^{2}}{V_{1}-(V_{2}/V_{1})}},\end{aligned}}}
5358:
The above formula was taken from
Sarndal et al. (1992) (also presented in Cochran 1977), but was written differently. The left side is how the variance was written and the right side is how we've developed the weighted version:
9618:{\displaystyle {\hat {R}}={\frac {\hat {Y}}{\hat {Z}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}z'_{i}}}\approx R+{\frac {1}{Z}}\sum _{i=1}^{n}\left({\frac {y'_{i}}{\pi _{i}}}-R{\frac {z'_{i}}{\pi _{i}}}\right)}
19737:
17174:
17055:
6576:
1741:{\displaystyle {\bar {x}}={\frac {\sum _{i=1}^{n}\left({\dfrac {x_{i}}{\sigma _{i}^{2}}}\right)}{\sum _{i=1}^{n}{\dfrac {1}{\sigma _{i}^{2}}}}}={\frac {\sum _{i=1}^{n}\left(x_{i}\cdot w_{i}\right)}{\sum _{i=1}^{n}w_{i}}},}
18245:
but also on its past values. Commonly, the strength of this dependence decreases as the separation of observations in time increases. To model this situation, one may replace the independent variable by its sliding mean
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4115:
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264:
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15958:
5122:
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5353:
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3297:
1032:
12393:
2539:
15841:
8933:
6285:{\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)}
15020:
10750:
10217:
2973:
451:
17303:
14705:
12213:
6355:
5105:{\displaystyle {\hat {Y}}_{pwr}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {y'_{i}}{p_{i}}}=\sum _{i=1}^{n}{\frac {y'_{i}}{np_{i}}}\approx \sum _{i=1}^{n}{\frac {y'_{i}}{\pi _{i}}}=\sum _{i=1}^{n}w_{i}y'_{i}}
3834:
3744:
14510:
1761:
1188:
13315:
4136:
3924:
1959:
16825:
12304:
9356:
11606:
19819:
7665:
604:
532:
19621:
18373:
16000:
13680:
12911:
12553:
12500:
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5369:
1496:
951:
11598:
269:
Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed.
10127:
171:
6752:
2221:{\displaystyle \sigma _{\bar {x}}^{2}=\sum _{i=1}^{n}{w_{i}'^{2}\sigma _{i}^{2}}={\frac {\sum _{i=1}^{n}{\sigma _{i}^{-4}\sigma _{i}^{2}}}{\left(\sum _{i=1}^{n}\sigma _{i}^{-2}\right)^{2}}}.}
18538:
18074:
12024:
2670:
14689:
14448:
3190:
is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The
125:(independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students):
15728:
14553:
16953:
12989:
19471:
19288:
13667:
9215:
7660:
14382:
14622:
20214:
9262:
57:), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in
19518:
18916:
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1999:
374:
330:
19557:
12864:
10331:
10278:
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4803:
3618:
3524:
10795:
6950:
19110:
The concept of weighted average can be extended to functions. Weighted averages of functions play an important role in the systems of weighted differential and integral calculus.
13541:
12540:
4459:{\displaystyle \operatorname {Var} \left({\hat {\bar {Y}}}_{{\text{known }}N}\right)={\frac {1}{N^{2}}}{\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}}
2449:
15877:
15143:
2619:
19060:
11967:
1260:
18420:
17668:
19276:
15151:
15070:
733:
18990:
16695:
13195:
12245:
11827:
1438:
18762:
4042:
perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.:
2572:
17772:
17690:
16760:
15987:
612:
18452:
16735:
15107:
12938:
1222:
18848:
17750:
7593:
7524:
3184:
2392:
19150:
13307:
13271:
13233:
11167:
6489:
12447:
2237:
16473:
12313:
Because one can always transform non-normalized weights to normalized weights, all formulas in this section can be adapted to non-normalized weights by replacing all
10367:
18946:
8767:
18595:
18568:
18243:
18216:
18169:
17721:
15758:
13588:
9289:
7854:
3471:
3444:
3409:
3223:
3100:
1399:
17824:
13635:
12941:
4828:
3767:
3103:
2786:
2653:
1194:
17945:
19125:
sources of error in calculating the variance of each data point. In this event, the variance in the weighted mean must be corrected to account for the fact that
19100:
19080:
19010:
18805:
18782:
18472:
18284:
18264:
18189:
18142:
18122:
18102:
17844:
17585:
which makes sense: the estimate is "compliant" in the second component and the estimate is compliant in the first component, so the weighted mean is nearly .
13561:
9295:, the variance calculation would look the same. When all weights are equal to one another, this formula is reduced to the standard unbiased variance estimator.
7613:
7564:
7544:
4853:
3854:
3155:
11781:
Because there is no closed analytical form for the variance of the weighted mean, it was proposed in the literature to rely on replication methods such as the
6581:
68:. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in
8644:
17856:
19158:
3929:
3302:
11161:
linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents):
17282:{\displaystyle \mathbf {x} _{2}:={\begin{bmatrix}0&1\end{bmatrix}}^{\top },\qquad \mathbf {C} _{2}:={\begin{bmatrix}100&0\\0&1\end{bmatrix}}}
17163:{\displaystyle \mathbf {x} _{1}:={\begin{bmatrix}1&0\end{bmatrix}}^{\top },\qquad \mathbf {C} _{1}:={\begin{bmatrix}1&0\\0&100\end{bmatrix}}}
12032:
19633:
11835:
1272:
20007:"Statistical Analysis of Precipitation Chemistry Measurements over the Eastern United States. Part I: Seasonal and Regional Patterns and Correlations"
17049:
For example, consider the weighted mean of the point with high variance in the second component and with high variance in the first component. Then
13617:), we can determine a correction factor to yield an unbiased estimator. Assuming each random variable is sampled from the same distribution with mean
10133:, it would include many combinations of covariances that will depend on the indicator variables. If the selection probability are uncorrelated (i.e.:
9291:
by some factor would lead to the same estimator. It also means that if we scale the sum of weights to be equal to a known-from-before population size
6893:{\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}}
16631:{\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{1-V_{2}}}.}
15622:{\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{V_{1}-1}}.}
6494:
3473:), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term:
2454:
15416:{\displaystyle \mathbf {C} ={\frac {\sum _{i=1}^{N}w_{i}\left(\mathbf {x} _{i}-\mu ^{*}\right)^{T}\left(\mathbf {x} _{i}-\mu ^{*}\right)}{V_{1}}}.}
4045:
181:
15885:
2978:
16775:
12247:, when all weights except one are zero. Its minimum value is found when all weights are equal (i.e., unweighted mean), in which case we have
5273:
18603:
6360:
3236:
1044:
175:
Or, this can be accomplished by weighting the class means by the number of students in each class. The larger class is given more "weight":
10333:. This helps illustrate that this formula incorporates the effect of correlation between y and z on the variance of the ratio estimators.
120:
Afternoon class = {81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99}
19844:
Technically, negatives may be used if all the values are either zero or negatives. This fills no function however as the weights work as
9116:{\displaystyle {\widehat {V({\bar {y}}_{w})}}={\frac {1}{(\sum _{i=1}^{n}w_{i})^{2}}}\sum _{i=1}^{n}w_{i}^{2}(y_{i}-{\bar {y}}_{w})^{2}}
128:
19966:
Gatz, Donald F.; Smith, Luther (June 1995). "The standard error of a weighted mean concentration—I. Bootstrapping vs other methods".
16935:{\displaystyle {\bar {\mathbf {x} }}=\mathbf {C} _{\bar {\mathbf {x} }}\left(\sum _{i=1}^{n}\mathbf {W} _{i}\mathbf {x} _{i}\right),}
4284:
959:
14961:
13197:
are drawn from the same distribution, then we can treat this set as an unweighted sample, or we can treat it as the weighted sample
12316:
10669:
10136:
281:
weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such a linear combination is called a
15766:
5263:{\displaystyle \operatorname {Var} ({\hat {Y}}_{pwr})={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}}
2906:
1886:{\displaystyle \sigma _{\bar {x}}={\sqrt {\frac {1}{\sum _{i=1}^{n}\sigma _{i}^{-2}}}}={\sqrt {\frac {1}{\sum _{i=1}^{n}w_{i}}}},}
385:
13485:{\displaystyle s^{2}\ ={\frac {\sum _{i=1}^{N}w_{i}}{\sum _{i=1}^{N}w_{i}-1}}\sum _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{*}\right)^{2}}
4265:{\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}\approx {\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}}
6297:
3776:
3623:
14453:
11762:{\displaystyle {\widehat {\sigma _{\bar {x}}^{2}}}={\frac {n}{(n-1)(n{\bar {w}})^{2}}}\sum w_{i}^{2}(x_{i}-{\bar {x}}_{w})^{2}}
2656:
random variables), then the variance of the weighted mean can be estimated as the multiplication of the unweighted variance by
20103:
7822:{\displaystyle {\hat {N}}=\sum _{i=1}^{n}w_{i}I_{i}=\sum _{i=1}^{n}{\frac {I_{i}}{\pi _{i}}}=\sum _{i=1}^{n}{\check {1}}'_{i}}
3863:
1899:
1133:
20267:
20201:
19934:
19880:
18384:
9305:
2638:
12250:
8914:
3229:
is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.:
13563:). In any case, the information on total number of samples is necessary in order to obtain an unbiased correction, even if
12138:
540:
468:
19566:
18292:
8641:. As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write:
2400:
15426:
Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights.
12873:
12462:
3111:
2577:
1446:
11549:
2773:{\displaystyle \operatorname {Var} ({\bar {y}}_{w})={\hat {\sigma }}_{y}^{2}{\frac {\overline {w^{2}}}{{\bar {w}}^{2}}}}
20072:"Weighted Standard Error and its Impact on Significance Testing (WinCross vs. Quantum & SPSS), Dr. Albert Madansky"
10070:
14691:, ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution.
6687:
20224:
20177:
19993:
18480:
18036:
13128:{\displaystyle s^{2}\ ={\frac {\sum \limits _{i=1}^{N}w_{i}\left(x_{i}-\mu ^{*}\right)^{2}}{\sum _{i=1}^{N}w_{i}-1}}}
11979:
11132:
A similar re-creation of the proof (up to some mistakes at the end) was provided by Thomas Lumley in crossvalidated.
19420:{\displaystyle \chi _{\nu }^{2}={\frac {1}{(n-1)}}\sum _{i=1}^{n}{\frac {(x_{i}-{\bar {x}})^{2}}{\sigma _{i}^{2}}};}
14631:
14387:
4855:-expanded with replacement estimator, or "probability with replacement" estimator). With the above notation, it is:
19779:
15666:
14515:
897:
19433:
13640:
9173:
7618:
20031:
20006:
14326:
20320:
14558:
4806:
14625:
9220:
19483:
18853:
15445:
2657:
1964:
338:
294:
17:
19523:
12815:
9130:
7497:
The previous section dealt with estimating the population mean as a ratio of an estimated population total (
4754:
3529:
10755:
8764:
for specific values of y and w, but the statistical properties comes when including the indicator variable
7546:), and the variance was estimated in that context. Another common case is that the population size itself (
6910:
2345:
20156:
20046:
13498:
12509:
3749:
When each element of the sample is inflated by the inverse of its selection probability, it is termed the
20325:
15853:
15258:{\displaystyle \mathbf {\mu ^{*}} ={\frac {\sum _{i=1}^{N}w_{i}\mathbf {x} _{i}}{\sum _{i=1}^{N}w_{i}}}.}
15119:
10130:
880:{\displaystyle {\bar {x}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}.}
20151:
Mark
Galassi, Jim Davies, James Theiler, Brian Gough, Gerard Jungman, Michael Booth, and Fabrice Rossi.
19015:
11938:
8923:
relatively accurate even for medium sample sizes. For when the sampling has a random sample size (as in
1440:, all having the same mean, one possible choice for the weights is given by the reciprocal of variance:
1231:
19560:
18393:
17606:
15637:
13598:
12307:
11786:
11154:
10283:
10230:
3476:
1363:
19249:
15046:
12963:). In the weighted setting, there are actually two different unbiased estimators, one for the case of
12449:
is used, the variance of the weighted sample is different from the variance of the unweighted sample.
19624:
18951:
17594:
17040:{\displaystyle \mathbf {C} _{\bar {\mathbf {x} }}=\left(\sum _{i=1}^{n}\mathbf {W} _{i}\right)^{-1},}
16943:
16644:
13144:
12218:
11800:
10129:
is the estimated covariance between the estimated sum of Y and estimated sum of Z. Since this is the
1411:
1038:
One can always normalize the weights by making the following transformation on the original weights:
717:{\displaystyle {\bar {x}}={\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}}{\sum \limits _{i=1}^{n}w_{i}}},}
19082:
observations matter and the effect of the remaining observations can be ignored safely, then choose
2661:
39:
18714:
17847:
16697:, then the weighted mean and covariance reduce to the unweighted sample mean and covariance above.
2544:
1402:
17755:
17673:
16743:
15970:
11536:{\displaystyle {\widehat {\sigma _{{\bar {x}}_{w}}^{2}}}={\frac {n}{(n-1)(n{\bar {w}})^{2}}}\left}
3075:
perspective, we are interested in estimating the variance of the weighted mean when the different
19804:
19784:
18425:
16713:
15079:
12916:
2334:{\displaystyle {\bar {x}}=\sigma _{\bar {x}}^{2}\sum _{i=1}^{n}{\frac {x_{i}}{\sigma _{i}^{2}}}.}
1263:
1200:
35:
19953:), How to estimate the (approximate) variance of the weighted mean?, URL (version: 2021-06-08):
18811:
17726:
7569:
7500:
3160:
2368:
2348:
of the mean of the probability distributions under the assumption that they are independent and
113:
Morning class = {62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98}
19794:
19789:
19128:
17598:
13279:
13238:
13200:
12960:
8892:{\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}}
6464:
3195:
1369:
58:
15035:
As a side note, other approaches have been described to compute the weighted sample variance.
12425:
894:
The formulas are simplified when the weights are normalized such that they sum up to 1, i.e.,
19950:
19809:
16445:
10339:
2896:{\displaystyle {\hat {\sigma }}_{y}^{2}={\frac {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}{n-1}}}
20300:
20260:
Data
Fitting and Uncertainty (A practical introduction to weighted least squares and beyond)
20071:
19875:
Cochran, W. G. (1977). Sampling
Techniques (3rd ed.). Nashville, TN: John Wiley & Sons.
18921:
18020:{\displaystyle {\bar {x}}=\sigma _{\bar {x}}^{2}(\mathbf {J} ^{T}\mathbf {W} \mathbf {X} ),}
20018:
19975:
18573:
18546:
18221:
18194:
18147:
17699:
15736:
15657:
13566:
11782:
9267:
7832:
3449:
3422:
3387:
3201:
3078:
2349:
1377:
69:
17784:
13620:
4813:
3752:
8:
19279:
6674:{\displaystyle {\check {\Delta }}_{ii}=1-{\frac {\pi _{i}\pi _{i}}{\pi _{i}}}=1-\pi _{i}}
20022:
19979:
8753:{\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y_{i}}{\sum _{i=1}^{n}w_{i}}}}
3106:
random variables. An alternative perspective for this problem is that of some arbitrary
3063:
observations. This has led to the development of alternative, more general, estimators.
20134:
20122:
19769:
19764:
19085:
19065:
18995:
18790:
18767:
18457:
18269:
18249:
18174:
18127:
18107:
18087:
17929:{\displaystyle \sigma _{\bar {x}}^{2}=(\mathbf {J} ^{T}\mathbf {W} \mathbf {J} )^{-1},}
17829:
16763:
16707:
15025:
The degrees of freedom of the weighted, unbiased sample variance vary accordingly from
13546:
12948:
12419:
7598:
7549:
7529:
4838:
3839:
3140:
3118:
282:
19236:{\displaystyle {\hat {\sigma }}_{\bar {x}}^{2}=\sigma _{\bar {x}}^{2}\chi _{\nu }^{2}}
4029:{\displaystyle {\check {y}}'_{i}=I_{i}{\check {y}}_{i}={\frac {I_{i}y_{i}}{\pi _{i}}}}
3377:{\displaystyle P(I_{i}=1|{\text{one sample draw}})=p_{i}\approx {\frac {\pi _{i}}{n}}}
20263:
20246:
20220:
20197:
20173:
20126:
20005:
Endlich, R. M.; Eymon, B. P.; Ferek, R. J.; Valdes, A. D.; Maxwell, C. (1988-12-01).
19987:
19930:
19876:
17693:
16738:
15641:
13602:
20138:
15733:(If they are not, divide the weights by their sum to normalize prior to calculating
20118:
20026:
19983:
19754:
13309:
are normalized to 1, then the correct expression after Bessel's correction becomes
12983:(where a weight equals the number of occurrences), then the unbiased estimator is:
11138:
8924:
7829:. With the above notation, the parameter we care about is the ratio of the sums of
6090:
3926:. As above, we can add a tick mark if multiplying by the indicator function. I.e.:
3413:
3230:
2641:
20286:
12125:{\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}\sum _{i=1}^{n}{w_{i}'^{2}},}
3446:) is fixed, and the randomness comes from it being included in the sample or not (
3059:
However, this estimation is rather limited due to the strong assumption about the
1347:{\textstyle \sigma _{\bar {x}}=\sigma {\sqrt {\sum \limits _{i=1}^{n}w_{i}'^{2}}}}
19845:
19824:
19799:
19774:
19732:{\displaystyle \sigma ^{2}={\frac {\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}}{n-1}}.}
12457:
11925:{\displaystyle \sigma _{\bar {x}}^{2}=\sum _{i=1}^{n}{w_{i}'^{2}\sigma _{i}^{2}}}
8903:
4276:
3191:
3107:
535:
65:
50:
31:
16770:(both denoted in the same way, via superscripts); the weight matrix then reads:
9302:
The Taylor linearization states that for a general ratio estimator of two sums (
20315:
17779:
16767:
15633:
13594:
6571:{\displaystyle {\check {\Delta }}_{ij}=1-{\frac {\pi _{i}\pi _{j}}{\pi _{ij}}}}
3233:). The probability of some element to be chosen, given a sample, is denoted as
2649:
15640:, these processes changing the data's mean and variance and thus leading to a
13601:, these processes changing the data's mean and variance and thus leading to a
61:
and also occurs in a more general form in several other areas of mathematics.
20309:
20250:
17775:
15847:
15113:
11158:
8919:
7488:
4110:{\displaystyle w_{i}={\frac {1}{\pi _{i}}}\approx {\frac {1}{n\times p_{i}}}}
1128:
19954:
6089:
An alternative term, for when the sampling has a random sample size (as in
259:{\displaystyle {\bar {x}}={\frac {(20\times 80)+(30\times 90)}{20+30}}=86.}
20152:
20130:
19924:
17850:
states that the estimate of the mean having minimum variance is given by:
10752:), and when assuming the probability of each element is very small (i.e.:
6754:), and when assuming the probability of each element is very small, then:
4120:
1190:
is a special case of the weighted mean where all data have equal weights.
16947:
15953:{\displaystyle \mathbf {\mu ^{*}} =\sum _{i=1}^{N}w_{i}\mathbf {x} _{i}.}
3050:{\displaystyle {\overline {w^{2}}}={\frac {\sum _{i=1}^{n}w_{i}^{2}}{n}}}
15644:(the population count, which is a requirement for Bessel's correction).
15032:
The standard deviation is simply the square root of the variance above.
13605:(the population count, which is a requirement for Bessel's correction).
64:
If all the weights are equal, then the weighted mean is the same as the
19475:
standard error of the weighted mean (variance weights, scale corrected)
5348:{\displaystyle {\overline {wy}}=\sum _{i=1}^{n}{\frac {w_{i}y_{i}}{n}}}
2574:, then the expectation of the weighted sample mean will be that value,
18701:{\displaystyle V_{1}=\sum _{i=1}^{m}{w^{i-1}}={\frac {1-w^{m}}{1-w}},}
6454:{\displaystyle C(I_{i},I_{j})=\pi _{ij}-\pi _{i}\pi _{j}=\Delta _{ij}}
3292:{\displaystyle P(I_{i}=1\mid {\text{Some sample of size }}n)=\pi _{i}}
1116:{\displaystyle w_{i}'={\frac {w_{i}}{\sum \limits _{j=1}^{n}{w_{j}}}}}
20291:
19814:
14628:). This means that to unbias our estimator we need to pre-divide by
13138:
This effectively applies Bessel's correction for frequency weights.
12955:
in the denominator (corresponding to the sample size) is changed to
12415:
8761:
2645:
2001:. It is a special case of the general formula in previous section,
1753:
standard error of the weighted mean (with inverse-variance weights)
1406:
19820:
Standard error of a proportion estimation when using weighted data
2344:
The significance of this choice is that this weighted mean is the
953:. For such normalized weights, the weighted mean is equivalently:
19749:
54:
20216:
The First
Systems of Weighted Differential and Integral Calculus
12414:
Typically when a mean is calculated it is important to know the
11157:
methods, the following (variance estimation of ratio-mean using
11153:
It has been shown, by Gatz et al. (1995), that in comparison to
288:
Using the previous example, we would get the following weights:
12951:
for the population variance. In normal unweighted samples, the
2633:
When treating the weights as constants, and having a sample of
1374:
For the weighted mean of a list of data for which each element
1027:{\displaystyle {\bar {x}}=\sum \limits _{i=1}^{n}{w_{i}'x_{i}}}
15015:{\displaystyle \operatorname {E} =\sigma _{\text{actual}}^{2}}
12388:{\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{n}{w_{i}}}}}
2534:{\displaystyle E({\bar {x}})=\sum _{i=1}^{n}{w_{i}'\mu _{i}}.}
20047:"GNU Scientific Library – Reference Manual: Weighted Samples"
12409:
10745:{\displaystyle \forall i\neq j:\Delta _{ij}=C(I_{i},I_{j})=0}
10212:{\displaystyle \forall i\neq j:\Delta _{ij}=C(I_{i},I_{j})=0}
4831:-estimator. This estimator can be itself estimated using the
462:
15836:{\displaystyle w_{i}'={\frac {w_{i}}{\sum _{i=1}^{N}w_{i}}}}
2968:{\displaystyle {\bar {w}}={\frac {\sum _{i=1}^{n}w_{i}}{n}}}
446:{\displaystyle {\bar {x}}=(0.4\times 80)+(0.6\times 90)=86.}
20243:
Data
Reduction and Error Analysis for the Physical Sciences
20032:
10.1175/1520-0450(1988)027<1322:SAOPCM>2.0.CO;2
19759:
19113:
16439:
The reasoning here is the same as in the previous section.
15632:
This estimator can be unbiased only if the weights are not
11976:
Consequently, if all the observations have equal variance,
9358:), they can be expanded around the true value R, and give:
30:"Weighted average" redirects here. Not to be confused with
20172:(2nd ed.). Singapore: World Scientific. p. 324.
19951:
https://stats.stackexchange.com/users/249135/thomas-lumley
13593:
The estimator can be unbiased only if the weights are not
6350:{\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}}
3829:{\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}}
3739:{\displaystyle V=y_{i}^{2}V=y_{i}^{2}\pi _{i}(1-\pi _{i})}
14505:{\displaystyle \left(1-{\frac {V_{2}}{V_{1}^{2}}}\right)}
3137:) and dividing it by the population size – either known (
1183:{\textstyle {\frac {1}{n}}\sum \limits _{i=1}^{n}{x_{i}}}
3919:{\displaystyle {\frac {y_{i}}{p_{i}}}=n{\check {y}}_{i}}
3125:, is calculated by taking an estimation of the total of
1954:{\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}/n}
1195:
independent and identically distributed random variables
20153:
GNU Scientific
Library - Reference manual, Version 1.15
20004:
19925:
Carl-Erik
Sarndal; Bengt Swensson; Jan Wretman (1992).
16820:{\displaystyle \mathbf {W} _{i}=\mathbf {C} _{i}^{-1}.}
16442:
Since we are assuming the weights are normalized, then
12870:(and thus are random variables), it can be shown that
12299:{\textstyle \sigma _{\bar {x}}=\sigma _{0}/{\sqrt {n}}}
9351:{\displaystyle {\hat {R}}={\frac {\hat {Y}}{\hat {Z}}}}
20283:
18079:
17547:
17518:
17482:
17248:
17199:
17129:
17080:
15449:
12253:
12208:{\textstyle 1/n\leq \sum _{i=1}^{n}{w_{i}'^{2}}\leq 1}
12141:
7566:) is unknown and is estimated using the sample (i.e.:
1275:
1136:
900:
19636:
19569:
19526:
19486:
19436:
19291:
19252:
19161:
19131:
19088:
19068:
19018:
18998:
18954:
18924:
18856:
18814:
18793:
18770:
18717:
18606:
18576:
18570:
is the sum of the unnormalized weights. In this case
18549:
18483:
18460:
18428:
18396:
18295:
18272:
18252:
18224:
18197:
18177:
18150:
18130:
18110:
18090:
18039:
17948:
17859:
17832:
17787:
17758:
17729:
17702:
17676:
17609:
17301:
17177:
17058:
16956:
16835:
16778:
16746:
16716:
16647:
16484:
16448:
15998:
15973:
15888:
15856:
15769:
15739:
15669:
15475:
15448:
15277:
15154:
15122:
15082:
15049:
14964:
14703:
14634:
14561:
14518:
14456:
14390:
14329:
13678:
13643:
13623:
13590:
has a different meaning other than frequency weight.
13569:
13549:
13501:
13318:
13282:
13241:
13203:
13147:
12992:
12919:
12876:
12818:
12551:
12512:
12465:
12428:
12319:
12221:
12035:
11982:
11941:
11838:
11803:
11609:
11552:
11170:
10803:
10758:
10672:
10666:
If the selection probability are uncorrelated (i.e.:
10377:
10342:
10286:
10233:
10139:
10073:
9635:
9366:
9308:
9270:
9223:
9176:
9133:
8936:
8770:
8647:
8165:
7862:
7835:
7668:
7621:
7601:
7572:
7552:
7532:
7503:
6958:
6913:
6763:
6690:
6684:
If the selection probability are uncorrelated (i.e.:
6584:
6497:
6467:
6363:
6300:
6101:
5367:
5276:
5125:
4861:
4841:
4816:
4757:
4482:
4296:
4139:
4048:
3932:
3866:
3842:
3779:
3755:
3626:
3532:
3479:
3452:
3425:
3390:
3305:
3239:
3204:
3163:
3143:
3081:
2981:
2909:
2789:
2673:
2580:
2547:
2457:
2403:
2371:
2240:
2010:
1967:
1902:
1764:
1613:
1555:
1510:
1449:
1414:
1380:
1234:
1203:
1047:
962:
736:
615:
599:{\displaystyle \left(w_{1},w_{2},\dots ,w_{n}\right)}
543:
527:{\displaystyle \left(x_{1},x_{2},\dots ,x_{n}\right)}
471:
388:
341:
297:
184:
131:
19616:{\displaystyle \sigma _{\bar {x}}^{2}=\sigma ^{2}/n}
18368:{\displaystyle z_{k}=\sum _{i=1}^{m}w_{i}x_{k+1-i}.}
18084:
Consider the time series of an independent variable
16950:), in terms of the covariance of the weighted mean:
3121:, the population mean, of some quantity of interest
18378:
14694:The final unbiased estimate of sample variance is:
14555:bias in the unweighted estimator (also notice that
12906:{\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}}
12495:{\displaystyle {\hat {\sigma }}_{\mathrm {w} }^{2}}
4133:is known we can estimate the population mean using
1491:{\displaystyle w_{i}={\frac {1}{\sigma _{i}^{2}}}.}
19731:
19615:
19551:
19512:
19465:
19419:
19270:
19235:
19152:is too large. The correction that must be made is
19144:
19094:
19074:
19054:
19004:
18984:
18940:
18910:
18842:
18799:
18776:
18756:
18700:
18589:
18562:
18532:
18466:
18446:
18414:
18367:
18278:
18258:
18237:
18210:
18183:
18163:
18136:
18116:
18096:
18068:
18019:
17928:
17838:
17818:
17766:
17744:
17715:
17684:
17662:
17574:
17281:
17162:
17039:
16934:
16819:
16754:
16729:
16689:
16630:
16467:
16428:
15981:
15952:
15871:
15835:
15752:
15722:
15621:
15458:
15415:
15257:
15137:
15101:
15064:
15014:
14947:
14683:
14616:
14547:
14504:
14442:
14376:
14312:
13661:
13629:
13582:
13555:
13535:
13484:
13301:
13265:
13227:
13189:
13127:
12932:
12905:
12858:
12801:
12534:
12494:
12441:
12387:
12298:
12239:
12207:
12124:
12018:
11971:standard error of the weighted mean (general case)
11961:
11924:
11821:
11761:
11593:{\displaystyle {\bar {w}}={\frac {\sum w_{i}}{n}}}
11592:
11535:
11122:
10789:
10744:
10656:
10361:
10325:
10272:
10211:
10121:
10056:
9617:
9350:
9283:
9256:
9209:
9162:
9115:
8891:
8752:
8633:
8151:
7848:
7821:
7654:
7607:
7587:
7558:
7538:
7518:
7477:
6944:
6892:
6746:
6673:
6570:
6491:is the probability of selecting both i and j. And
6483:
6453:
6349:
6284:
6075:
5347:
5262:
5104:
4847:
4822:
4797:
4740:
4458:
4264:
4109:
4028:
3918:
3848:
3828:
3761:
3738:
3612:
3518:
3465:
3438:
3403:
3376:
3291:
3217:
3178:
3149:
3094:
3049:
2967:
2895:
2772:
2613:
2566:
2533:
2443:
2386:
2333:
2220:
1993:
1953:
1885:
1740:
1490:
1432:
1393:
1346:
1254:
1216:
1182:
1115:
1026:
945:
879:
716:
598:
526:
461:Formally, the weighted mean of a non-empty finite
445:
368:
324:
258:
165:
19520:, they cancel out in the weighted mean variance,
19105:
10122:{\displaystyle {\hat {C}}({\hat {Y}},{\hat {Z}})}
9264:would give the same estimator, since multiplying
166:{\displaystyle {\bar {x}}={\frac {4300}{50}}=86.}
20307:
20145:
15268:And the weighted covariance matrix is given by:
15072:(each set of single observations on each of the
10219:), this term would still include a summation of
7615:can be described as the sum of weights. So when
6747:{\displaystyle \forall i\neq j:C(I_{i},I_{j})=0}
4473:The general formula can be developed like this:
20104:"Extension of covariance selection mathematics"
19102:such that the tail area is sufficiently small.
18533:{\displaystyle w_{i}={\frac {w^{i-1}}{V_{1}}},}
18069:{\displaystyle \mathbf {W} =\mathbf {C} ^{-1}.}
12410:§ Correcting for over- or under-dispersion
12019:{\displaystyle \sigma _{i}^{2}=\sigma _{0}^{2}}
11776:
3299:, and the one-draw probability of selection is
2231:The equations above can be combined to obtain:
20213:Jane Grossman, Michael Grossman, Robert Katz.
20011:Journal of Applied Meteorology and Climatology
19920:
19918:
19916:
19914:
19912:
19910:
19908:
17588:
14684:{\displaystyle 1-\left(V_{2}/V_{1}^{2}\right)}
14443:{\displaystyle V_{2}=\sum _{i=1}^{N}w_{i}^{2}}
12026:, the weighted sample mean will have variance
11829:, the variance of the weighted sample mean is
3066:
2451:then the weighted sample mean has expectation
946:{\textstyle \sum \limits _{i=1}^{n}{w_{i}'}=1}
20192:G. H. Hardy, J. E. Littlewood, and G. Pólya.
19906:
19904:
19902:
19900:
19898:
19896:
19894:
19892:
19890:
19888:
19871:
19869:
19867:
19865:
19119:
15723:{\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}=1.}
15038:
14548:{\displaystyle \left({\frac {N-1}{N}}\right)}
12947:For small samples, it is customary to use an
11797:For uncorrelated observations with variances
6093:), is presented in Sarndal et al. (1992) as:
272:
20097:
20095:
13296:
13283:
13260:
13242:
13222:
13204:
13184:
13148:
10797:), then the above reduced to the following:
8159:. We can estimate it using our sample with:
20170:Statistical Methods in Experimental Physics
19466:{\displaystyle {\hat {\sigma }}_{\bar {x}}}
17599:Variance § Sum of correlated variables
15967:weighted estimate of the covariance matrix
15442:weighted estimate of the covariance matrix
13662:{\displaystyle \sigma _{\text{actual}}^{2}}
12866:for normalized weights. If the weights are
12403:
11148:
9210:{\displaystyle w_{i}={\frac {1}{\pi _{i}}}}
7655:{\displaystyle w_{i}={\frac {1}{\pi _{i}}}}
4287:), then the variance of this estimator is:
4279:is one that results in a fixed sample size
1357:
20257:
19885:
19862:
18171:. In many common situations, the value of
16700:
14450:. Therefore, the bias in our estimator is
14377:{\displaystyle V_{1}=\sum _{i=1}^{N}w_{i}}
12215:. The variance attains its maximum value,
10131:covariance of two sums of random variables
4805:and it may be estimated by the (unbiased)
456:
20240:
20092:
20030:
19965:
16710:estimate. We simply replace the variance
15466:, with Bessel's correction, is given by:
14617:{\displaystyle \ V_{1}^{2}/V_{2}=N_{eff}}
13273:, and we get the same result either way.
9627:And the variance can be approximated by:
2397:If the observations have expected values
2355:
19955:https://stats.stackexchange.com/q/525770
19114:Correcting for over- or under-dispersion
17752:is the common mean to be estimated, and
15647:
9257:{\displaystyle w_{i}={\frac {1}{p_{i}}}}
20196:(2nd ed.), Cambridge University Press,
19513:{\displaystyle \sigma _{i}=\sigma _{0}}
18911:{\displaystyle {e^{-1}}(1-w)=0.39(1-w)}
18144:observations sampled at discrete times
15459:{\displaystyle \textstyle \mathbf {C} }
15076:random variables) is assigned a weight
13613:If the weights are instead non-random (
12913:is the maximum likelihood estimator of
2541:In particular, if the means are equal,
1994:{\displaystyle \sigma _{i}=\sigma _{0}}
369:{\displaystyle {\frac {30}{20+30}}=0.6}
325:{\displaystyle {\frac {20}{20+30}}=0.4}
14:
20308:
19552:{\displaystyle \sigma _{\bar {x}}^{2}}
15429:
15043:In a weighted sample, each row vector
13608:
12859:{\displaystyle \sum _{i=1}^{N}w_{i}=1}
12422:about that mean. When a weighted mean
9163:{\displaystyle \pi _{i}\approx p_{i}n}
6085:And we got to the formula from above.
4798:{\displaystyle Y=\sum _{i=1}^{N}y_{i}}
3613:{\displaystyle E=y_{i}E=y_{i}\pi _{i}}
109:test grades in each class as follows:
20284:
20167:
20101:
18385:Exponentially weighted moving average
13495:where the total number of samples is
10790:{\displaystyle (1-\pi _{i})\approx 1}
8913:In this case, the variability of the
8906:and it is approximately unbiased for
6945:{\displaystyle (1-\pi _{i})\approx 1}
3129:over all elements in the population (
2628:
13536:{\displaystyle \sum _{i=1}^{N}w_{i}}
12974:
12535:{\displaystyle {\hat {\sigma }}^{2}}
2444:{\displaystyle E(x_{i})={\mu _{i}},}
19480:When all data variances are equal,
18080:Decreasing strength of interactions
16829:The weighted mean in this case is:
15872:{\displaystyle \mathbf {\mu ^{*}} }
15138:{\displaystyle \mathbf {\mu ^{*}} }
14823:
14011:
13725:
13013:
12689:
12589:
12502:is defined similarly to the normal
12398:
4751:The population total is denoted as
3112:selected with unequal probabilities
2614:{\displaystyle E({\bar {x}})=\mu .}
1501:The weighted mean in this case is:
1401:potentially comes from a different
1302:
1226:standard error of the weighted mean
1148:
1078:
979:
902:
678:
635:
379:Then, apply the weights like this:
24:
20301:Tool to calculate Weighted Average
20234:
20123:10.1111/j.1469-1809.1957.tb01874.x
19055:{\displaystyle \leq {e^{-n(1-w)}}}
18850:, the weight approximately equals
18441:
18403:
17603:In the general case, suppose that
17219:
17100:
16641:If all weights are the same, i.e.
14980:
14965:
14752:
14714:
14208:
14190:
14130:
14112:
14041:
13986:
13962:
13879:
13861:
13820:
13802:
13745:
13683:
12892:
12667:
12481:
11962:{\displaystyle \sigma _{\bar {x}}}
11600:. Further simplification leads to
10689:
10673:
10532:
10156:
10140:
9748:
7197:
7079:
6691:
6589:
6502:
6439:
6221:
3186:). In this context, each value of
1255:{\displaystyle \sigma _{\bar {x}}}
534:, with corresponding non-negative
25:
20337:
20277:
18415:{\displaystyle 0<\Delta <1}
17663:{\displaystyle \mathbf {X} =^{T}}
10326:{\displaystyle z'_{i}=I_{i}z_{i}}
10273:{\displaystyle y'_{i}=I_{i}y_{i}}
3526:. With the following expectancy:
3519:{\displaystyle y'_{i}=y_{i}I_{i}}
3225:) that get 1 if some observation
19780:Weighted average cost of capital
19271:{\displaystyle \chi _{\nu }^{2}}
18379:Exponentially decreasing weights
18050:
18041:
18007:
18002:
17991:
17906:
17901:
17890:
17760:
17678:
17611:
17452:
17432:
17417:
17397:
17358:
17335:
17310:
17230:
17180:
17111:
17061:
17010:
16967:
16959:
16914:
16902:
16863:
16855:
16840:
16796:
16781:
16748:
16576:
16535:
16486:
16342:
16301:
16221:
16180:
16004:
15975:
15937:
15895:
15863:
15567:
15526:
15477:
15451:
15369:
15328:
15279:
15206:
15161:
15129:
15065:{\displaystyle \mathbf {x} _{i}}
15052:
12306:, i.e., it degenerates into the
7526:) with a known population size (
80:
20245:. New York, N.Y.: McGraw-Hill.
20207:
20186:
20161:
20102:Price, George R. (April 1972).
19120:§ Weighted sample variance
18985:{\displaystyle {1-e^{-1}}=0.61}
17227:
17108:
16690:{\displaystyle w_{i}/V_{1}=1/N}
15029: − 1 down to 0.
13669:, taking expectations we have,
13190:{\displaystyle \{2,2,4,5,5,5\}}
12240:{\displaystyle \sigma _{0}^{2}}
11822:{\displaystyle \sigma _{i}^{2}}
7489:Variance of the weighted mean (
3110:of the data in which units are
1433:{\displaystyle \sigma _{i}^{2}}
20064:
20039:
19998:
19959:
19943:
19927:Model Assisted Survey Sampling
19838:
19703:
19696:
19674:
19580:
19537:
19456:
19444:
19387:
19380:
19358:
19328:
19316:
19206:
19181:
19169:
19106:Weighted averages of functions
19062:. Where primarily the closest
19046:
19034:
18905:
18893:
18884:
18872:
18828:
18815:
18751:
18739:
18011:
17986:
17974:
17955:
17911:
17885:
17870:
17807:
17788:
17736:
17651:
17618:
17314:
16971:
16867:
16844:
16413:
16385:
14991:
14971:
14932:
14904:
14880:
14853:
14803:
14770:
14744:
14233:
14224:
14220:
14214:
14199:
14196:
14155:
14146:
14142:
14136:
14121:
14118:
14086:
14077:
14050:
14047:
13997:
13978:
13968:
13904:
13895:
13891:
13885:
13870:
13867:
13845:
13836:
13832:
13826:
13811:
13808:
13783:
13774:
13754:
13751:
13711:
13699:
13689:
12884:
12659:
12563:
12520:
12473:
12264:
12046:
11952:
11849:
11792:
11750:
11737:
11714:
11684:
11677:
11665:
11662:
11650:
11623:
11559:
11519:
11512:
11490:
11470:
11457:
11445:
11432:
11400:
11397:
11391:
11369:
11354:
11332:
11319:
11306:
11274:
11254:
11247:
11235:
11232:
11220:
11186:
11108:
11095:
11072:
11024:
10989:
10944:
10918:
10899:
10858:
10833:
10821:
10811:
10778:
10759:
10733:
10707:
10621:
10573:
10535:
10467:
10442:
10430:
10420:
10400:
10394:
10385:
10200:
10174:
10116:
10110:
10095:
10086:
10080:
10046:
10040:
10025:
10016:
10010:
9998:
9976:
9970:
9961:
9948:
9929:
9923:
9914:
9888:
9839:
9788:
9751:
9683:
9658:
9652:
9643:
9399:
9389:
9373:
9341:
9331:
9315:
9104:
9091:
9068:
9020:
8985:
8966:
8954:
8944:
8778:
8655:
8619:
8393:
8348:
8194:
8189:
8172:
8051:
8009:
7875:
7804:
7675:
7579:
7510:
7330:
7311:
7241:
7222:
7200:
7123:
7104:
7082:
7001:
6979:
6969:
6933:
6914:
6809:
6787:
6782:
6770:
6735:
6709:
6592:
6505:
6393:
6367:
6308:
6265:
6246:
6224:
6147:
6125:
6120:
6108:
5499:
5400:
5388:
5378:
5160:
5142:
5132:
5115:The estimated variance of the
4869:
4526:
4497:
4492:
4321:
4316:
4183:
4154:
4149:
4121:Variance of the weighted sum (
3975:
3940:
3904:
3787:
3733:
3714:
3683:
3670:
3646:
3630:
3584:
3571:
3552:
3536:
3338:
3329:
3309:
3273:
3243:
3170:
2916:
2870:
2863:
2841:
2797:
2756:
2715:
2702:
2690:
2680:
2599:
2593:
2584:
2476:
2470:
2461:
2420:
2407:
2378:
2266:
2247:
2021:
1913:
1775:
1517:
1286:
1245:
969:
743:
622:
434:
422:
416:
404:
395:
233:
221:
215:
203:
191:
138:
97:with 20 students, one with 30
13:
1:
19855:
19623:, formulated in terms of the
19559:, which again reduces to the
18757:{\displaystyle V_{1}=1/(1-w)}
10223:covariances for each element
3384:(If N is very large and each
3194:procedure yields a series of
2567:{\displaystyle \mu _{i}=\mu }
2360:
20241:Bevington, Philip R (1969).
19988:10.1016/1352-2310(94)00210-C
17767:{\displaystyle \mathbf {J} }
17685:{\displaystyle \mathbf {C} }
16755:{\displaystyle \mathbf {C} }
15982:{\displaystyle \mathbf {C} }
12967:and another for the case of
11777:Replication-based estimators
6053:
5949:
5287:
5244:
4440:
2994:
2745:
2346:maximum likelihood estimator
7:
19742:
18447:{\displaystyle w=1-\Delta }
18422:at each time step. Setting
17589:Accounting for correlations
17292:then the weighted mean is:
16730:{\displaystyle \sigma ^{2}}
15102:{\displaystyle w_{i}\geq 0}
13235:with corresponding weights
12933:{\displaystyle \sigma ^{2}}
3067:Survey sampling perspective
2623:
1217:{\displaystyle \sigma ^{2}}
75:
10:
20342:
20157:Sec. 21.7 Weighted Samples
19561:standard error of the mean
19117:
18918:, the tail area the value
18843:{\displaystyle (1-w)^{-1}}
18382:
17745:{\displaystyle {\bar {x}}}
17592:
15039:Weighted sample covariance
12407:
12308:standard error of the mean
7588:{\displaystyle {\hat {N}}}
7519:{\displaystyle {\hat {Y}}}
7493:-estimator for ratio-mean)
4807:Horvitz–Thompson estimator
3179:{\displaystyle {\hat {N}}}
2387:{\displaystyle {\bar {x}}}
2365:The weighted sample mean,
1367:
1364:Inverse-variance weighting
1361:
273:Convex combination example
49:is similar to an ordinary
29:
20168:James, Frederick (2006).
19810:Weighted sum of variables
19625:sample standard deviation
19145:{\displaystyle \chi ^{2}}
18104:and a dependent variable
17595:Generalized least squares
13302:{\displaystyle \{w_{i}\}}
13276:If the frequency weights
13266:{\displaystyle \{2,1,3\}}
13228:{\displaystyle \{2,4,5\}}
6484:{\displaystyle \pi _{ij}}
3267:Some sample of size
1193:If the data elements are
53:(the most common type of
20111:Annals of Human Genetics
19831:
18992:. The tail area at step
17696:relating the quantities
16942:(where the order of the
12442:{\displaystyle \mu ^{*}}
12404:Weighted sample variance
11149:Bootstrapping validation
5119:-estimator is given by:
3836:. A related quantity is
1403:probability distribution
1358:Variance-defined weights
47:weighted arithmetic mean
19968:Atmospheric Environment
19805:Weighted moving average
19785:Weighted geometric mean
16701:Vector-valued estimates
16468:{\displaystyle V_{1}=1}
13141:For example, if values
12944:Gaussian observations.
10362:{\displaystyle z_{i}=1}
4129:If the population size
1264:uncertainty propagation
457:Mathematical definition
36:Weighted geometric mean
19795:Weighted least squares
19790:Weighted harmonic mean
19733:
19673:
19617:
19553:
19514:
19467:
19421:
19354:
19272:
19237:
19146:
19096:
19076:
19056:
19006:
18986:
18942:
18941:{\displaystyle e^{-1}}
18912:
18844:
18801:
18778:
18758:
18702:
18640:
18591:
18564:
18534:
18474:normalized weights by
18468:
18448:
18416:
18369:
18329:
18280:
18260:
18239:
18212:
18185:
18165:
18138:
18118:
18098:
18070:
18021:
17930:
17840:
17820:
17768:
17746:
17717:
17686:
17664:
17576:
17283:
17164:
17041:
17007:
16936:
16899:
16821:
16756:
16731:
16691:
16632:
16516:
16469:
16430:
16282:
16161:
16122:
16077:
16038:
15983:
15954:
15924:
15873:
15837:
15819:
15754:
15724:
15703:
15623:
15507:
15460:
15417:
15309:
15259:
15238:
15193:
15139:
15103:
15066:
15016:
14949:
14842:
14685:
14618:
14549:
14506:
14444:
14424:
14378:
14363:
14314:
14030:
13744:
13663:
13631:
13584:
13557:
13537:
13522:
13486:
13431:
13391:
13358:
13303:
13267:
13229:
13191:
13129:
13105:
13032:
12934:
12907:
12860:
12839:
12803:
12781:
12708:
12608:
12536:
12496:
12443:
12389:
12369:
12300:
12241:
12209:
12176:
12126:
12096:
12020:
11963:
11926:
11884:
11823:
11763:
11594:
11537:
11124:
11056:
11012:
10892:
10791:
10746:
10658:
10522:
10501:
10363:
10327:
10274:
10213:
10123:
10058:
9738:
9717:
9619:
9544:
9478:
9432:
9352:
9285:
9258:
9211:
9164:
9117:
9052:
9008:
8893:
8862:
8816:
8754:
8736:
8693:
8635:
8582:
8536:
8483:
8437:
8385:
8340:
8283:
8226:
8153:
8135:
8092:
8043:
8001:
7954:
7907:
7850:
7823:
7796:
7748:
7704:
7656:
7609:
7589:
7560:
7540:
7520:
7479:
7433:
7305:
7187:
7069:
7048:
6946:
6894:
6852:
6748:
6675:
6572:
6485:
6455:
6351:
6286:
6211:
6190:
6077:
6013:
5909:
5806:
5743:
5660:
5586:
5458:
5349:
5315:
5264:
5204:
5106:
5078:
5027:
4971:
4920:
4849:
4824:
4799:
4784:
4742:
4705:
4645:
4585:
4460:
4400:
4266:
4232:
4125:-estimator for totals)
4111:
4030:
3920:
3850:
3830:
3763:
3740:
3614:
3520:
3467:
3440:
3405:
3378:
3293:
3219:
3180:
3151:
3096:
3051:
3025:
2969:
2948:
2897:
2840:
2774:
2615:
2568:
2535:
2502:
2445:
2388:
2356:Statistical properties
2335:
2298:
2222:
2183:
2120:
2056:
1995:
1955:
1887:
1865:
1812:
1742:
1721:
1665:
1611:
1549:
1492:
1434:
1395:
1370:Weighted least squares
1348:
1321:
1256:
1218:
1184:
1167:
1117:
1097:
1028:
998:
947:
921:
881:
718:
697:
654:
600:
528:
447:
370:
326:
260:
167:
59:descriptive statistics
40:Weighted harmonic mean
20321:Mathematical analysis
19734:
19653:
19618:
19554:
19515:
19468:
19422:
19334:
19273:
19238:
19147:
19097:
19077:
19057:
19007:
18987:
18943:
18913:
18845:
18802:
18787:The damping constant
18779:
18759:
18703:
18620:
18592:
18590:{\displaystyle V_{1}}
18565:
18563:{\displaystyle V_{1}}
18535:
18469:
18449:
18417:
18370:
18309:
18281:
18261:
18240:
18238:{\displaystyle x_{i}}
18213:
18211:{\displaystyle t_{i}}
18186:
18166:
18164:{\displaystyle t_{i}}
18139:
18119:
18099:
18071:
18022:
17931:
17841:
17821:
17769:
17747:
17718:
17716:{\displaystyle x_{i}}
17687:
17665:
17577:
17284:
17165:
17042:
16987:
16944:matrix–vector product
16937:
16879:
16822:
16757:
16732:
16692:
16633:
16496:
16475:and this reduces to:
16470:
16431:
16262:
16141:
16102:
16057:
16018:
15984:
15955:
15904:
15879:can be simplified to
15874:
15838:
15799:
15755:
15753:{\displaystyle V_{1}}
15725:
15683:
15642:loss of the base rate
15624:
15487:
15461:
15418:
15289:
15260:
15218:
15173:
15140:
15104:
15067:
15017:
14950:
14822:
14686:
14626:effective sample size
14619:
14550:
14507:
14445:
14404:
14379:
14343:
14315:
14010:
13724:
13664:
13632:
13603:loss of the base rate
13585:
13583:{\displaystyle w_{i}}
13558:
13538:
13502:
13487:
13411:
13371:
13338:
13304:
13268:
13230:
13192:
13130:
13085:
13012:
12935:
12908:
12861:
12819:
12804:
12761:
12688:
12588:
12537:
12497:
12444:
12390:
12349:
12301:
12242:
12210:
12156:
12127:
12076:
12021:
11964:
11927:
11864:
11824:
11764:
11595:
11538:
11125:
11036:
10992:
10872:
10792:
10747:
10659:
10502:
10481:
10364:
10328:
10275:
10214:
10124:
10059:
9718:
9697:
9620:
9524:
9458:
9412:
9353:
9286:
9284:{\displaystyle w_{i}}
9259:
9212:
9165:
9118:
9032:
8988:
8927:), it is as follows:
8894:
8842:
8796:
8755:
8716:
8673:
8636:
8562:
8516:
8463:
8417:
8365:
8320:
8263:
8206:
8154:
8115:
8072:
8023:
7981:
7934:
7887:
7851:
7849:{\displaystyle y_{i}}
7824:
7776:
7728:
7684:
7657:
7610:
7595:). The estimation of
7590:
7561:
7541:
7521:
7480:
7413:
7285:
7167:
7049:
7028:
6947:
6895:
6832:
6749:
6676:
6573:
6486:
6456:
6352:
6287:
6191:
6170:
6078:
5993:
5889:
5786:
5723:
5640:
5566:
5438:
5350:
5295:
5265:
5184:
5107:
5058:
5007:
4951:
4900:
4850:
4825:
4800:
4764:
4743:
4685:
4625:
4565:
4461:
4380:
4267:
4212:
4112:
4031:
3921:
3851:
3831:
3764:
3741:
3615:
3521:
3468:
3466:{\displaystyle I_{i}}
3441:
3439:{\displaystyle y_{i}}
3406:
3404:{\displaystyle p_{i}}
3379:
3294:
3220:
3218:{\displaystyle I_{i}}
3181:
3152:
3097:
3095:{\displaystyle y_{i}}
3052:
3005:
2970:
2928:
2898:
2820:
2775:
2616:
2569:
2536:
2482:
2446:
2389:
2336:
2278:
2223:
2163:
2100:
2036:
1996:
1956:
1896:Note this reduces to
1888:
1845:
1792:
1743:
1701:
1645:
1591:
1529:
1493:
1435:
1396:
1394:{\displaystyle x_{i}}
1349:
1301:
1257:
1219:
1185:
1147:
1118:
1077:
1029:
978:
948:
901:
882:
719:
677:
634:
601:
529:
448:
371:
327:
261:
168:
19634:
19567:
19524:
19484:
19434:
19289:
19250:
19159:
19129:
19086:
19066:
19016:
18996:
18952:
18922:
18854:
18812:
18791:
18768:
18764:for large values of
18715:
18604:
18574:
18547:
18481:
18458:
18426:
18394:
18293:
18270:
18250:
18222:
18218:depends not only on
18195:
18175:
18148:
18128:
18108:
18088:
18037:
17946:
17857:
17848:Gauss–Markov theorem
17830:
17819:{\displaystyle ^{T}}
17785:
17756:
17727:
17700:
17674:
17607:
17299:
17175:
17056:
16954:
16833:
16776:
16744:
16714:
16645:
16482:
16446:
15996:
15971:
15886:
15854:
15767:
15737:
15667:
15473:
15446:
15275:
15152:
15120:
15080:
15047:
14962:
14701:
14632:
14559:
14516:
14454:
14388:
14327:
13676:
13641:
13637:and actual variance
13630:{\displaystyle \mu }
13621:
13567:
13547:
13499:
13316:
13280:
13239:
13201:
13145:
12990:
12959: − 1 (see
12917:
12874:
12816:
12549:
12510:
12463:
12426:
12317:
12251:
12219:
12139:
12033:
11980:
11939:
11836:
11801:
11607:
11550:
11168:
10801:
10756:
10670:
10375:
10340:
10284:
10231:
10137:
10071:
9633:
9364:
9306:
9268:
9221:
9174:
9170:, then either using
9131:
8934:
8768:
8645:
8163:
7860:
7833:
7666:
7619:
7599:
7570:
7550:
7530:
7501:
6956:
6911:
6761:
6688:
6582:
6495:
6465:
6361:
6298:
6099:
5365:
5274:
5123:
4859:
4839:
4823:{\displaystyle \pi }
4814:
4755:
4480:
4294:
4137:
4046:
3930:
3864:
3840:
3777:
3762:{\displaystyle \pi }
3753:
3624:
3530:
3477:
3450:
3423:
3419:Since each element (
3388:
3303:
3237:
3202:
3161:
3141:
3114:(with replacement).
3079:
2979:
2907:
2787:
2671:
2658:Kish's design effect
2652:(as is the case for
2644:, all with the same
2578:
2545:
2455:
2401:
2369:
2352:with the same mean.
2350:normally distributed
2238:
2008:
1965:
1900:
1762:
1508:
1447:
1412:
1378:
1273:
1232:
1201:
1134:
1045:
960:
898:
734:
613:
541:
469:
386:
339:
295:
182:
129:
20258:Strutz, T. (2010).
20079:Analyticalgroup.com
20023:1988JApMe..27.1322E
19980:1995AtmEn..29.1185G
19591:
19548:
19411:
19306:
19280:reduced chi-squared
19267:
19232:
19217:
19192:
17985:
17881:
17449:
17414:
17375:
17352:
16813:
16137:
15782:
15654:reliability weights
15648:Reliability weights
15434:If the weights are
15011:
14990:
14802:
14762:
14724:
14675:
14579:
14512:, analogous to the
14494:
14439:
14305:
14283:
14187:
13996:
13957:
13658:
13615:reliability weights
13609:Reliability weights
12979:If the weights are
12969:reliability weights
12961:Bessel's correction
12902:
12677:
12491:
12332:
12236:
12197:
12117:
12075:
12057:
12015:
11997:
11920:
11905:
11860:
11818:
11713:
11634:
11486:
11204:
11071:
10369:the above becomes:
10299:
10246:
9597:
9564:
9501:
9455:
9067:
8885:
8839:
8760:. This will be the
8605:
8559:
8506:
8460:
8407:
8362:
7818:
5101:
5042:
4986:
4935:
4728:
4660:
4600:
4255:
3954:
3703:
3666:
3645:
3551:
3492:
3040:
2813:
2731:
2516:
2325:
2277:
2201:
2154:
2139:
2092:
2077:
2032:
1942:
1924:
1830:
1632:
1581:
1482:
1429:
1341:
1262:, can be shown via
1060:
1012:
935:
20326:Summary statistics
20262:. Vieweg+Teubner.
19770:Summary statistics
19765:Standard deviation
19729:
19613:
19570:
19549:
19527:
19510:
19473:can be called the
19463:
19417:
19397:
19292:
19268:
19253:
19233:
19218:
19196:
19162:
19142:
19092:
19072:
19052:
19002:
18982:
18938:
18908:
18840:
18797:
18774:
18754:
18698:
18587:
18560:
18530:
18464:
18444:
18412:
18365:
18276:
18266:for a window size
18256:
18235:
18208:
18181:
18161:
18134:
18114:
18094:
18066:
18017:
17964:
17926:
17860:
17836:
17816:
17764:
17742:
17713:
17682:
17660:
17572:
17570:
17562:
17533:
17507:
17430:
17395:
17356:
17333:
17279:
17273:
17212:
17160:
17154:
17093:
17037:
16932:
16817:
16794:
16764:arithmetic inverse
16752:
16727:
16708:maximum likelihood
16687:
16628:
16465:
16426:
16424:
16123:
15979:
15950:
15869:
15833:
15770:
15750:
15720:
15656:, the weights are
15619:
15456:
15455:
15413:
15255:
15135:
15099:
15062:
15012:
14997:
14974:
14945:
14943:
14788:
14737:
14708:
14681:
14661:
14614:
14565:
14545:
14502:
14480:
14440:
14425:
14374:
14310:
14308:
14291:
14269:
14173:
13971:
13943:
13659:
13644:
13627:
13580:
13553:
13533:
13482:
13299:
13263:
13225:
13187:
13125:
12949:unbiased estimator
12930:
12903:
12877:
12856:
12799:
12797:
12652:
12532:
12492:
12466:
12439:
12420:standard deviation
12385:
12320:
12296:
12237:
12222:
12205:
12178:
12122:
12098:
12061:
12036:
12016:
12001:
11983:
11969:can be called the
11959:
11935:whose square root
11922:
11906:
11886:
11839:
11819:
11804:
11759:
11699:
11613:
11590:
11533:
11463:
11174:
11120:
11057:
10787:
10742:
10654:
10359:
10323:
10287:
10270:
10234:
10209:
10119:
10054:
9615:
9585:
9552:
9489:
9443:
9348:
9300:
9281:
9254:
9207:
9160:
9113:
9053:
8889:
8873:
8827:
8750:
8631:
8593:
8547:
8494:
8448:
8386:
8341:
8149:
7846:
7819:
7797:
7652:
7605:
7585:
7556:
7536:
7516:
7475:
7473:
6942:
6905:
6890:
6744:
6671:
6568:
6481:
6451:
6347:
6282:
6073:
6071:
5345:
5260:
5102:
5089:
5030:
4974:
4923:
4845:
4835:-estimator (i.e.:
4820:
4809:, also called the
4795:
4738:
4716:
4648:
4588:
4471:
4456:
4262:
4243:
4107:
4026:
3933:
3916:
3846:
3826:
3759:
3736:
3689:
3652:
3633:
3610:
3539:
3516:
3480:
3463:
3436:
3401:
3374:
3289:
3215:
3198:indicator values (
3176:
3147:
3119:Survey methodology
3092:
3047:
3026:
2965:
2893:
2790:
2770:
2708:
2637:observations from
2629:Simple i.i.d. case
2611:
2564:
2531:
2504:
2441:
2384:
2331:
2311:
2256:
2218:
2184:
2140:
2122:
2078:
2058:
2011:
1991:
1951:
1928:
1903:
1883:
1813:
1738:
1634:
1618:
1583:
1567:
1488:
1468:
1430:
1415:
1391:
1344:
1322:
1252:
1214:
1180:
1113:
1048:
1024:
1000:
943:
923:
877:
727:which expands to:
714:
596:
524:
443:
366:
322:
283:convex combination
256:
163:
27:Statistical amount
20269:978-3-8348-1022-9
20202:978-0-521-35880-4
20017:(12): 1322–1333.
19974:(11): 1185–1193.
19936:978-0-387-97528-3
19881:978-0-471-16240-7
19724:
19699:
19583:
19540:
19459:
19447:
19412:
19383:
19332:
19209:
19184:
19172:
19095:{\displaystyle w}
19075:{\displaystyle n}
19005:{\displaystyle n}
18800:{\displaystyle w}
18777:{\displaystyle m}
18693:
18525:
18467:{\displaystyle m}
18279:{\displaystyle m}
18259:{\displaystyle z}
18184:{\displaystyle y}
18137:{\displaystyle n}
18117:{\displaystyle y}
18097:{\displaystyle x}
17977:
17958:
17873:
17839:{\displaystyle n}
17739:
17694:covariance matrix
17317:
16974:
16870:
16847:
16739:covariance matrix
16623:
16417:
16139:
15831:
15614:
15436:frequency weights
15430:Frequency weights
15408:
15250:
15004:
14936:
14807:
14747:
14727:
14564:
14539:
14495:
14298:
14284:
14188:
14100:
13981:
13950:
13937:
13859:
13790:
13702:
13651:
13556:{\displaystyle N}
13409:
13331:
13123:
13005:
12981:frequency weights
12975:Frequency weights
12965:frequency weights
12887:
12868:frequency weights
12793:
12662:
12646:
12577:
12566:
12523:
12476:
12383:
12294:
12267:
12049:
11955:
11852:
11740:
11694:
11680:
11638:
11626:
11588:
11562:
11515:
11473:
11448:
11435:
11394:
11357:
11322:
11309:
11264:
11250:
11208:
11189:
11098:
11034:
10967:
10947:
10870:
10861:
10840:
10824:
10644:
10624:
10596:
10576:
10538:
10479:
10470:
10449:
10433:
10407:
10397:
10113:
10098:
10083:
10043:
10028:
10013:
10001:
9983:
9973:
9951:
9936:
9926:
9900:
9891:
9866:
9842:
9815:
9791:
9754:
9695:
9686:
9665:
9655:
9608:
9575:
9522:
9503:
9404:
9402:
9392:
9376:
9346:
9344:
9334:
9318:
9298:
9252:
9205:
9094:
9030:
8973:
8957:
8902:This is called a
8887:
8781:
8748:
8658:
8622:
8607:
8508:
8409:
8396:
8351:
8312:
8309:
8259:
8197:
8192:
8175:
8147:
8064:
8054:
8012:
7973:
7970:
7930:
7878:
7856:s, and 1s. I.e.:
7807:
7771:
7678:
7650:
7608:{\displaystyle N}
7582:
7559:{\displaystyle N}
7539:{\displaystyle N}
7513:
7411:
7379:
7355:
7283:
7244:
7225:
7203:
7165:
7126:
7107:
7085:
7026:
6997:
6989:
6982:
6903:
6830:
6805:
6797:
6790:
6785:
6650:
6595:
6566:
6508:
6345:
6311:
6268:
6249:
6227:
6168:
6143:
6135:
6128:
6123:
6056:
5991:
5952:
5887:
5869:
5831:
5775:
5721:
5703:
5638:
5625:
5601:
5564:
5546:
5502:
5487:
5436:
5418:
5397:
5391:
5343:
5290:
5247:
5182:
5145:
5053:
5002:
4946:
4898:
4872:
4848:{\displaystyle p}
4733:
4677:
4671:
4617:
4611:
4563:
4547:
4529:
4507:
4500:
4495:
4469:
4443:
4378:
4360:
4331:
4324:
4319:
4260:
4204:
4186:
4164:
4157:
4152:
4105:
4077:
4024:
3978:
3943:
3907:
3889:
3849:{\displaystyle p}
3824:
3790:
3372:
3336:
3268:
3173:
3150:{\displaystyle N}
3045:
2997:
2963:
2919:
2891:
2866:
2800:
2768:
2759:
2748:
2718:
2693:
2596:
2473:
2381:
2326:
2269:
2250:
2213:
2024:
1916:
1878:
1877:
1833:
1832:
1778:
1733:
1637:
1633:
1582:
1520:
1483:
1342:
1289:
1248:
1145:
1111:
972:
872:
746:
709:
625:
398:
358:
314:
248:
194:
155:
141:
85:Given two school
70:Simpson's paradox
16:(Redirected from
20333:
20297:
20296:
20273:
20254:
20228:
20211:
20205:
20190:
20184:
20183:
20165:
20159:
20149:
20143:
20142:
20108:
20099:
20090:
20089:
20087:
20085:
20076:
20068:
20062:
20061:
20059:
20057:
20043:
20037:
20036:
20034:
20002:
19996:
19991:
19963:
19957:
19947:
19941:
19940:
19922:
19883:
19873:
19849:
19842:
19755:Central tendency
19738:
19736:
19735:
19730:
19725:
19723:
19712:
19711:
19710:
19701:
19700:
19692:
19686:
19685:
19672:
19667:
19651:
19646:
19645:
19622:
19620:
19619:
19614:
19609:
19604:
19603:
19590:
19585:
19584:
19576:
19558:
19556:
19555:
19550:
19547:
19542:
19541:
19533:
19519:
19517:
19516:
19511:
19509:
19508:
19496:
19495:
19472:
19470:
19469:
19464:
19462:
19461:
19460:
19452:
19449:
19448:
19440:
19430:The square root
19426:
19424:
19423:
19418:
19413:
19410:
19405:
19396:
19395:
19394:
19385:
19384:
19376:
19370:
19369:
19356:
19353:
19348:
19333:
19331:
19311:
19305:
19300:
19277:
19275:
19274:
19269:
19266:
19261:
19242:
19240:
19239:
19234:
19231:
19226:
19216:
19211:
19210:
19202:
19191:
19186:
19185:
19177:
19174:
19173:
19165:
19151:
19149:
19148:
19143:
19141:
19140:
19101:
19099:
19098:
19093:
19081:
19079:
19078:
19073:
19061:
19059:
19058:
19053:
19051:
19050:
19049:
19011:
19009:
19008:
19003:
18991:
18989:
18988:
18983:
18975:
18974:
18973:
18948:, the head area
18947:
18945:
18944:
18939:
18937:
18936:
18917:
18915:
18914:
18909:
18871:
18870:
18869:
18849:
18847:
18846:
18841:
18839:
18838:
18806:
18804:
18803:
18798:
18783:
18781:
18780:
18775:
18763:
18761:
18760:
18755:
18738:
18727:
18726:
18707:
18705:
18704:
18699:
18694:
18692:
18681:
18680:
18679:
18663:
18658:
18657:
18656:
18639:
18634:
18616:
18615:
18596:
18594:
18593:
18588:
18586:
18585:
18569:
18567:
18566:
18561:
18559:
18558:
18539:
18537:
18536:
18531:
18526:
18524:
18523:
18514:
18513:
18498:
18493:
18492:
18473:
18471:
18470:
18465:
18453:
18451:
18450:
18445:
18421:
18419:
18418:
18413:
18374:
18372:
18371:
18366:
18361:
18360:
18339:
18338:
18328:
18323:
18305:
18304:
18285:
18283:
18282:
18277:
18265:
18263:
18262:
18257:
18244:
18242:
18241:
18236:
18234:
18233:
18217:
18215:
18214:
18209:
18207:
18206:
18190:
18188:
18187:
18182:
18170:
18168:
18167:
18162:
18160:
18159:
18143:
18141:
18140:
18135:
18123:
18121:
18120:
18115:
18103:
18101:
18100:
18095:
18075:
18073:
18072:
18067:
18062:
18061:
18053:
18044:
18026:
18024:
18023:
18018:
18010:
18005:
18000:
17999:
17994:
17984:
17979:
17978:
17970:
17960:
17959:
17951:
17935:
17933:
17932:
17927:
17922:
17921:
17909:
17904:
17899:
17898:
17893:
17880:
17875:
17874:
17866:
17845:
17843:
17842:
17837:
17825:
17823:
17822:
17817:
17815:
17814:
17773:
17771:
17770:
17765:
17763:
17751:
17749:
17748:
17743:
17741:
17740:
17732:
17722:
17720:
17719:
17714:
17712:
17711:
17691:
17689:
17688:
17683:
17681:
17669:
17667:
17666:
17661:
17659:
17658:
17649:
17648:
17630:
17629:
17614:
17581:
17579:
17578:
17573:
17571:
17567:
17566:
17538:
17537:
17512:
17511:
17470:
17466:
17462:
17461:
17460:
17455:
17448:
17440:
17435:
17426:
17425:
17420:
17413:
17405:
17400:
17389:
17388:
17380:
17376:
17374:
17366:
17361:
17351:
17343:
17338:
17319:
17318:
17313:
17308:
17288:
17286:
17285:
17280:
17278:
17277:
17239:
17238:
17233:
17223:
17222:
17217:
17216:
17189:
17188:
17183:
17169:
17167:
17166:
17161:
17159:
17158:
17120:
17119:
17114:
17104:
17103:
17098:
17097:
17070:
17069:
17064:
17046:
17044:
17043:
17038:
17033:
17032:
17024:
17020:
17019:
17018:
17013:
17006:
17001:
16977:
16976:
16975:
16970:
16965:
16962:
16941:
16939:
16938:
16933:
16928:
16924:
16923:
16922:
16917:
16911:
16910:
16905:
16898:
16893:
16873:
16872:
16871:
16866:
16861:
16858:
16849:
16848:
16843:
16838:
16826:
16824:
16823:
16818:
16812:
16804:
16799:
16790:
16789:
16784:
16761:
16759:
16758:
16753:
16751:
16736:
16734:
16733:
16728:
16726:
16725:
16696:
16694:
16693:
16688:
16683:
16672:
16671:
16662:
16657:
16656:
16637:
16635:
16634:
16629:
16624:
16622:
16621:
16620:
16604:
16603:
16599:
16598:
16597:
16585:
16584:
16579:
16568:
16567:
16562:
16558:
16557:
16556:
16544:
16543:
16538:
16526:
16525:
16515:
16510:
16494:
16489:
16474:
16472:
16471:
16466:
16458:
16457:
16435:
16433:
16432:
16427:
16425:
16418:
16416:
16412:
16411:
16402:
16397:
16396:
16381:
16380:
16370:
16369:
16365:
16364:
16363:
16351:
16350:
16345:
16334:
16333:
16328:
16324:
16323:
16322:
16310:
16309:
16304:
16292:
16291:
16281:
16276:
16260:
16252:
16248:
16244:
16243:
16242:
16230:
16229:
16224:
16213:
16212:
16207:
16203:
16202:
16201:
16189:
16188:
16183:
16171:
16170:
16160:
16155:
16140:
16138:
16136:
16131:
16121:
16116:
16098:
16097:
16092:
16088:
16087:
16086:
16076:
16071:
16049:
16048:
16047:
16037:
16032:
16016:
16007:
15988:
15986:
15985:
15980:
15978:
15959:
15957:
15956:
15951:
15946:
15945:
15940:
15934:
15933:
15923:
15918:
15900:
15899:
15898:
15878:
15876:
15875:
15870:
15868:
15867:
15866:
15842:
15840:
15839:
15834:
15832:
15830:
15829:
15828:
15818:
15813:
15797:
15796:
15787:
15778:
15759:
15757:
15756:
15751:
15749:
15748:
15729:
15727:
15726:
15721:
15713:
15712:
15702:
15697:
15679:
15678:
15628:
15626:
15625:
15620:
15615:
15613:
15606:
15605:
15595:
15594:
15590:
15589:
15588:
15576:
15575:
15570:
15559:
15558:
15553:
15549:
15548:
15547:
15535:
15534:
15529:
15517:
15516:
15506:
15501:
15485:
15480:
15465:
15463:
15462:
15457:
15454:
15422:
15420:
15419:
15414:
15409:
15407:
15406:
15397:
15396:
15392:
15391:
15390:
15378:
15377:
15372:
15361:
15360:
15355:
15351:
15350:
15349:
15337:
15336:
15331:
15319:
15318:
15308:
15303:
15287:
15282:
15264:
15262:
15261:
15256:
15251:
15249:
15248:
15247:
15237:
15232:
15216:
15215:
15214:
15209:
15203:
15202:
15192:
15187:
15171:
15166:
15165:
15164:
15144:
15142:
15141:
15136:
15134:
15133:
15132:
15108:
15106:
15105:
15100:
15092:
15091:
15071:
15069:
15068:
15063:
15061:
15060:
15055:
15021:
15019:
15018:
15013:
15010:
15005:
15002:
14989:
14984:
14983:
14954:
14952:
14951:
14946:
14944:
14937:
14935:
14931:
14930:
14921:
14916:
14915:
14900:
14899:
14889:
14888:
14887:
14878:
14877:
14865:
14864:
14852:
14851:
14841:
14836:
14820:
14812:
14808:
14806:
14801:
14796:
14787:
14782:
14781:
14761:
14756:
14755:
14749:
14748:
14740:
14736:
14725:
14723:
14718:
14717:
14690:
14688:
14687:
14682:
14680:
14676:
14674:
14669:
14660:
14655:
14654:
14623:
14621:
14620:
14615:
14613:
14612:
14594:
14593:
14584:
14578:
14573:
14562:
14554:
14552:
14551:
14546:
14544:
14540:
14535:
14524:
14511:
14509:
14508:
14503:
14501:
14497:
14496:
14493:
14488:
14479:
14478:
14469:
14449:
14447:
14446:
14441:
14438:
14433:
14423:
14418:
14400:
14399:
14383:
14381:
14380:
14375:
14373:
14372:
14362:
14357:
14339:
14338:
14319:
14317:
14316:
14311:
14309:
14304:
14299:
14296:
14290:
14286:
14285:
14282:
14277:
14268:
14267:
14258:
14239:
14232:
14231:
14189:
14186:
14181:
14172:
14171:
14162:
14154:
14153:
14105:
14101:
14099:
14098:
14089:
14085:
14084:
14075:
14074:
14062:
14061:
14040:
14039:
14029:
14024:
14008:
13995:
13990:
13989:
13983:
13982:
13974:
13956:
13951:
13948:
13942:
13938:
13933:
13922:
13910:
13903:
13902:
13860:
13852:
13844:
13843:
13795:
13791:
13786:
13782:
13781:
13766:
13765:
13743:
13738:
13722:
13710:
13709:
13704:
13703:
13695:
13668:
13666:
13665:
13660:
13657:
13652:
13649:
13636:
13634:
13633:
13628:
13589:
13587:
13586:
13581:
13579:
13578:
13562:
13560:
13559:
13554:
13542:
13540:
13539:
13534:
13532:
13531:
13521:
13516:
13491:
13489:
13488:
13483:
13481:
13480:
13475:
13471:
13470:
13469:
13457:
13456:
13441:
13440:
13430:
13425:
13410:
13408:
13401:
13400:
13390:
13385:
13369:
13368:
13367:
13357:
13352:
13336:
13329:
13328:
13327:
13308:
13306:
13305:
13300:
13295:
13294:
13272:
13270:
13269:
13264:
13234:
13232:
13231:
13226:
13196:
13194:
13193:
13188:
13134:
13132:
13131:
13126:
13124:
13122:
13115:
13114:
13104:
13099:
13083:
13082:
13081:
13076:
13072:
13071:
13070:
13058:
13057:
13042:
13041:
13031:
13026:
13010:
13003:
13002:
13001:
12939:
12937:
12936:
12931:
12929:
12928:
12912:
12910:
12909:
12904:
12901:
12896:
12895:
12889:
12888:
12880:
12865:
12863:
12862:
12857:
12849:
12848:
12838:
12833:
12808:
12806:
12805:
12800:
12798:
12794:
12792:
12791:
12790:
12780:
12775:
12759:
12758:
12757:
12752:
12748:
12747:
12746:
12734:
12733:
12718:
12717:
12707:
12702:
12686:
12676:
12671:
12670:
12664:
12663:
12655:
12647:
12642:
12641:
12640:
12635:
12631:
12624:
12623:
12607:
12602:
12586:
12575:
12574:
12573:
12568:
12567:
12559:
12541:
12539:
12538:
12533:
12531:
12530:
12525:
12524:
12516:
12506:sample variance
12501:
12499:
12498:
12493:
12490:
12485:
12484:
12478:
12477:
12469:
12448:
12446:
12445:
12440:
12438:
12437:
12399:Related concepts
12394:
12392:
12391:
12386:
12384:
12382:
12381:
12380:
12379:
12368:
12363:
12347:
12346:
12337:
12328:
12305:
12303:
12302:
12297:
12295:
12290:
12288:
12283:
12282:
12270:
12269:
12268:
12260:
12246:
12244:
12243:
12238:
12235:
12230:
12214:
12212:
12211:
12206:
12198:
12196:
12195:
12186:
12175:
12170:
12149:
12131:
12129:
12128:
12123:
12118:
12116:
12115:
12106:
12095:
12090:
12074:
12069:
12056:
12051:
12050:
12042:
12025:
12023:
12022:
12017:
12014:
12009:
11996:
11991:
11968:
11966:
11965:
11960:
11958:
11957:
11956:
11948:
11931:
11929:
11928:
11923:
11921:
11919:
11914:
11904:
11903:
11894:
11883:
11878:
11859:
11854:
11853:
11845:
11828:
11826:
11825:
11820:
11817:
11812:
11768:
11766:
11765:
11760:
11758:
11757:
11748:
11747:
11742:
11741:
11733:
11726:
11725:
11712:
11707:
11695:
11693:
11692:
11691:
11682:
11681:
11673:
11645:
11640:
11639:
11633:
11628:
11627:
11619:
11612:
11599:
11597:
11596:
11591:
11589:
11584:
11583:
11582:
11569:
11564:
11563:
11555:
11542:
11540:
11539:
11534:
11532:
11528:
11527:
11526:
11517:
11516:
11508:
11502:
11501:
11485:
11480:
11475:
11474:
11466:
11456:
11455:
11450:
11449:
11441:
11437:
11436:
11428:
11422:
11421:
11412:
11411:
11396:
11395:
11387:
11381:
11380:
11365:
11364:
11359:
11358:
11350:
11340:
11339:
11330:
11329:
11324:
11323:
11315:
11311:
11310:
11302:
11296:
11295:
11286:
11285:
11265:
11263:
11262:
11261:
11252:
11251:
11243:
11215:
11210:
11209:
11203:
11198:
11197:
11196:
11191:
11190:
11182:
11173:
11139:Poisson sampling
11129:
11127:
11126:
11121:
11116:
11115:
11106:
11105:
11100:
11099:
11091:
11084:
11083:
11070:
11065:
11055:
11050:
11035:
11033:
11032:
11031:
11022:
11021:
11011:
11006:
10984:
10979:
10978:
10973:
10969:
10968:
10966:
10965:
10956:
10955:
10954:
10949:
10948:
10940:
10933:
10932:
10922:
10917:
10916:
10891:
10886:
10871:
10869:
10868:
10863:
10862:
10854:
10847:
10842:
10841:
10836:
10832:
10831:
10826:
10825:
10817:
10806:
10796:
10794:
10793:
10788:
10777:
10776:
10751:
10749:
10748:
10743:
10732:
10731:
10719:
10718:
10700:
10699:
10663:
10661:
10660:
10655:
10650:
10646:
10645:
10643:
10642:
10633:
10632:
10631:
10626:
10625:
10617:
10610:
10609:
10599:
10597:
10595:
10594:
10585:
10584:
10583:
10578:
10577:
10569:
10562:
10561:
10551:
10549:
10548:
10540:
10539:
10531:
10521:
10516:
10500:
10495:
10480:
10478:
10477:
10472:
10471:
10463:
10456:
10451:
10450:
10445:
10441:
10440:
10435:
10434:
10426:
10415:
10409:
10408:
10403:
10399:
10398:
10390:
10380:
10368:
10366:
10365:
10360:
10352:
10351:
10332:
10330:
10329:
10324:
10322:
10321:
10312:
10311:
10295:
10279:
10277:
10276:
10271:
10269:
10268:
10259:
10258:
10242:
10218:
10216:
10215:
10210:
10199:
10198:
10186:
10185:
10167:
10166:
10128:
10126:
10125:
10120:
10115:
10114:
10106:
10100:
10099:
10091:
10085:
10084:
10076:
10063:
10061:
10060:
10055:
10053:
10049:
10045:
10044:
10036:
10030:
10029:
10021:
10015:
10014:
10006:
10003:
10002:
9994:
9985:
9984:
9979:
9975:
9974:
9966:
9956:
9953:
9952:
9944:
9938:
9937:
9932:
9928:
9927:
9919:
9909:
9901:
9899:
9898:
9893:
9892:
9884:
9877:
9872:
9868:
9867:
9865:
9864:
9855:
9854:
9853:
9844:
9843:
9835:
9829:
9828:
9818:
9816:
9814:
9813:
9804:
9803:
9802:
9793:
9792:
9784:
9778:
9777:
9767:
9765:
9764:
9756:
9755:
9747:
9737:
9732:
9716:
9711:
9696:
9694:
9693:
9688:
9687:
9679:
9672:
9667:
9666:
9661:
9657:
9656:
9648:
9638:
9624:
9622:
9621:
9616:
9614:
9610:
9609:
9607:
9606:
9593:
9584:
9576:
9574:
9573:
9560:
9551:
9543:
9538:
9523:
9515:
9504:
9502:
9497:
9488:
9487:
9477:
9472:
9456:
9451:
9442:
9441:
9431:
9426:
9410:
9405:
9403:
9395:
9393:
9385:
9383:
9378:
9377:
9369:
9357:
9355:
9354:
9349:
9347:
9345:
9337:
9335:
9327:
9325:
9320:
9319:
9311:
9290:
9288:
9287:
9282:
9280:
9279:
9263:
9261:
9260:
9255:
9253:
9251:
9250:
9238:
9233:
9232:
9216:
9214:
9213:
9208:
9206:
9204:
9203:
9191:
9186:
9185:
9169:
9167:
9166:
9161:
9156:
9155:
9143:
9142:
9122:
9120:
9119:
9114:
9112:
9111:
9102:
9101:
9096:
9095:
9087:
9080:
9079:
9066:
9061:
9051:
9046:
9031:
9029:
9028:
9027:
9018:
9017:
9007:
9002:
8980:
8975:
8974:
8969:
8965:
8964:
8959:
8958:
8950:
8939:
8925:Poisson sampling
8898:
8896:
8895:
8890:
8888:
8886:
8881:
8872:
8871:
8861:
8856:
8840:
8835:
8826:
8825:
8815:
8810:
8794:
8789:
8788:
8783:
8782:
8774:
8759:
8757:
8756:
8751:
8749:
8747:
8746:
8745:
8735:
8730:
8714:
8713:
8712:
8703:
8702:
8692:
8687:
8671:
8666:
8665:
8660:
8659:
8651:
8640:
8638:
8637:
8632:
8630:
8629:
8624:
8623:
8615:
8608:
8606:
8601:
8592:
8591:
8581:
8576:
8560:
8555:
8546:
8545:
8535:
8530:
8514:
8509:
8507:
8502:
8493:
8492:
8482:
8477:
8461:
8456:
8447:
8446:
8436:
8431:
8415:
8410:
8408:
8403:
8398:
8397:
8389:
8384:
8379:
8363:
8358:
8353:
8352:
8344:
8339:
8334:
8318:
8313:
8311:
8310:
8308:
8307:
8295:
8293:
8292:
8282:
8277:
8261:
8260:
8258:
8257:
8248:
8247:
8238:
8236:
8235:
8225:
8220:
8204:
8199:
8198:
8193:
8185:
8183:
8177:
8176:
8168:
8158:
8156:
8155:
8150:
8148:
8146:
8145:
8144:
8134:
8129:
8113:
8112:
8111:
8102:
8101:
8091:
8086:
8070:
8065:
8063:
8062:
8061:
8056:
8055:
8047:
8042:
8037:
8021:
8020:
8019:
8014:
8013:
8005:
8000:
7995:
7979:
7974:
7972:
7971:
7969:
7968:
7956:
7953:
7948:
7932:
7931:
7929:
7928:
7919:
7918:
7909:
7906:
7901:
7885:
7880:
7879:
7871:
7855:
7853:
7852:
7847:
7845:
7844:
7828:
7826:
7825:
7820:
7814:
7809:
7808:
7800:
7795:
7790:
7772:
7770:
7769:
7760:
7759:
7750:
7747:
7742:
7724:
7723:
7714:
7713:
7703:
7698:
7680:
7679:
7671:
7661:
7659:
7658:
7653:
7651:
7649:
7648:
7636:
7631:
7630:
7614:
7612:
7611:
7606:
7594:
7592:
7591:
7586:
7584:
7583:
7575:
7565:
7563:
7562:
7557:
7545:
7543:
7542:
7537:
7525:
7523:
7522:
7517:
7515:
7514:
7506:
7492:
7484:
7482:
7481:
7476:
7474:
7470:
7469:
7464:
7460:
7459:
7458:
7449:
7448:
7432:
7427:
7412:
7410:
7409:
7397:
7389:
7385:
7381:
7380:
7378:
7377:
7368:
7367:
7358:
7356:
7354:
7353:
7344:
7343:
7334:
7329:
7328:
7304:
7299:
7284:
7282:
7281:
7269:
7261:
7257:
7253:
7252:
7251:
7246:
7245:
7237:
7233:
7232:
7227:
7226:
7218:
7214:
7213:
7205:
7204:
7196:
7186:
7181:
7166:
7164:
7163:
7151:
7143:
7139:
7135:
7134:
7133:
7128:
7127:
7119:
7115:
7114:
7109:
7108:
7100:
7096:
7095:
7087:
7086:
7078:
7068:
7063:
7047:
7042:
7027:
7025:
7024:
7012:
7000:
6999:
6998:
6995:
6990:
6988:pwr (known
6987:
6984:
6983:
6975:
6951:
6949:
6948:
6943:
6932:
6931:
6899:
6897:
6896:
6891:
6889:
6888:
6883:
6879:
6878:
6877:
6868:
6867:
6851:
6846:
6831:
6829:
6828:
6816:
6808:
6807:
6806:
6803:
6798:
6796:pwr (known
6795:
6792:
6791:
6786:
6778:
6776:
6753:
6751:
6750:
6745:
6734:
6733:
6721:
6720:
6680:
6678:
6677:
6672:
6670:
6669:
6651:
6649:
6648:
6639:
6638:
6637:
6628:
6627:
6617:
6606:
6605:
6597:
6596:
6588:
6577:
6575:
6574:
6569:
6567:
6565:
6564:
6552:
6551:
6550:
6541:
6540:
6530:
6519:
6518:
6510:
6509:
6501:
6490:
6488:
6487:
6482:
6480:
6479:
6460:
6458:
6457:
6452:
6450:
6449:
6434:
6433:
6424:
6423:
6411:
6410:
6392:
6391:
6379:
6378:
6356:
6354:
6353:
6348:
6346:
6344:
6343:
6334:
6333:
6324:
6319:
6318:
6313:
6312:
6304:
6291:
6289:
6288:
6283:
6281:
6277:
6276:
6275:
6270:
6269:
6261:
6257:
6256:
6251:
6250:
6242:
6238:
6237:
6229:
6228:
6220:
6210:
6205:
6189:
6184:
6169:
6167:
6166:
6154:
6146:
6145:
6144:
6141:
6136:
6134:pwr (known
6133:
6130:
6129:
6124:
6116:
6114:
6091:Poisson sampling
6082:
6080:
6079:
6074:
6072:
6068:
6067:
6062:
6058:
6057:
6052:
6044:
6039:
6038:
6029:
6028:
6012:
6007:
5992:
5990:
5976:
5968:
5964:
5963:
5958:
5954:
5953:
5948:
5940:
5935:
5934:
5925:
5924:
5908:
5903:
5888:
5886:
5872:
5870:
5865:
5864:
5855:
5847:
5843:
5842:
5837:
5833:
5832:
5827:
5826:
5825:
5816:
5815:
5805:
5800:
5784:
5776:
5774:
5773:
5764:
5763:
5754:
5742:
5737:
5722:
5720:
5706:
5704:
5696:
5691:
5690:
5685:
5681:
5680:
5679:
5670:
5669:
5659:
5654:
5639:
5631:
5626:
5624:
5623:
5614:
5613:
5604:
5602:
5594:
5585:
5580:
5565:
5563:
5549:
5547:
5539:
5531:
5527:
5526:
5521:
5517:
5516:
5515:
5504:
5503:
5495:
5488:
5486:
5485:
5476:
5475:
5466:
5457:
5452:
5437:
5435:
5421:
5419:
5411:
5399:
5398:
5395:
5393:
5392:
5384:
5354:
5352:
5351:
5346:
5344:
5339:
5338:
5337:
5328:
5327:
5317:
5314:
5309:
5291:
5286:
5278:
5269:
5267:
5266:
5261:
5259:
5258:
5253:
5249:
5248:
5243:
5235:
5230:
5229:
5220:
5219:
5203:
5198:
5183:
5181:
5167:
5159:
5158:
5147:
5146:
5138:
5111:
5109:
5108:
5103:
5097:
5088:
5087:
5077:
5072:
5054:
5052:
5051:
5038:
5029:
5026:
5021:
5003:
5001:
5000:
4999:
4982:
4973:
4970:
4965:
4947:
4945:
4944:
4931:
4922:
4919:
4914:
4899:
4891:
4886:
4885:
4874:
4873:
4865:
4854:
4852:
4851:
4846:
4829:
4827:
4826:
4821:
4804:
4802:
4801:
4796:
4794:
4793:
4783:
4778:
4747:
4745:
4744:
4739:
4734:
4729:
4724:
4715:
4714:
4704:
4699:
4683:
4678:
4673:
4672:
4670:
4669:
4656:
4647:
4644:
4639:
4623:
4618:
4613:
4612:
4610:
4609:
4596:
4587:
4584:
4579:
4564:
4556:
4553:
4548:
4543:
4542:
4531:
4530:
4522:
4518:
4513:
4512:
4508:
4505:
4502:
4501:
4496:
4488:
4486:
4465:
4463:
4462:
4457:
4455:
4454:
4449:
4445:
4444:
4439:
4431:
4426:
4425:
4416:
4415:
4399:
4394:
4379:
4377:
4363:
4361:
4359:
4358:
4346:
4341:
4337:
4336:
4332:
4329:
4326:
4325:
4320:
4312:
4310:
4271:
4269:
4268:
4263:
4261:
4256:
4251:
4242:
4241:
4231:
4226:
4210:
4205:
4200:
4199:
4188:
4187:
4179:
4175:
4170:
4169:
4165:
4162:
4159:
4158:
4153:
4145:
4143:
4116:
4114:
4113:
4108:
4106:
4104:
4103:
4102:
4083:
4078:
4076:
4075:
4063:
4058:
4057:
4035:
4033:
4032:
4027:
4025:
4023:
4022:
4013:
4012:
4011:
4002:
4001:
3991:
3986:
3985:
3980:
3979:
3971:
3967:
3966:
3950:
3945:
3944:
3936:
3925:
3923:
3922:
3917:
3915:
3914:
3909:
3908:
3900:
3890:
3888:
3887:
3878:
3877:
3868:
3855:
3853:
3852:
3847:
3835:
3833:
3832:
3827:
3825:
3823:
3822:
3813:
3812:
3803:
3798:
3797:
3792:
3791:
3783:
3768:
3766:
3765:
3760:
3745:
3743:
3742:
3737:
3732:
3731:
3713:
3712:
3702:
3697:
3682:
3681:
3665:
3660:
3641:
3620:; and variance:
3619:
3617:
3616:
3611:
3609:
3608:
3599:
3598:
3583:
3582:
3567:
3566:
3547:
3525:
3523:
3522:
3517:
3515:
3514:
3505:
3504:
3488:
3472:
3470:
3469:
3464:
3462:
3461:
3445:
3443:
3442:
3437:
3435:
3434:
3414:cluster sampling
3410:
3408:
3407:
3402:
3400:
3399:
3383:
3381:
3380:
3375:
3373:
3368:
3367:
3358:
3353:
3352:
3337:
3334:
3332:
3321:
3320:
3298:
3296:
3295:
3290:
3288:
3287:
3269:
3266:
3255:
3254:
3231:Poisson sampling
3224:
3222:
3221:
3216:
3214:
3213:
3185:
3183:
3182:
3177:
3175:
3174:
3166:
3157:) or estimated (
3156:
3154:
3153:
3148:
3101:
3099:
3098:
3093:
3091:
3090:
3056:
3054:
3053:
3048:
3046:
3041:
3039:
3034:
3024:
3019:
3003:
2998:
2993:
2992:
2983:
2974:
2972:
2971:
2966:
2964:
2959:
2958:
2957:
2947:
2942:
2926:
2921:
2920:
2912:
2902:
2900:
2899:
2894:
2892:
2890:
2879:
2878:
2877:
2868:
2867:
2859:
2853:
2852:
2839:
2834:
2818:
2812:
2807:
2802:
2801:
2793:
2779:
2777:
2776:
2771:
2769:
2767:
2766:
2761:
2760:
2752:
2744:
2743:
2734:
2733:
2730:
2725:
2720:
2719:
2711:
2701:
2700:
2695:
2694:
2686:
2642:random variables
2620:
2618:
2617:
2612:
2598:
2597:
2589:
2573:
2571:
2570:
2565:
2557:
2556:
2540:
2538:
2537:
2532:
2527:
2526:
2525:
2512:
2501:
2496:
2475:
2474:
2466:
2450:
2448:
2447:
2442:
2437:
2436:
2435:
2419:
2418:
2393:
2391:
2390:
2385:
2383:
2382:
2374:
2340:
2338:
2337:
2332:
2327:
2324:
2319:
2310:
2309:
2300:
2297:
2292:
2276:
2271:
2270:
2262:
2252:
2251:
2243:
2227:
2225:
2224:
2219:
2214:
2212:
2211:
2206:
2202:
2200:
2192:
2182:
2177:
2156:
2155:
2153:
2148:
2138:
2130:
2119:
2114:
2098:
2093:
2091:
2086:
2076:
2075:
2066:
2055:
2050:
2031:
2026:
2025:
2017:
2000:
1998:
1997:
1992:
1990:
1989:
1977:
1976:
1960:
1958:
1957:
1952:
1947:
1941:
1936:
1923:
1918:
1917:
1909:
1892:
1890:
1889:
1884:
1879:
1876:
1875:
1874:
1864:
1859:
1840:
1839:
1834:
1831:
1829:
1821:
1811:
1806:
1787:
1786:
1781:
1780:
1779:
1771:
1747:
1745:
1744:
1739:
1734:
1732:
1731:
1730:
1720:
1715:
1699:
1698:
1694:
1693:
1692:
1680:
1679:
1664:
1659:
1643:
1638:
1636:
1635:
1631:
1626:
1614:
1610:
1605:
1589:
1588:
1584:
1580:
1575:
1566:
1565:
1556:
1548:
1543:
1527:
1522:
1521:
1513:
1497:
1495:
1494:
1489:
1484:
1481:
1476:
1464:
1459:
1458:
1439:
1437:
1436:
1431:
1428:
1423:
1400:
1398:
1397:
1392:
1390:
1389:
1353:
1351:
1350:
1345:
1343:
1340:
1339:
1330:
1320:
1315:
1300:
1292:
1291:
1290:
1282:
1261:
1259:
1258:
1253:
1251:
1250:
1249:
1241:
1223:
1221:
1220:
1215:
1213:
1212:
1189:
1187:
1186:
1181:
1179:
1178:
1177:
1166:
1161:
1146:
1138:
1122:
1120:
1119:
1114:
1112:
1110:
1109:
1108:
1107:
1096:
1091:
1075:
1074:
1065:
1056:
1033:
1031:
1030:
1025:
1023:
1022:
1021:
1008:
997:
992:
974:
973:
965:
952:
950:
949:
944:
936:
931:
920:
915:
886:
884:
883:
878:
873:
871:
870:
869:
851:
850:
838:
837:
827:
826:
825:
816:
815:
797:
796:
787:
786:
774:
773:
764:
763:
753:
748:
747:
739:
723:
721:
720:
715:
710:
708:
707:
706:
696:
691:
675:
674:
673:
664:
663:
653:
648:
632:
627:
626:
618:
605:
603:
602:
597:
595:
591:
590:
589:
571:
570:
558:
557:
533:
531:
530:
525:
523:
519:
518:
517:
499:
498:
486:
485:
452:
450:
449:
444:
400:
399:
391:
375:
373:
372:
367:
359:
357:
343:
331:
329:
328:
323:
315:
313:
299:
265:
263:
262:
257:
249:
247:
236:
201:
196:
195:
187:
172:
170:
169:
164:
156:
148:
143:
142:
134:
108:
106:
102:
96:
94:
90:
21:
20341:
20340:
20336:
20335:
20334:
20332:
20331:
20330:
20306:
20305:
20287:"Weighted Mean"
20280:
20270:
20237:
20235:Further reading
20232:
20231:
20212:
20208:
20191:
20187:
20180:
20166:
20162:
20150:
20146:
20106:
20100:
20093:
20083:
20081:
20074:
20070:
20069:
20065:
20055:
20053:
20045:
20044:
20040:
20003:
19999:
19964:
19960:
19949:Thomas Lumley (
19948:
19944:
19937:
19923:
19886:
19874:
19863:
19858:
19853:
19852:
19846:absolute values
19843:
19839:
19834:
19829:
19825:Ratio estimator
19800:Weighted median
19775:Weight function
19745:
19713:
19706:
19702:
19691:
19690:
19681:
19677:
19668:
19657:
19652:
19650:
19641:
19637:
19635:
19632:
19631:
19605:
19599:
19595:
19586:
19575:
19574:
19568:
19565:
19564:
19543:
19532:
19531:
19525:
19522:
19521:
19504:
19500:
19491:
19487:
19485:
19482:
19481:
19451:
19450:
19439:
19438:
19437:
19435:
19432:
19431:
19406:
19401:
19390:
19386:
19375:
19374:
19365:
19361:
19357:
19355:
19349:
19338:
19315:
19310:
19301:
19296:
19290:
19287:
19286:
19262:
19257:
19251:
19248:
19247:
19227:
19222:
19212:
19201:
19200:
19187:
19176:
19175:
19164:
19163:
19160:
19157:
19156:
19136:
19132:
19130:
19127:
19126:
19122:
19116:
19108:
19087:
19084:
19083:
19067:
19064:
19063:
19027:
19023:
19022:
19017:
19014:
19013:
18997:
18994:
18993:
18966:
18962:
18955:
18953:
18950:
18949:
18929:
18925:
18923:
18920:
18919:
18862:
18858:
18857:
18855:
18852:
18851:
18831:
18827:
18813:
18810:
18809:
18792:
18789:
18788:
18769:
18766:
18765:
18734:
18722:
18718:
18716:
18713:
18712:
18682:
18675:
18671:
18664:
18662:
18646:
18642:
18641:
18635:
18624:
18611:
18607:
18605:
18602:
18601:
18581:
18577:
18575:
18572:
18571:
18554:
18550:
18548:
18545:
18544:
18519:
18515:
18503:
18499:
18497:
18488:
18484:
18482:
18479:
18478:
18459:
18456:
18455:
18427:
18424:
18423:
18395:
18392:
18391:
18387:
18381:
18344:
18340:
18334:
18330:
18324:
18313:
18300:
18296:
18294:
18291:
18290:
18271:
18268:
18267:
18251:
18248:
18247:
18229:
18225:
18223:
18220:
18219:
18202:
18198:
18196:
18193:
18192:
18176:
18173:
18172:
18155:
18151:
18149:
18146:
18145:
18129:
18126:
18125:
18109:
18106:
18105:
18089:
18086:
18085:
18082:
18054:
18049:
18048:
18040:
18038:
18035:
18034:
18006:
18001:
17995:
17990:
17989:
17980:
17969:
17968:
17950:
17949:
17947:
17944:
17943:
17914:
17910:
17905:
17900:
17894:
17889:
17888:
17876:
17865:
17864:
17858:
17855:
17854:
17831:
17828:
17827:
17810:
17806:
17786:
17783:
17782:
17759:
17757:
17754:
17753:
17731:
17730:
17728:
17725:
17724:
17707:
17703:
17701:
17698:
17697:
17677:
17675:
17672:
17671:
17654:
17650:
17644:
17640:
17625:
17621:
17610:
17608:
17605:
17604:
17601:
17591:
17569:
17568:
17561:
17560:
17554:
17553:
17543:
17542:
17532:
17531:
17525:
17524:
17514:
17513:
17506:
17505:
17500:
17494:
17493:
17488:
17478:
17477:
17468:
17467:
17456:
17451:
17450:
17441:
17436:
17431:
17421:
17416:
17415:
17406:
17401:
17396:
17394:
17390:
17381:
17367:
17362:
17357:
17344:
17339:
17334:
17332:
17328:
17327:
17320:
17309:
17307:
17306:
17302:
17300:
17297:
17296:
17272:
17271:
17266:
17260:
17259:
17254:
17244:
17243:
17234:
17229:
17228:
17218:
17211:
17210:
17205:
17195:
17194:
17193:
17184:
17179:
17178:
17176:
17173:
17172:
17153:
17152:
17147:
17141:
17140:
17135:
17125:
17124:
17115:
17110:
17109:
17099:
17092:
17091:
17086:
17076:
17075:
17074:
17065:
17060:
17059:
17057:
17054:
17053:
17025:
17014:
17009:
17008:
17002:
16991:
16986:
16982:
16981:
16966:
16964:
16963:
16958:
16957:
16955:
16952:
16951:
16918:
16913:
16912:
16906:
16901:
16900:
16894:
16883:
16878:
16874:
16862:
16860:
16859:
16854:
16853:
16839:
16837:
16836:
16834:
16831:
16830:
16805:
16800:
16795:
16785:
16780:
16779:
16777:
16774:
16773:
16747:
16745:
16742:
16741:
16721:
16717:
16715:
16712:
16711:
16703:
16679:
16667:
16663:
16658:
16652:
16648:
16646:
16643:
16642:
16616:
16612:
16605:
16593:
16589:
16580:
16575:
16574:
16573:
16569:
16563:
16552:
16548:
16539:
16534:
16533:
16532:
16528:
16527:
16521:
16517:
16511:
16500:
16495:
16493:
16485:
16483:
16480:
16479:
16453:
16449:
16447:
16444:
16443:
16423:
16422:
16407:
16403:
16398:
16392:
16388:
16376:
16372:
16371:
16359:
16355:
16346:
16341:
16340:
16339:
16335:
16329:
16318:
16314:
16305:
16300:
16299:
16298:
16294:
16293:
16287:
16283:
16277:
16266:
16261:
16259:
16250:
16249:
16238:
16234:
16225:
16220:
16219:
16218:
16214:
16208:
16197:
16193:
16184:
16179:
16178:
16177:
16173:
16172:
16166:
16162:
16156:
16145:
16132:
16127:
16117:
16106:
16093:
16082:
16078:
16072:
16061:
16056:
16052:
16051:
16050:
16043:
16039:
16033:
16022:
16017:
16015:
16008:
16003:
15999:
15997:
15994:
15993:
15974:
15972:
15969:
15968:
15941:
15936:
15935:
15929:
15925:
15919:
15908:
15894:
15890:
15889:
15887:
15884:
15883:
15862:
15858:
15857:
15855:
15852:
15851:
15824:
15820:
15814:
15803:
15798:
15792:
15788:
15786:
15774:
15768:
15765:
15764:
15744:
15740:
15738:
15735:
15734:
15708:
15704:
15698:
15687:
15674:
15670:
15668:
15665:
15664:
15652:In the case of
15650:
15601:
15597:
15596:
15584:
15580:
15571:
15566:
15565:
15564:
15560:
15554:
15543:
15539:
15530:
15525:
15524:
15523:
15519:
15518:
15512:
15508:
15502:
15491:
15486:
15484:
15476:
15474:
15471:
15470:
15450:
15447:
15444:
15443:
15432:
15402:
15398:
15386:
15382:
15373:
15368:
15367:
15366:
15362:
15356:
15345:
15341:
15332:
15327:
15326:
15325:
15321:
15320:
15314:
15310:
15304:
15293:
15288:
15286:
15278:
15276:
15273:
15272:
15243:
15239:
15233:
15222:
15217:
15210:
15205:
15204:
15198:
15194:
15188:
15177:
15172:
15170:
15160:
15156:
15155:
15153:
15150:
15149:
15128:
15124:
15123:
15121:
15118:
15117:
15087:
15083:
15081:
15078:
15077:
15056:
15051:
15050:
15048:
15045:
15044:
15041:
15006:
15001:
14985:
14979:
14978:
14963:
14960:
14959:
14942:
14941:
14926:
14922:
14917:
14911:
14907:
14895:
14891:
14890:
14883:
14879:
14873:
14869:
14860:
14856:
14847:
14843:
14837:
14826:
14821:
14819:
14810:
14809:
14797:
14792:
14783:
14777:
14773:
14763:
14757:
14751:
14750:
14739:
14738:
14735:
14728:
14719:
14713:
14712:
14704:
14702:
14699:
14698:
14670:
14665:
14656:
14650:
14646:
14645:
14641:
14633:
14630:
14629:
14602:
14598:
14589:
14585:
14580:
14574:
14569:
14560:
14557:
14556:
14525:
14523:
14519:
14517:
14514:
14513:
14489:
14484:
14474:
14470:
14468:
14461:
14457:
14455:
14452:
14451:
14434:
14429:
14419:
14408:
14395:
14391:
14389:
14386:
14385:
14368:
14364:
14358:
14347:
14334:
14330:
14328:
14325:
14324:
14307:
14306:
14300:
14295:
14278:
14273:
14263:
14259:
14257:
14250:
14246:
14237:
14236:
14227:
14223:
14182:
14177:
14167:
14163:
14161:
14149:
14145:
14103:
14102:
14094:
14090:
14080:
14076:
14070:
14066:
14057:
14053:
14035:
14031:
14025:
14014:
14009:
14007:
14000:
13991:
13985:
13984:
13973:
13972:
13959:
13958:
13952:
13947:
13923:
13921:
13917:
13908:
13907:
13898:
13894:
13851:
13839:
13835:
13793:
13792:
13777:
13773:
13761:
13757:
13739:
13728:
13723:
13721:
13714:
13705:
13694:
13693:
13692:
13679:
13677:
13674:
13673:
13653:
13648:
13642:
13639:
13638:
13622:
13619:
13618:
13611:
13574:
13570:
13568:
13565:
13564:
13548:
13545:
13544:
13527:
13523:
13517:
13506:
13500:
13497:
13496:
13476:
13465:
13461:
13452:
13448:
13447:
13443:
13442:
13436:
13432:
13426:
13415:
13396:
13392:
13386:
13375:
13370:
13363:
13359:
13353:
13342:
13337:
13335:
13323:
13319:
13317:
13314:
13313:
13290:
13286:
13281:
13278:
13277:
13240:
13237:
13236:
13202:
13199:
13198:
13146:
13143:
13142:
13110:
13106:
13100:
13089:
13084:
13077:
13066:
13062:
13053:
13049:
13048:
13044:
13043:
13037:
13033:
13027:
13016:
13011:
13009:
12997:
12993:
12991:
12988:
12987:
12977:
12924:
12920:
12918:
12915:
12914:
12897:
12891:
12890:
12879:
12878:
12875:
12872:
12871:
12844:
12840:
12834:
12823:
12817:
12814:
12813:
12796:
12795:
12786:
12782:
12776:
12765:
12760:
12753:
12742:
12738:
12729:
12725:
12724:
12720:
12719:
12713:
12709:
12703:
12692:
12687:
12685:
12678:
12672:
12666:
12665:
12654:
12653:
12649:
12648:
12636:
12619:
12615:
12614:
12610:
12609:
12603:
12592:
12587:
12585:
12578:
12569:
12558:
12557:
12556:
12552:
12550:
12547:
12546:
12526:
12515:
12514:
12513:
12511:
12508:
12507:
12486:
12480:
12479:
12468:
12467:
12464:
12461:
12460:
12458:sample variance
12433:
12429:
12427:
12424:
12423:
12412:
12406:
12401:
12375:
12371:
12370:
12364:
12353:
12348:
12342:
12338:
12336:
12324:
12318:
12315:
12314:
12289:
12284:
12278:
12274:
12259:
12258:
12254:
12252:
12249:
12248:
12231:
12226:
12220:
12217:
12216:
12191:
12187:
12182:
12177:
12171:
12160:
12145:
12140:
12137:
12136:
12111:
12107:
12102:
12097:
12091:
12080:
12070:
12065:
12052:
12041:
12040:
12034:
12031:
12030:
12010:
12005:
11992:
11987:
11981:
11978:
11977:
11947:
11946:
11942:
11940:
11937:
11936:
11915:
11910:
11899:
11895:
11890:
11885:
11879:
11868:
11855:
11844:
11843:
11837:
11834:
11833:
11813:
11808:
11802:
11799:
11798:
11795:
11779:
11753:
11749:
11743:
11732:
11731:
11730:
11721:
11717:
11708:
11703:
11687:
11683:
11672:
11671:
11649:
11644:
11629:
11618:
11617:
11611:
11610:
11608:
11605:
11604:
11578:
11574:
11570:
11568:
11554:
11553:
11551:
11548:
11547:
11522:
11518:
11507:
11506:
11497:
11493:
11481:
11476:
11465:
11464:
11451:
11440:
11439:
11438:
11427:
11426:
11417:
11413:
11407:
11403:
11386:
11385:
11376:
11372:
11360:
11349:
11348:
11347:
11335:
11331:
11325:
11314:
11313:
11312:
11301:
11300:
11291:
11287:
11281:
11277:
11270:
11266:
11257:
11253:
11242:
11241:
11219:
11214:
11199:
11192:
11181:
11180:
11179:
11178:
11172:
11171:
11169:
11166:
11165:
11151:
11134:
11111:
11107:
11101:
11090:
11089:
11088:
11079:
11075:
11066:
11061:
11051:
11040:
11027:
11023:
11017:
11013:
11007:
10996:
10988:
10983:
10974:
10961:
10957:
10950:
10939:
10938:
10937:
10928:
10924:
10923:
10921:
10912:
10908:
10898:
10894:
10893:
10887:
10876:
10864:
10853:
10852:
10851:
10846:
10827:
10816:
10815:
10814:
10807:
10805:
10804:
10802:
10799:
10798:
10772:
10768:
10757:
10754:
10753:
10727:
10723:
10714:
10710:
10692:
10688:
10671:
10668:
10667:
10638:
10634:
10627:
10616:
10615:
10614:
10605:
10601:
10600:
10598:
10590:
10586:
10579:
10568:
10567:
10566:
10557:
10553:
10552:
10550:
10541:
10530:
10529:
10528:
10527:
10523:
10517:
10506:
10496:
10485:
10473:
10462:
10461:
10460:
10455:
10436:
10425:
10424:
10423:
10416:
10414:
10413:
10389:
10388:
10381:
10379:
10378:
10376:
10373:
10372:
10347:
10343:
10341:
10338:
10337:
10317:
10313:
10307:
10303:
10291:
10285:
10282:
10281:
10264:
10260:
10254:
10250:
10238:
10232:
10229:
10228:
10194:
10190:
10181:
10177:
10159:
10155:
10138:
10135:
10134:
10105:
10104:
10090:
10089:
10075:
10074:
10072:
10069:
10068:
10035:
10034:
10020:
10019:
10005:
10004:
9993:
9992:
9965:
9964:
9957:
9955:
9954:
9943:
9942:
9918:
9917:
9910:
9908:
9907:
9906:
9902:
9894:
9883:
9882:
9881:
9876:
9860:
9856:
9849:
9845:
9834:
9833:
9824:
9820:
9819:
9817:
9809:
9805:
9798:
9794:
9783:
9782:
9773:
9769:
9768:
9766:
9757:
9746:
9745:
9744:
9743:
9739:
9733:
9722:
9712:
9701:
9689:
9678:
9677:
9676:
9671:
9647:
9646:
9639:
9637:
9636:
9634:
9631:
9630:
9602:
9598:
9589:
9583:
9569:
9565:
9556:
9550:
9549:
9545:
9539:
9528:
9514:
9493:
9483:
9479:
9473:
9462:
9457:
9447:
9437:
9433:
9427:
9416:
9411:
9409:
9394:
9384:
9382:
9368:
9367:
9365:
9362:
9361:
9336:
9326:
9324:
9310:
9309:
9307:
9304:
9303:
9275:
9271:
9269:
9266:
9265:
9246:
9242:
9237:
9228:
9224:
9222:
9219:
9218:
9199:
9195:
9190:
9181:
9177:
9175:
9172:
9171:
9151:
9147:
9138:
9134:
9132:
9129:
9128:
9107:
9103:
9097:
9086:
9085:
9084:
9075:
9071:
9062:
9057:
9047:
9036:
9023:
9019:
9013:
9009:
9003:
8992:
8984:
8979:
8960:
8949:
8948:
8947:
8940:
8938:
8937:
8935:
8932:
8931:
8904:Ratio estimator
8877:
8867:
8863:
8857:
8846:
8841:
8831:
8821:
8817:
8811:
8800:
8795:
8793:
8784:
8773:
8772:
8771:
8769:
8766:
8765:
8741:
8737:
8731:
8720:
8715:
8708:
8704:
8698:
8694:
8688:
8677:
8672:
8670:
8661:
8650:
8649:
8648:
8646:
8643:
8642:
8625:
8614:
8613:
8612:
8597:
8587:
8583:
8577:
8566:
8561:
8551:
8541:
8537:
8531:
8520:
8515:
8513:
8498:
8488:
8484:
8478:
8467:
8462:
8452:
8442:
8438:
8432:
8421:
8416:
8414:
8399:
8388:
8387:
8380:
8369:
8364:
8354:
8343:
8342:
8335:
8324:
8319:
8317:
8303:
8299:
8294:
8288:
8284:
8278:
8267:
8262:
8253:
8249:
8243:
8239:
8237:
8231:
8227:
8221:
8210:
8205:
8203:
8184:
8182:
8181:
8167:
8166:
8164:
8161:
8160:
8140:
8136:
8130:
8119:
8114:
8107:
8103:
8097:
8093:
8087:
8076:
8071:
8069:
8057:
8046:
8045:
8044:
8038:
8027:
8022:
8015:
8004:
8003:
8002:
7996:
7985:
7980:
7978:
7964:
7960:
7955:
7949:
7938:
7933:
7924:
7920:
7914:
7910:
7908:
7902:
7891:
7886:
7884:
7870:
7869:
7861:
7858:
7857:
7840:
7836:
7834:
7831:
7830:
7810:
7799:
7798:
7791:
7780:
7765:
7761:
7755:
7751:
7749:
7743:
7732:
7719:
7715:
7709:
7705:
7699:
7688:
7670:
7669:
7667:
7664:
7663:
7644:
7640:
7635:
7626:
7622:
7620:
7617:
7616:
7600:
7597:
7596:
7574:
7573:
7571:
7568:
7567:
7551:
7548:
7547:
7531:
7528:
7527:
7505:
7504:
7502:
7499:
7498:
7495:
7490:
7486:
7472:
7471:
7465:
7454:
7450:
7444:
7440:
7439:
7435:
7434:
7428:
7417:
7405:
7401:
7396:
7387:
7386:
7373:
7369:
7363:
7359:
7357:
7349:
7345:
7339:
7335:
7333:
7324:
7320:
7310:
7306:
7300:
7289:
7277:
7273:
7268:
7259:
7258:
7247:
7236:
7235:
7234:
7228:
7217:
7216:
7215:
7206:
7195:
7194:
7193:
7192:
7188:
7182:
7171:
7159:
7155:
7150:
7141:
7140:
7129:
7118:
7117:
7116:
7110:
7099:
7098:
7097:
7088:
7077:
7076:
7075:
7074:
7070:
7064:
7053:
7043:
7032:
7020:
7016:
7011:
7004:
6994:
6986:
6985:
6974:
6973:
6972:
6959:
6957:
6954:
6953:
6927:
6923:
6912:
6909:
6908:
6907:We assume that
6884:
6873:
6869:
6863:
6859:
6858:
6854:
6853:
6847:
6836:
6824:
6820:
6815:
6802:
6794:
6793:
6777:
6775:
6774:
6773:
6762:
6759:
6758:
6729:
6725:
6716:
6712:
6689:
6686:
6685:
6665:
6661:
6644:
6640:
6633:
6629:
6623:
6619:
6618:
6616:
6598:
6587:
6586:
6585:
6583:
6580:
6579:
6578:, and for i=j:
6557:
6553:
6546:
6542:
6536:
6532:
6531:
6529:
6511:
6500:
6499:
6498:
6496:
6493:
6492:
6472:
6468:
6466:
6463:
6462:
6442:
6438:
6429:
6425:
6419:
6415:
6403:
6399:
6387:
6383:
6374:
6370:
6362:
6359:
6358:
6339:
6335:
6329:
6325:
6323:
6314:
6303:
6302:
6301:
6299:
6296:
6295:
6271:
6260:
6259:
6258:
6252:
6241:
6240:
6239:
6230:
6219:
6218:
6217:
6216:
6212:
6206:
6195:
6185:
6174:
6162:
6158:
6153:
6140:
6132:
6131:
6115:
6113:
6112:
6111:
6100:
6097:
6096:
6087:
6070:
6069:
6063:
6045:
6043:
6034:
6030:
6024:
6020:
6019:
6015:
6014:
6008:
5997:
5980:
5975:
5966:
5965:
5959:
5941:
5939:
5930:
5926:
5920:
5916:
5915:
5911:
5910:
5904:
5893:
5876:
5871:
5860:
5856:
5854:
5845:
5844:
5838:
5821:
5817:
5811:
5807:
5801:
5790:
5785:
5783:
5769:
5765:
5759:
5755:
5753:
5749:
5745:
5744:
5738:
5727:
5710:
5705:
5695:
5686:
5675:
5671:
5665:
5661:
5655:
5644:
5630:
5619:
5615:
5609:
5605:
5603:
5593:
5592:
5588:
5587:
5581:
5570:
5553:
5548:
5538:
5529:
5528:
5522:
5505:
5494:
5493:
5492:
5481:
5477:
5471:
5467:
5465:
5464:
5460:
5459:
5453:
5442:
5425:
5420:
5410:
5403:
5394:
5383:
5382:
5381:
5368:
5366:
5363:
5362:
5333:
5329:
5323:
5319:
5318:
5316:
5310:
5299:
5279:
5277:
5275:
5272:
5271:
5254:
5236:
5234:
5225:
5221:
5215:
5211:
5210:
5206:
5205:
5199:
5188:
5171:
5166:
5148:
5137:
5136:
5135:
5124:
5121:
5120:
5093:
5083:
5079:
5073:
5062:
5047:
5043:
5034:
5028:
5022:
5011:
4995:
4991:
4987:
4978:
4972:
4966:
4955:
4940:
4936:
4927:
4921:
4915:
4904:
4890:
4875:
4864:
4863:
4862:
4860:
4857:
4856:
4840:
4837:
4836:
4815:
4812:
4811:
4789:
4785:
4779:
4768:
4756:
4753:
4752:
4720:
4710:
4706:
4700:
4689:
4684:
4682:
4665:
4661:
4652:
4646:
4640:
4629:
4624:
4622:
4605:
4601:
4592:
4586:
4580:
4569:
4555:
4554:
4552:
4532:
4521:
4520:
4519:
4517:
4504:
4503:
4487:
4485:
4484:
4483:
4481:
4478:
4477:
4450:
4432:
4430:
4421:
4417:
4411:
4407:
4406:
4402:
4401:
4395:
4384:
4367:
4362:
4354:
4350:
4345:
4328:
4327:
4311:
4309:
4308:
4307:
4303:
4295:
4292:
4291:
4277:sampling design
4247:
4237:
4233:
4227:
4216:
4211:
4209:
4189:
4178:
4177:
4176:
4174:
4161:
4160:
4144:
4142:
4141:
4140:
4138:
4135:
4134:
4127:
4098:
4094:
4087:
4082:
4071:
4067:
4062:
4053:
4049:
4047:
4044:
4043:
4018:
4014:
4007:
4003:
3997:
3993:
3992:
3990:
3981:
3970:
3969:
3968:
3962:
3958:
3946:
3935:
3934:
3931:
3928:
3927:
3910:
3899:
3898:
3897:
3883:
3879:
3873:
3869:
3867:
3865:
3862:
3861:
3841:
3838:
3837:
3818:
3814:
3808:
3804:
3802:
3793:
3782:
3781:
3780:
3778:
3775:
3774:
3754:
3751:
3750:
3727:
3723:
3708:
3704:
3698:
3693:
3677:
3673:
3661:
3656:
3637:
3625:
3622:
3621:
3604:
3600:
3594:
3590:
3578:
3574:
3562:
3558:
3543:
3531:
3528:
3527:
3510:
3506:
3500:
3496:
3484:
3478:
3475:
3474:
3457:
3453:
3451:
3448:
3447:
3430:
3426:
3424:
3421:
3420:
3395:
3391:
3389:
3386:
3385:
3363:
3359:
3357:
3348:
3344:
3335:one sample draw
3333:
3328:
3316:
3312:
3304:
3301:
3300:
3283:
3279:
3265:
3250:
3246:
3238:
3235:
3234:
3209:
3205:
3203:
3200:
3199:
3192:survey sampling
3165:
3164:
3162:
3159:
3158:
3142:
3139:
3138:
3108:sampling design
3086:
3082:
3080:
3077:
3076:
3069:
3035:
3030:
3020:
3009:
3004:
3002:
2988:
2984:
2982:
2980:
2977:
2976:
2953:
2949:
2943:
2932:
2927:
2925:
2911:
2910:
2908:
2905:
2904:
2880:
2873:
2869:
2858:
2857:
2848:
2844:
2835:
2824:
2819:
2817:
2808:
2803:
2792:
2791:
2788:
2785:
2784:
2762:
2751:
2750:
2749:
2739:
2735:
2732:
2726:
2721:
2710:
2709:
2696:
2685:
2684:
2683:
2672:
2669:
2668:
2631:
2626:
2588:
2587:
2579:
2576:
2575:
2552:
2548:
2546:
2543:
2542:
2521:
2517:
2508:
2503:
2497:
2486:
2465:
2464:
2456:
2453:
2452:
2431:
2427:
2426:
2414:
2410:
2402:
2399:
2398:
2373:
2372:
2370:
2367:
2366:
2363:
2358:
2320:
2315:
2305:
2301:
2299:
2293:
2282:
2272:
2261:
2260:
2242:
2241:
2239:
2236:
2235:
2207:
2193:
2188:
2178:
2167:
2162:
2158:
2157:
2149:
2144:
2131:
2126:
2121:
2115:
2104:
2099:
2097:
2087:
2082:
2071:
2067:
2062:
2057:
2051:
2040:
2027:
2016:
2015:
2009:
2006:
2005:
1985:
1981:
1972:
1968:
1966:
1963:
1962:
1943:
1937:
1932:
1919:
1908:
1907:
1901:
1898:
1897:
1870:
1866:
1860:
1849:
1844:
1838:
1822:
1817:
1807:
1796:
1791:
1785:
1770:
1769:
1765:
1763:
1760:
1759:
1726:
1722:
1716:
1705:
1700:
1688:
1684:
1675:
1671:
1670:
1666:
1660:
1649:
1644:
1642:
1627:
1622:
1612:
1606:
1595:
1590:
1576:
1571:
1561:
1557:
1554:
1550:
1544:
1533:
1528:
1526:
1512:
1511:
1509:
1506:
1505:
1477:
1472:
1463:
1454:
1450:
1448:
1445:
1444:
1424:
1419:
1413:
1410:
1409:
1385:
1381:
1379:
1376:
1375:
1372:
1366:
1360:
1335:
1331:
1326:
1316:
1305:
1299:
1281:
1280:
1276:
1274:
1271:
1270:
1240:
1239:
1235:
1233:
1230:
1229:
1208:
1204:
1202:
1199:
1198:
1173:
1169:
1168:
1162:
1151:
1137:
1135:
1132:
1131:
1103:
1099:
1098:
1092:
1081:
1076:
1070:
1066:
1064:
1052:
1046:
1043:
1042:
1017:
1013:
1004:
999:
993:
982:
964:
963:
961:
958:
957:
927:
922:
916:
905:
899:
896:
895:
865:
861:
846:
842:
833:
829:
828:
821:
817:
811:
807:
792:
788:
782:
778:
769:
765:
759:
755:
754:
752:
738:
737:
735:
732:
731:
702:
698:
692:
681:
676:
669:
665:
659:
655:
649:
638:
633:
631:
617:
616:
614:
611:
610:
585:
581:
566:
562:
553:
549:
548:
544:
542:
539:
538:
513:
509:
494:
490:
481:
477:
476:
472:
470:
467:
466:
459:
390:
389:
387:
384:
383:
347:
342:
340:
337:
336:
303:
298:
296:
293:
292:
277:Since only the
275:
237:
202:
200:
186:
185:
183:
180:
179:
147:
133:
132:
130:
127:
126:
104:
100:
98:
92:
88:
86:
83:
78:
66:arithmetic mean
51:arithmetic mean
43:
32:Weighted median
28:
23:
22:
15:
12:
11:
5:
20339:
20329:
20328:
20323:
20318:
20304:
20303:
20298:
20279:
20278:External links
20276:
20275:
20274:
20268:
20255:
20236:
20233:
20230:
20229:
20206:
20185:
20178:
20160:
20144:
20117:(4): 485–490.
20091:
20063:
20038:
19997:
19958:
19942:
19935:
19884:
19860:
19859:
19857:
19854:
19851:
19850:
19836:
19835:
19833:
19830:
19828:
19827:
19822:
19817:
19812:
19807:
19802:
19797:
19792:
19787:
19782:
19777:
19772:
19767:
19762:
19757:
19752:
19746:
19744:
19741:
19740:
19739:
19728:
19722:
19719:
19716:
19709:
19705:
19698:
19695:
19689:
19684:
19680:
19676:
19671:
19666:
19663:
19660:
19656:
19649:
19644:
19640:
19612:
19608:
19602:
19598:
19594:
19589:
19582:
19579:
19573:
19546:
19539:
19536:
19530:
19507:
19503:
19499:
19494:
19490:
19458:
19455:
19446:
19443:
19428:
19427:
19416:
19409:
19404:
19400:
19393:
19389:
19382:
19379:
19373:
19368:
19364:
19360:
19352:
19347:
19344:
19341:
19337:
19330:
19327:
19324:
19321:
19318:
19314:
19309:
19304:
19299:
19295:
19265:
19260:
19256:
19244:
19243:
19230:
19225:
19221:
19215:
19208:
19205:
19199:
19195:
19190:
19183:
19180:
19171:
19168:
19139:
19135:
19115:
19112:
19107:
19104:
19091:
19071:
19048:
19045:
19042:
19039:
19036:
19033:
19030:
19026:
19021:
19001:
18981:
18978:
18972:
18969:
18965:
18961:
18958:
18935:
18932:
18928:
18907:
18904:
18901:
18898:
18895:
18892:
18889:
18886:
18883:
18880:
18877:
18874:
18868:
18865:
18861:
18837:
18834:
18830:
18826:
18823:
18820:
18817:
18796:
18773:
18753:
18750:
18747:
18744:
18741:
18737:
18733:
18730:
18725:
18721:
18709:
18708:
18697:
18691:
18688:
18685:
18678:
18674:
18670:
18667:
18661:
18655:
18652:
18649:
18645:
18638:
18633:
18630:
18627:
18623:
18619:
18614:
18610:
18584:
18580:
18557:
18553:
18541:
18540:
18529:
18522:
18518:
18512:
18509:
18506:
18502:
18496:
18491:
18487:
18463:
18454:we can define
18443:
18440:
18437:
18434:
18431:
18411:
18408:
18405:
18402:
18399:
18380:
18377:
18376:
18375:
18364:
18359:
18356:
18353:
18350:
18347:
18343:
18337:
18333:
18327:
18322:
18319:
18316:
18312:
18308:
18303:
18299:
18275:
18255:
18232:
18228:
18205:
18201:
18180:
18158:
18154:
18133:
18113:
18093:
18081:
18078:
18077:
18076:
18065:
18060:
18057:
18052:
18047:
18043:
18028:
18027:
18016:
18013:
18009:
18004:
17998:
17993:
17988:
17983:
17976:
17973:
17967:
17963:
17957:
17954:
17937:
17936:
17925:
17920:
17917:
17913:
17908:
17903:
17897:
17892:
17887:
17884:
17879:
17872:
17869:
17863:
17835:
17813:
17809:
17805:
17802:
17799:
17796:
17793:
17790:
17780:vector of ones
17762:
17738:
17735:
17710:
17706:
17680:
17657:
17653:
17647:
17643:
17639:
17636:
17633:
17628:
17624:
17620:
17617:
17613:
17590:
17587:
17583:
17582:
17565:
17559:
17556:
17555:
17552:
17549:
17548:
17546:
17541:
17536:
17530:
17527:
17526:
17523:
17520:
17519:
17517:
17510:
17504:
17501:
17499:
17496:
17495:
17492:
17489:
17487:
17484:
17483:
17481:
17476:
17473:
17471:
17469:
17465:
17459:
17454:
17447:
17444:
17439:
17434:
17429:
17424:
17419:
17412:
17409:
17404:
17399:
17393:
17387:
17384:
17379:
17373:
17370:
17365:
17360:
17355:
17350:
17347:
17342:
17337:
17331:
17326:
17323:
17321:
17316:
17312:
17305:
17304:
17290:
17289:
17276:
17270:
17267:
17265:
17262:
17261:
17258:
17255:
17253:
17250:
17249:
17247:
17242:
17237:
17232:
17226:
17221:
17215:
17209:
17206:
17204:
17201:
17200:
17198:
17192:
17187:
17182:
17170:
17157:
17151:
17148:
17146:
17143:
17142:
17139:
17136:
17134:
17131:
17130:
17128:
17123:
17118:
17113:
17107:
17102:
17096:
17090:
17087:
17085:
17082:
17081:
17079:
17073:
17068:
17063:
17036:
17031:
17028:
17023:
17017:
17012:
17005:
17000:
16997:
16994:
16990:
16985:
16980:
16973:
16969:
16961:
16931:
16927:
16921:
16916:
16909:
16904:
16897:
16892:
16889:
16886:
16882:
16877:
16869:
16865:
16857:
16852:
16846:
16842:
16816:
16811:
16808:
16803:
16798:
16793:
16788:
16783:
16768:matrix inverse
16750:
16724:
16720:
16702:
16699:
16686:
16682:
16678:
16675:
16670:
16666:
16661:
16655:
16651:
16639:
16638:
16627:
16619:
16615:
16611:
16608:
16602:
16596:
16592:
16588:
16583:
16578:
16572:
16566:
16561:
16555:
16551:
16547:
16542:
16537:
16531:
16524:
16520:
16514:
16509:
16506:
16503:
16499:
16492:
16488:
16464:
16461:
16456:
16452:
16437:
16436:
16421:
16415:
16410:
16406:
16401:
16395:
16391:
16387:
16384:
16379:
16375:
16368:
16362:
16358:
16354:
16349:
16344:
16338:
16332:
16327:
16321:
16317:
16313:
16308:
16303:
16297:
16290:
16286:
16280:
16275:
16272:
16269:
16265:
16258:
16255:
16253:
16251:
16247:
16241:
16237:
16233:
16228:
16223:
16217:
16211:
16206:
16200:
16196:
16192:
16187:
16182:
16176:
16169:
16165:
16159:
16154:
16151:
16148:
16144:
16135:
16130:
16126:
16120:
16115:
16112:
16109:
16105:
16101:
16096:
16091:
16085:
16081:
16075:
16070:
16067:
16064:
16060:
16055:
16046:
16042:
16036:
16031:
16028:
16025:
16021:
16014:
16011:
16009:
16006:
16002:
16001:
15977:
15961:
15960:
15949:
15944:
15939:
15932:
15928:
15922:
15917:
15914:
15911:
15907:
15903:
15897:
15893:
15865:
15861:
15844:
15843:
15827:
15823:
15817:
15812:
15809:
15806:
15802:
15795:
15791:
15785:
15781:
15777:
15773:
15747:
15743:
15731:
15730:
15719:
15716:
15711:
15707:
15701:
15696:
15693:
15690:
15686:
15682:
15677:
15673:
15649:
15646:
15630:
15629:
15618:
15612:
15609:
15604:
15600:
15593:
15587:
15583:
15579:
15574:
15569:
15563:
15557:
15552:
15546:
15542:
15538:
15533:
15528:
15522:
15515:
15511:
15505:
15500:
15497:
15494:
15490:
15483:
15479:
15453:
15431:
15428:
15424:
15423:
15412:
15405:
15401:
15395:
15389:
15385:
15381:
15376:
15371:
15365:
15359:
15354:
15348:
15344:
15340:
15335:
15330:
15324:
15317:
15313:
15307:
15302:
15299:
15296:
15292:
15285:
15281:
15266:
15265:
15254:
15246:
15242:
15236:
15231:
15228:
15225:
15221:
15213:
15208:
15201:
15197:
15191:
15186:
15183:
15180:
15176:
15169:
15163:
15159:
15131:
15127:
15098:
15095:
15090:
15086:
15059:
15054:
15040:
15037:
15009:
15000:
14996:
14993:
14988:
14982:
14977:
14973:
14970:
14967:
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5779:
5772:
5768:
5762:
5758:
5752:
5748:
5741:
5736:
5733:
5730:
5726:
5719:
5716:
5713:
5709:
5702:
5699:
5694:
5689:
5684:
5678:
5674:
5668:
5664:
5658:
5653:
5650:
5647:
5643:
5637:
5634:
5629:
5622:
5618:
5612:
5608:
5600:
5597:
5591:
5584:
5579:
5576:
5573:
5569:
5562:
5559:
5556:
5552:
5545:
5542:
5537:
5534:
5532:
5530:
5525:
5520:
5514:
5511:
5508:
5501:
5498:
5491:
5484:
5480:
5474:
5470:
5463:
5456:
5451:
5448:
5445:
5441:
5434:
5431:
5428:
5424:
5417:
5414:
5409:
5406:
5404:
5402:
5390:
5387:
5380:
5377:
5374:
5371:
5370:
5342:
5336:
5332:
5326:
5322:
5313:
5308:
5305:
5302:
5298:
5294:
5289:
5285:
5282:
5257:
5252:
5246:
5242:
5239:
5233:
5228:
5224:
5218:
5214:
5209:
5202:
5197:
5194:
5191:
5187:
5180:
5177:
5174:
5170:
5165:
5162:
5157:
5154:
5151:
5144:
5141:
5134:
5131:
5128:
5100:
5096:
5092:
5086:
5082:
5076:
5071:
5068:
5065:
5061:
5057:
5050:
5046:
5041:
5037:
5033:
5025:
5020:
5017:
5014:
5010:
5006:
4998:
4994:
4990:
4985:
4981:
4977:
4969:
4964:
4961:
4958:
4954:
4950:
4943:
4939:
4934:
4930:
4926:
4918:
4913:
4910:
4907:
4903:
4897:
4894:
4889:
4884:
4881:
4878:
4871:
4868:
4844:
4819:
4792:
4788:
4782:
4777:
4774:
4771:
4767:
4763:
4760:
4749:
4748:
4737:
4732:
4727:
4723:
4719:
4713:
4709:
4703:
4698:
4695:
4692:
4688:
4681:
4676:
4668:
4664:
4659:
4655:
4651:
4643:
4638:
4635:
4632:
4628:
4621:
4616:
4608:
4604:
4599:
4595:
4591:
4583:
4578:
4575:
4572:
4568:
4562:
4559:
4551:
4546:
4541:
4538:
4535:
4528:
4525:
4516:
4511:
4499:
4494:
4491:
4468:
4467:
4466:
4453:
4448:
4442:
4438:
4435:
4429:
4424:
4420:
4414:
4410:
4405:
4398:
4393:
4390:
4387:
4383:
4376:
4373:
4370:
4366:
4357:
4353:
4349:
4344:
4340:
4335:
4323:
4318:
4315:
4306:
4302:
4299:
4259:
4254:
4250:
4246:
4240:
4236:
4230:
4225:
4222:
4219:
4215:
4208:
4203:
4198:
4195:
4192:
4185:
4182:
4173:
4168:
4156:
4151:
4148:
4126:
4119:
4101:
4097:
4093:
4090:
4086:
4081:
4074:
4070:
4066:
4061:
4056:
4052:
4021:
4017:
4010:
4006:
4000:
3996:
3989:
3984:
3977:
3974:
3965:
3961:
3957:
3953:
3949:
3942:
3939:
3913:
3906:
3903:
3896:
3893:
3886:
3882:
3876:
3872:
3845:
3821:
3817:
3811:
3807:
3801:
3796:
3789:
3786:
3773:values, i.e.:
3758:
3735:
3730:
3726:
3722:
3719:
3716:
3711:
3707:
3701:
3696:
3692:
3688:
3685:
3680:
3676:
3672:
3669:
3664:
3659:
3655:
3651:
3648:
3644:
3640:
3636:
3632:
3629:
3607:
3603:
3597:
3593:
3589:
3586:
3581:
3577:
3573:
3570:
3565:
3561:
3557:
3554:
3550:
3546:
3542:
3538:
3535:
3513:
3509:
3503:
3499:
3495:
3491:
3487:
3483:
3460:
3456:
3433:
3429:
3398:
3394:
3371:
3366:
3362:
3356:
3351:
3347:
3343:
3340:
3331:
3327:
3324:
3319:
3315:
3311:
3308:
3286:
3282:
3278:
3275:
3272:
3264:
3261:
3258:
3253:
3249:
3245:
3242:
3212:
3208:
3172:
3169:
3146:
3089:
3085:
3068:
3065:
3044:
3038:
3033:
3029:
3023:
3018:
3015:
3012:
3008:
3001:
2996:
2991:
2987:
2962:
2956:
2952:
2946:
2941:
2938:
2935:
2931:
2924:
2918:
2915:
2889:
2886:
2883:
2876:
2872:
2865:
2862:
2856:
2851:
2847:
2843:
2838:
2833:
2830:
2827:
2823:
2816:
2811:
2806:
2799:
2796:
2781:
2780:
2765:
2758:
2755:
2747:
2742:
2738:
2729:
2724:
2717:
2714:
2707:
2704:
2699:
2692:
2689:
2682:
2679:
2676:
2630:
2627:
2625:
2622:
2610:
2607:
2604:
2601:
2595:
2592:
2586:
2583:
2563:
2560:
2555:
2551:
2530:
2524:
2520:
2515:
2511:
2507:
2500:
2495:
2492:
2489:
2485:
2481:
2478:
2472:
2469:
2463:
2460:
2440:
2434:
2430:
2425:
2422:
2417:
2413:
2409:
2406:
2380:
2377:
2362:
2359:
2357:
2354:
2342:
2341:
2330:
2323:
2318:
2314:
2308:
2304:
2296:
2291:
2288:
2285:
2281:
2275:
2268:
2265:
2259:
2255:
2249:
2246:
2229:
2228:
2217:
2210:
2205:
2199:
2196:
2191:
2187:
2181:
2176:
2173:
2170:
2166:
2161:
2152:
2147:
2143:
2137:
2134:
2129:
2125:
2118:
2113:
2110:
2107:
2103:
2096:
2090:
2085:
2081:
2074:
2070:
2065:
2061:
2054:
2049:
2046:
2043:
2039:
2035:
2030:
2023:
2020:
2014:
1988:
1984:
1980:
1975:
1971:
1950:
1946:
1940:
1935:
1931:
1927:
1922:
1915:
1912:
1906:
1894:
1893:
1882:
1873:
1869:
1863:
1858:
1855:
1852:
1848:
1843:
1837:
1828:
1825:
1820:
1816:
1810:
1805:
1802:
1799:
1795:
1790:
1784:
1777:
1774:
1768:
1749:
1748:
1737:
1729:
1725:
1719:
1714:
1711:
1708:
1704:
1697:
1691:
1687:
1683:
1678:
1674:
1669:
1663:
1658:
1655:
1652:
1648:
1641:
1630:
1625:
1621:
1617:
1609:
1604:
1601:
1598:
1594:
1587:
1579:
1574:
1570:
1564:
1560:
1553:
1547:
1542:
1539:
1536:
1532:
1525:
1519:
1516:
1499:
1498:
1487:
1480:
1475:
1471:
1467:
1462:
1457:
1453:
1427:
1422:
1418:
1388:
1384:
1362:Main article:
1359:
1356:
1355:
1354:
1338:
1334:
1329:
1325:
1319:
1314:
1311:
1308:
1304:
1298:
1295:
1288:
1285:
1279:
1247:
1244:
1238:
1211:
1207:
1197:with variance
1176:
1172:
1165:
1160:
1157:
1154:
1150:
1144:
1141:
1125:
1124:
1106:
1102:
1095:
1090:
1087:
1084:
1080:
1073:
1069:
1063:
1059:
1055:
1051:
1036:
1035:
1020:
1016:
1011:
1007:
1003:
996:
991:
988:
985:
981:
977:
971:
968:
942:
939:
934:
930:
926:
919:
914:
911:
908:
904:
888:
887:
876:
868:
864:
860:
857:
854:
849:
845:
841:
836:
832:
824:
820:
814:
810:
806:
803:
800:
795:
791:
785:
781:
777:
772:
768:
762:
758:
751:
745:
742:
725:
724:
713:
705:
701:
695:
690:
687:
684:
680:
672:
668:
662:
658:
652:
647:
644:
641:
637:
630:
624:
621:
594:
588:
584:
580:
577:
574:
569:
565:
561:
556:
552:
547:
522:
516:
512:
508:
505:
502:
497:
493:
489:
484:
480:
475:
458:
455:
454:
453:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
397:
394:
377:
376:
365:
362:
356:
353:
350:
346:
333:
332:
321:
318:
312:
309:
306:
302:
274:
271:
267:
266:
255:
252:
246:
243:
240:
235:
232:
229:
226:
223:
220:
217:
214:
211:
208:
205:
199:
193:
190:
162:
159:
154:
151:
146:
140:
137:
122:
121:
115:
114:
82:
79:
77:
74:
26:
9:
6:
4:
3:
2:
20338:
20327:
20324:
20322:
20319:
20317:
20314:
20313:
20311:
20302:
20299:
20294:
20293:
20288:
20282:
20281:
20271:
20265:
20261:
20256:
20252:
20248:
20244:
20239:
20238:
20226:
20225:0-9771170-1-4
20222:
20218:
20217:
20210:
20203:
20199:
20195:
20189:
20181:
20179:981-270-527-9
20175:
20171:
20164:
20158:
20154:
20148:
20140:
20136:
20132:
20128:
20124:
20120:
20116:
20112:
20105:
20098:
20096:
20080:
20073:
20067:
20052:
20048:
20042:
20033:
20028:
20024:
20020:
20016:
20012:
20008:
20001:
19995:
19989:
19985:
19981:
19977:
19973:
19969:
19962:
19956:
19952:
19946:
19938:
19932:
19928:
19921:
19919:
19917:
19915:
19913:
19911:
19909:
19907:
19905:
19903:
19901:
19899:
19897:
19895:
19893:
19891:
19889:
19882:
19878:
19872:
19870:
19868:
19866:
19861:
19847:
19841:
19837:
19826:
19823:
19821:
19818:
19816:
19813:
19811:
19808:
19806:
19803:
19801:
19798:
19796:
19793:
19791:
19788:
19786:
19783:
19781:
19778:
19776:
19773:
19771:
19768:
19766:
19763:
19761:
19758:
19756:
19753:
19751:
19748:
19747:
19726:
19720:
19717:
19714:
19707:
19693:
19687:
19682:
19678:
19669:
19664:
19661:
19658:
19654:
19647:
19642:
19638:
19630:
19629:
19628:
19626:
19610:
19606:
19600:
19596:
19592:
19587:
19577:
19571:
19562:
19544:
19534:
19528:
19505:
19501:
19497:
19492:
19488:
19478:
19476:
19453:
19441:
19414:
19407:
19402:
19398:
19391:
19377:
19371:
19366:
19362:
19350:
19345:
19342:
19339:
19335:
19325:
19322:
19319:
19312:
19307:
19302:
19297:
19293:
19285:
19284:
19283:
19281:
19263:
19258:
19254:
19228:
19223:
19219:
19213:
19203:
19197:
19193:
19188:
19178:
19166:
19155:
19154:
19153:
19137:
19133:
19121:
19111:
19103:
19089:
19069:
19043:
19040:
19037:
19031:
19028:
19024:
19019:
18999:
18979:
18976:
18970:
18967:
18963:
18959:
18956:
18933:
18930:
18926:
18902:
18899:
18896:
18890:
18887:
18881:
18878:
18875:
18866:
18863:
18859:
18835:
18832:
18824:
18821:
18818:
18794:
18785:
18771:
18748:
18745:
18742:
18735:
18731:
18728:
18723:
18719:
18695:
18689:
18686:
18683:
18676:
18672:
18668:
18665:
18659:
18653:
18650:
18647:
18643:
18636:
18631:
18628:
18625:
18621:
18617:
18612:
18608:
18600:
18599:
18598:
18582:
18578:
18555:
18551:
18527:
18520:
18516:
18510:
18507:
18504:
18500:
18494:
18489:
18485:
18477:
18476:
18475:
18461:
18438:
18435:
18432:
18429:
18409:
18406:
18400:
18397:
18386:
18362:
18357:
18354:
18351:
18348:
18345:
18341:
18335:
18331:
18325:
18320:
18317:
18314:
18310:
18306:
18301:
18297:
18289:
18288:
18287:
18273:
18253:
18230:
18226:
18203:
18199:
18178:
18156:
18152:
18131:
18111:
18091:
18063:
18058:
18055:
18045:
18033:
18032:
18031:
18014:
17996:
17981:
17971:
17965:
17961:
17952:
17942:
17941:
17940:
17923:
17918:
17915:
17895:
17882:
17877:
17867:
17861:
17853:
17852:
17851:
17849:
17833:
17811:
17803:
17800:
17797:
17794:
17791:
17781:
17777:
17776:design matrix
17733:
17708:
17704:
17695:
17655:
17645:
17641:
17637:
17634:
17631:
17626:
17622:
17615:
17600:
17596:
17586:
17563:
17557:
17550:
17544:
17539:
17534:
17528:
17521:
17515:
17508:
17502:
17497:
17490:
17485:
17479:
17474:
17472:
17463:
17457:
17445:
17442:
17437:
17427:
17422:
17410:
17407:
17402:
17391:
17385:
17382:
17377:
17371:
17368:
17363:
17353:
17348:
17345:
17340:
17329:
17324:
17322:
17295:
17294:
17293:
17274:
17268:
17263:
17256:
17251:
17245:
17240:
17235:
17224:
17213:
17207:
17202:
17196:
17190:
17185:
17171:
17155:
17149:
17144:
17137:
17132:
17126:
17121:
17116:
17105:
17094:
17088:
17083:
17077:
17071:
17066:
17052:
17051:
17050:
17047:
17034:
17029:
17026:
17021:
17015:
17003:
16998:
16995:
16992:
16988:
16983:
16978:
16949:
16945:
16929:
16925:
16919:
16907:
16895:
16890:
16887:
16884:
16880:
16875:
16850:
16827:
16814:
16809:
16806:
16801:
16791:
16786:
16771:
16769:
16765:
16740:
16722:
16718:
16709:
16698:
16684:
16680:
16676:
16673:
16668:
16664:
16659:
16653:
16649:
16625:
16617:
16613:
16609:
16606:
16600:
16594:
16590:
16586:
16581:
16570:
16564:
16559:
16553:
16549:
16545:
16540:
16529:
16522:
16518:
16512:
16507:
16504:
16501:
16497:
16490:
16478:
16477:
16476:
16462:
16459:
16454:
16450:
16440:
16419:
16408:
16404:
16399:
16393:
16389:
16382:
16377:
16373:
16366:
16360:
16356:
16352:
16347:
16336:
16330:
16325:
16319:
16315:
16311:
16306:
16295:
16288:
16284:
16278:
16273:
16270:
16267:
16263:
16256:
16254:
16245:
16239:
16235:
16231:
16226:
16215:
16209:
16204:
16198:
16194:
16190:
16185:
16174:
16167:
16163:
16157:
16152:
16149:
16146:
16142:
16133:
16128:
16124:
16118:
16113:
16110:
16107:
16103:
16099:
16094:
16089:
16083:
16079:
16073:
16068:
16065:
16062:
16058:
16053:
16044:
16040:
16034:
16029:
16026:
16023:
16019:
16012:
16010:
15992:
15991:
15990:
15966:
15947:
15942:
15930:
15926:
15920:
15915:
15912:
15909:
15905:
15901:
15891:
15882:
15881:
15880:
15859:
15849:
15848:weighted mean
15825:
15821:
15815:
15810:
15807:
15804:
15800:
15793:
15789:
15783:
15779:
15775:
15771:
15763:
15762:
15761:
15745:
15741:
15717:
15714:
15709:
15705:
15699:
15694:
15691:
15688:
15684:
15680:
15675:
15671:
15663:
15662:
15661:
15659:
15655:
15645:
15643:
15639:
15635:
15616:
15610:
15607:
15602:
15598:
15591:
15585:
15581:
15577:
15572:
15561:
15555:
15550:
15544:
15540:
15536:
15531:
15520:
15513:
15509:
15503:
15498:
15495:
15492:
15488:
15481:
15469:
15468:
15467:
15441:
15437:
15427:
15410:
15403:
15399:
15393:
15387:
15383:
15379:
15374:
15363:
15357:
15352:
15346:
15342:
15338:
15333:
15322:
15315:
15311:
15305:
15300:
15297:
15294:
15290:
15283:
15271:
15270:
15269:
15252:
15244:
15240:
15234:
15229:
15226:
15223:
15219:
15211:
15199:
15195:
15189:
15184:
15181:
15178:
15174:
15167:
15157:
15148:
15147:
15146:
15125:
15115:
15114:weighted mean
15110:
15096:
15093:
15088:
15084:
15075:
15057:
15036:
15033:
15030:
15028:
15023:
15007:
14998:
14994:
14986:
14975:
14968:
14938:
14927:
14923:
14918:
14912:
14908:
14901:
14896:
14892:
14884:
14874:
14870:
14866:
14861:
14857:
14848:
14844:
14838:
14833:
14830:
14827:
14816:
14814:
14798:
14793:
14789:
14784:
14778:
14774:
14767:
14764:
14758:
14741:
14732:
14730:
14720:
14709:
14697:
14696:
14695:
14692:
14677:
14671:
14666:
14662:
14657:
14651:
14647:
14642:
14638:
14635:
14627:
14609:
14606:
14603:
14599:
14595:
14590:
14586:
14581:
14575:
14570:
14566:
14541:
14536:
14532:
14529:
14526:
14520:
14498:
14490:
14485:
14481:
14475:
14471:
14465:
14462:
14458:
14435:
14430:
14426:
14420:
14415:
14412:
14409:
14405:
14401:
14396:
14392:
14369:
14365:
14359:
14354:
14351:
14348:
14344:
14340:
14335:
14331:
14301:
14292:
14287:
14279:
14274:
14270:
14264:
14260:
14254:
14251:
14247:
14243:
14241:
14228:
14217:
14211:
14205:
14202:
14193:
14183:
14178:
14174:
14168:
14164:
14158:
14150:
14139:
14133:
14127:
14124:
14115:
14109:
14107:
14095:
14091:
14081:
14071:
14067:
14063:
14058:
14054:
14044:
14036:
14032:
14026:
14021:
14018:
14015:
14004:
14002:
13992:
13975:
13965:
13953:
13944:
13939:
13934:
13930:
13927:
13924:
13918:
13914:
13912:
13899:
13888:
13882:
13876:
13873:
13864:
13856:
13853:
13848:
13840:
13829:
13823:
13817:
13814:
13805:
13799:
13797:
13787:
13778:
13770:
13767:
13762:
13758:
13748:
13740:
13735:
13732:
13729:
13718:
13716:
13706:
13696:
13686:
13672:
13671:
13670:
13654:
13645:
13624:
13616:
13606:
13604:
13600:
13596:
13591:
13575:
13571:
13550:
13528:
13524:
13518:
13513:
13510:
13507:
13503:
13477:
13472:
13466:
13462:
13458:
13453:
13449:
13444:
13437:
13433:
13427:
13422:
13419:
13416:
13412:
13405:
13402:
13397:
13393:
13387:
13382:
13379:
13376:
13372:
13364:
13360:
13354:
13349:
13346:
13343:
13339:
13332:
13324:
13320:
13312:
13311:
13310:
13291:
13287:
13274:
13257:
13254:
13251:
13248:
13245:
13219:
13216:
13213:
13210:
13207:
13181:
13178:
13175:
13172:
13169:
13166:
13163:
13160:
13157:
13154:
13151:
13139:
13119:
13116:
13111:
13107:
13101:
13096:
13093:
13090:
13086:
13078:
13073:
13067:
13063:
13059:
13054:
13050:
13045:
13038:
13034:
13028:
13023:
13020:
13017:
13006:
12998:
12994:
12986:
12985:
12984:
12982:
12972:
12970:
12966:
12962:
12958:
12954:
12950:
12945:
12943:
12925:
12921:
12898:
12881:
12869:
12853:
12850:
12845:
12841:
12835:
12830:
12827:
12824:
12820:
12787:
12783:
12777:
12772:
12769:
12766:
12762:
12754:
12749:
12743:
12739:
12735:
12730:
12726:
12721:
12714:
12710:
12704:
12699:
12696:
12693:
12682:
12680:
12673:
12656:
12643:
12637:
12632:
12628:
12625:
12620:
12616:
12611:
12604:
12599:
12596:
12593:
12582:
12580:
12570:
12560:
12545:
12544:
12543:
12527:
12517:
12505:
12487:
12470:
12459:
12455:
12450:
12434:
12430:
12421:
12417:
12411:
12396:
12376:
12372:
12365:
12360:
12357:
12354:
12350:
12343:
12339:
12333:
12329:
12325:
12321:
12311:
12309:
12291:
12285:
12279:
12275:
12271:
12261:
12255:
12232:
12227:
12223:
12202:
12199:
12192:
12188:
12183:
12179:
12172:
12167:
12164:
12161:
12157:
12153:
12150:
12146:
12142:
12119:
12112:
12108:
12103:
12099:
12092:
12087:
12084:
12081:
12077:
12071:
12066:
12062:
12058:
12053:
12043:
12037:
12029:
12028:
12027:
12011:
12006:
12002:
11998:
11993:
11988:
11984:
11974:
11972:
11949:
11943:
11916:
11911:
11907:
11900:
11896:
11891:
11887:
11880:
11875:
11872:
11869:
11865:
11861:
11856:
11846:
11840:
11832:
11831:
11830:
11814:
11809:
11805:
11790:
11788:
11787:Bootstrapping
11784:
11774:
11754:
11744:
11734:
11727:
11722:
11718:
11709:
11704:
11700:
11696:
11688:
11674:
11668:
11659:
11656:
11653:
11646:
11641:
11635:
11630:
11620:
11614:
11603:
11602:
11601:
11585:
11579:
11575:
11571:
11565:
11556:
11529:
11523:
11509:
11503:
11498:
11494:
11487:
11482:
11477:
11467:
11460:
11452:
11442:
11429:
11423:
11418:
11414:
11408:
11404:
11388:
11382:
11377:
11373:
11366:
11361:
11351:
11344:
11341:
11336:
11326:
11316:
11303:
11297:
11292:
11288:
11282:
11278:
11271:
11267:
11258:
11244:
11238:
11229:
11226:
11223:
11216:
11211:
11205:
11200:
11193:
11183:
11175:
11164:
11163:
11162:
11160:
11159:Taylor series
11156:
11155:bootstrapping
11146:
11142:
11140:
11133:
11130:
11117:
11112:
11102:
11092:
11085:
11080:
11076:
11067:
11062:
11058:
11052:
11047:
11044:
11041:
11037:
11028:
11018:
11014:
11008:
11003:
11000:
10997:
10993:
10985:
10980:
10975:
10970:
10962:
10958:
10951:
10941:
10934:
10929:
10925:
10913:
10909:
10905:
10902:
10895:
10888:
10883:
10880:
10877:
10873:
10865:
10855:
10848:
10843:
10837:
10828:
10818:
10808:
10784:
10781:
10773:
10769:
10765:
10762:
10739:
10736:
10728:
10724:
10720:
10715:
10711:
10704:
10701:
10696:
10693:
10685:
10682:
10679:
10676:
10664:
10651:
10647:
10639:
10635:
10628:
10618:
10611:
10606:
10602:
10591:
10587:
10580:
10570:
10563:
10558:
10554:
10545:
10542:
10524:
10518:
10513:
10510:
10507:
10503:
10497:
10492:
10489:
10486:
10482:
10474:
10464:
10457:
10452:
10446:
10437:
10427:
10417:
10410:
10404:
10391:
10382:
10370:
10356:
10353:
10348:
10344:
10334:
10318:
10314:
10308:
10304:
10300:
10296:
10292:
10288:
10265:
10261:
10255:
10251:
10247:
10243:
10239:
10235:
10226:
10222:
10206:
10203:
10195:
10191:
10187:
10182:
10178:
10171:
10168:
10163:
10160:
10152:
10149:
10146:
10143:
10132:
10107:
10101:
10092:
10077:
10065:
10050:
10037:
10031:
10022:
10007:
9995:
9989:
9986:
9980:
9967:
9958:
9945:
9939:
9933:
9920:
9911:
9903:
9895:
9885:
9878:
9873:
9869:
9861:
9857:
9850:
9846:
9836:
9830:
9825:
9821:
9810:
9806:
9799:
9795:
9785:
9779:
9774:
9770:
9761:
9758:
9740:
9734:
9729:
9726:
9723:
9719:
9713:
9708:
9705:
9702:
9698:
9690:
9680:
9673:
9668:
9662:
9649:
9640:
9628:
9625:
9611:
9603:
9599:
9594:
9590:
9586:
9580:
9577:
9570:
9566:
9561:
9557:
9553:
9546:
9540:
9535:
9532:
9529:
9525:
9519:
9516:
9511:
9508:
9505:
9498:
9494:
9490:
9484:
9480:
9474:
9469:
9466:
9463:
9459:
9452:
9448:
9444:
9438:
9434:
9428:
9423:
9420:
9417:
9413:
9406:
9396:
9386:
9379:
9370:
9359:
9338:
9328:
9321:
9312:
9296:
9294:
9276:
9272:
9247:
9243:
9239:
9234:
9229:
9225:
9200:
9196:
9192:
9187:
9182:
9178:
9157:
9152:
9148:
9144:
9139:
9135:
9108:
9098:
9088:
9081:
9076:
9072:
9063:
9058:
9054:
9048:
9043:
9040:
9037:
9033:
9024:
9014:
9010:
9004:
8999:
8996:
8993:
8989:
8981:
8976:
8970:
8961:
8951:
8941:
8930:
8929:
8928:
8926:
8921:
8920:Taylor series
8916:
8911:
8909:
8905:
8900:
8882:
8878:
8874:
8868:
8864:
8858:
8853:
8850:
8847:
8843:
8836:
8832:
8828:
8822:
8818:
8812:
8807:
8804:
8801:
8797:
8790:
8785:
8775:
8763:
8742:
8738:
8732:
8727:
8724:
8721:
8717:
8709:
8705:
8699:
8695:
8689:
8684:
8681:
8678:
8674:
8667:
8662:
8652:
8626:
8616:
8609:
8602:
8598:
8594:
8588:
8584:
8578:
8573:
8570:
8567:
8563:
8556:
8552:
8548:
8542:
8538:
8532:
8527:
8524:
8521:
8517:
8510:
8503:
8499:
8495:
8489:
8485:
8479:
8474:
8471:
8468:
8464:
8457:
8453:
8449:
8443:
8439:
8433:
8428:
8425:
8422:
8418:
8411:
8404:
8400:
8390:
8381:
8376:
8373:
8370:
8366:
8359:
8355:
8345:
8336:
8331:
8328:
8325:
8321:
8314:
8304:
8300:
8296:
8289:
8285:
8279:
8274:
8271:
8268:
8264:
8254:
8250:
8244:
8240:
8232:
8228:
8222:
8217:
8214:
8211:
8207:
8200:
8186:
8178:
8169:
8141:
8137:
8131:
8126:
8123:
8120:
8116:
8108:
8104:
8098:
8094:
8088:
8083:
8080:
8077:
8073:
8066:
8058:
8048:
8039:
8034:
8031:
8028:
8024:
8016:
8006:
7997:
7992:
7989:
7986:
7982:
7975:
7965:
7961:
7957:
7950:
7945:
7942:
7939:
7935:
7925:
7921:
7915:
7911:
7903:
7898:
7895:
7892:
7888:
7881:
7872:
7866:
7863:
7841:
7837:
7815:
7811:
7801:
7792:
7787:
7784:
7781:
7777:
7773:
7766:
7762:
7756:
7752:
7744:
7739:
7736:
7733:
7729:
7725:
7720:
7716:
7710:
7706:
7700:
7695:
7692:
7689:
7685:
7681:
7672:
7645:
7641:
7637:
7632:
7627:
7623:
7602:
7576:
7553:
7533:
7507:
7485:
7466:
7461:
7455:
7451:
7445:
7441:
7436:
7429:
7424:
7421:
7418:
7414:
7406:
7402:
7398:
7393:
7391:
7382:
7374:
7370:
7364:
7360:
7350:
7346:
7340:
7336:
7325:
7321:
7317:
7314:
7307:
7301:
7296:
7293:
7290:
7286:
7278:
7274:
7270:
7265:
7263:
7254:
7248:
7238:
7229:
7219:
7210:
7207:
7189:
7183:
7178:
7175:
7172:
7168:
7160:
7156:
7152:
7147:
7145:
7136:
7130:
7120:
7111:
7101:
7092:
7089:
7071:
7065:
7060:
7057:
7054:
7050:
7044:
7039:
7036:
7033:
7029:
7021:
7017:
7013:
7008:
7006:
6991:
6976:
6966:
6963:
6939:
6936:
6928:
6924:
6920:
6917:
6885:
6880:
6874:
6870:
6864:
6860:
6855:
6848:
6843:
6840:
6837:
6833:
6825:
6821:
6817:
6812:
6799:
6779:
6767:
6764:
6757:
6756:
6755:
6741:
6738:
6730:
6726:
6722:
6717:
6713:
6706:
6703:
6700:
6697:
6694:
6682:
6666:
6662:
6658:
6655:
6652:
6645:
6641:
6634:
6630:
6624:
6620:
6613:
6610:
6607:
6602:
6599:
6561:
6558:
6554:
6547:
6543:
6537:
6533:
6526:
6523:
6520:
6515:
6512:
6476:
6473:
6469:
6446:
6443:
6435:
6430:
6426:
6420:
6416:
6412:
6407:
6404:
6400:
6396:
6388:
6384:
6380:
6375:
6371:
6364:
6340:
6336:
6330:
6326:
6320:
6315:
6305:
6292:
6278:
6272:
6262:
6253:
6243:
6234:
6231:
6213:
6207:
6202:
6199:
6196:
6192:
6186:
6181:
6178:
6175:
6171:
6163:
6159:
6155:
6150:
6137:
6117:
6105:
6102:
6094:
6092:
6086:
6083:
6064:
6059:
6049:
6046:
6040:
6035:
6031:
6025:
6021:
6016:
6009:
6004:
6001:
5998:
5994:
5987:
5984:
5981:
5977:
5972:
5970:
5960:
5955:
5945:
5942:
5936:
5931:
5927:
5921:
5917:
5912:
5905:
5900:
5897:
5894:
5890:
5883:
5880:
5877:
5873:
5866:
5861:
5857:
5851:
5849:
5839:
5834:
5828:
5822:
5818:
5812:
5808:
5802:
5797:
5794:
5791:
5787:
5780:
5777:
5770:
5766:
5760:
5756:
5750:
5746:
5739:
5734:
5731:
5728:
5724:
5717:
5714:
5711:
5707:
5700:
5697:
5692:
5687:
5682:
5676:
5672:
5666:
5662:
5656:
5651:
5648:
5645:
5641:
5635:
5632:
5627:
5620:
5616:
5610:
5606:
5598:
5595:
5589:
5582:
5577:
5574:
5571:
5567:
5560:
5557:
5554:
5550:
5543:
5540:
5535:
5533:
5523:
5518:
5512:
5509:
5506:
5496:
5489:
5482:
5478:
5472:
5468:
5461:
5454:
5449:
5446:
5443:
5439:
5432:
5429:
5426:
5422:
5415:
5412:
5407:
5405:
5385:
5375:
5372:
5360:
5356:
5340:
5334:
5330:
5324:
5320:
5311:
5306:
5303:
5300:
5296:
5292:
5283:
5280:
5255:
5250:
5240:
5237:
5231:
5226:
5222:
5216:
5212:
5207:
5200:
5195:
5192:
5189:
5185:
5178:
5175:
5172:
5168:
5163:
5155:
5152:
5149:
5139:
5129:
5126:
5118:
5113:
5098:
5094:
5090:
5084:
5080:
5074:
5069:
5066:
5063:
5059:
5055:
5048:
5044:
5039:
5035:
5031:
5023:
5018:
5015:
5012:
5008:
5004:
4996:
4992:
4988:
4983:
4979:
4975:
4967:
4962:
4959:
4956:
4952:
4948:
4941:
4937:
4932:
4928:
4924:
4916:
4911:
4908:
4905:
4901:
4895:
4892:
4887:
4882:
4879:
4876:
4866:
4842:
4834:
4830:
4817:
4808:
4790:
4786:
4780:
4775:
4772:
4769:
4765:
4761:
4758:
4735:
4730:
4725:
4721:
4717:
4711:
4707:
4701:
4696:
4693:
4690:
4686:
4679:
4674:
4666:
4662:
4657:
4653:
4649:
4641:
4636:
4633:
4630:
4626:
4619:
4614:
4606:
4602:
4597:
4593:
4589:
4581:
4576:
4573:
4570:
4566:
4560:
4557:
4549:
4544:
4539:
4536:
4533:
4523:
4514:
4509:
4489:
4476:
4475:
4474:
4451:
4446:
4436:
4433:
4427:
4422:
4418:
4412:
4408:
4403:
4396:
4391:
4388:
4385:
4381:
4374:
4371:
4368:
4364:
4355:
4351:
4347:
4342:
4338:
4333:
4313:
4304:
4300:
4297:
4290:
4289:
4288:
4286:
4282:
4278:
4273:
4257:
4252:
4248:
4244:
4238:
4234:
4228:
4223:
4220:
4217:
4213:
4206:
4201:
4196:
4193:
4190:
4180:
4171:
4166:
4146:
4132:
4124:
4118:
4099:
4095:
4091:
4088:
4084:
4079:
4072:
4068:
4064:
4059:
4054:
4050:
4041:
4036:
4019:
4015:
4008:
4004:
3998:
3994:
3987:
3982:
3972:
3963:
3959:
3955:
3951:
3947:
3937:
3911:
3901:
3894:
3891:
3884:
3880:
3874:
3870:
3859:
3843:
3819:
3815:
3809:
3805:
3799:
3794:
3784:
3772:
3756:
3747:
3728:
3724:
3720:
3717:
3709:
3705:
3699:
3694:
3690:
3686:
3678:
3674:
3667:
3662:
3657:
3653:
3649:
3642:
3638:
3634:
3627:
3605:
3601:
3595:
3591:
3587:
3579:
3575:
3568:
3563:
3559:
3555:
3548:
3544:
3540:
3533:
3511:
3507:
3501:
3497:
3493:
3489:
3485:
3481:
3458:
3454:
3431:
3427:
3417:
3415:
3396:
3392:
3369:
3364:
3360:
3354:
3349:
3345:
3341:
3325:
3322:
3317:
3313:
3306:
3284:
3280:
3276:
3270:
3262:
3259:
3256:
3251:
3247:
3240:
3232:
3228:
3210:
3206:
3197:
3193:
3189:
3167:
3144:
3136:
3133:or sometimes
3132:
3128:
3124:
3120:
3115:
3113:
3109:
3105:
3087:
3083:
3074:
3064:
3062:
3057:
3042:
3036:
3031:
3027:
3021:
3016:
3013:
3010:
3006:
2999:
2989:
2985:
2960:
2954:
2950:
2944:
2939:
2936:
2933:
2929:
2922:
2913:
2887:
2884:
2881:
2874:
2860:
2854:
2849:
2845:
2836:
2831:
2828:
2825:
2821:
2814:
2809:
2804:
2794:
2763:
2753:
2740:
2736:
2727:
2722:
2712:
2705:
2697:
2687:
2677:
2674:
2667:
2666:
2665:
2663:
2659:
2655:
2651:
2647:
2643:
2640:
2636:
2621:
2608:
2605:
2602:
2590:
2581:
2561:
2558:
2553:
2549:
2528:
2522:
2518:
2513:
2509:
2505:
2498:
2493:
2490:
2487:
2483:
2479:
2467:
2458:
2438:
2432:
2428:
2423:
2415:
2411:
2404:
2395:
2375:
2353:
2351:
2347:
2328:
2321:
2316:
2312:
2306:
2302:
2294:
2289:
2286:
2283:
2279:
2273:
2263:
2257:
2253:
2244:
2234:
2233:
2232:
2215:
2208:
2203:
2197:
2194:
2189:
2185:
2179:
2174:
2171:
2168:
2164:
2159:
2150:
2145:
2141:
2135:
2132:
2127:
2123:
2116:
2111:
2108:
2105:
2101:
2094:
2088:
2083:
2079:
2072:
2068:
2063:
2059:
2052:
2047:
2044:
2041:
2037:
2033:
2028:
2018:
2012:
2004:
2003:
2002:
1986:
1982:
1978:
1973:
1969:
1948:
1944:
1938:
1933:
1929:
1925:
1920:
1910:
1904:
1880:
1871:
1867:
1861:
1856:
1853:
1850:
1846:
1841:
1835:
1826:
1823:
1818:
1814:
1808:
1803:
1800:
1797:
1793:
1788:
1782:
1772:
1766:
1758:
1757:
1756:
1754:
1735:
1727:
1723:
1717:
1712:
1709:
1706:
1702:
1695:
1689:
1685:
1681:
1676:
1672:
1667:
1661:
1656:
1653:
1650:
1646:
1639:
1628:
1623:
1619:
1615:
1607:
1602:
1599:
1596:
1592:
1585:
1577:
1572:
1568:
1562:
1558:
1551:
1545:
1540:
1537:
1534:
1530:
1523:
1514:
1504:
1503:
1502:
1485:
1478:
1473:
1469:
1465:
1460:
1455:
1451:
1443:
1442:
1441:
1425:
1420:
1416:
1408:
1404:
1386:
1382:
1371:
1365:
1336:
1332:
1327:
1323:
1317:
1312:
1309:
1306:
1296:
1293:
1283:
1277:
1269:
1268:
1267:
1265:
1242:
1236:
1227:
1209:
1205:
1196:
1191:
1174:
1170:
1163:
1158:
1155:
1152:
1142:
1139:
1130:
1129:ordinary mean
1104:
1100:
1093:
1088:
1085:
1082:
1071:
1067:
1061:
1057:
1053:
1049:
1041:
1040:
1039:
1018:
1014:
1009:
1005:
1001:
994:
989:
986:
983:
975:
966:
956:
955:
954:
940:
937:
932:
928:
924:
917:
912:
909:
906:
892:
874:
866:
862:
858:
855:
852:
847:
843:
839:
834:
830:
822:
818:
812:
808:
804:
801:
798:
793:
789:
783:
779:
775:
770:
766:
760:
756:
749:
740:
730:
729:
728:
711:
703:
699:
693:
688:
685:
682:
670:
666:
660:
656:
650:
645:
642:
639:
628:
619:
609:
608:
607:
592:
586:
582:
578:
575:
572:
567:
563:
559:
554:
550:
545:
537:
520:
514:
510:
506:
503:
500:
495:
491:
487:
482:
478:
473:
464:
440:
437:
431:
428:
425:
419:
413:
410:
407:
401:
392:
382:
381:
380:
363:
360:
354:
351:
348:
344:
335:
334:
319:
316:
310:
307:
304:
300:
291:
290:
289:
286:
284:
280:
270:
253:
250:
244:
241:
238:
230:
227:
224:
218:
212:
209:
206:
197:
188:
178:
177:
176:
173:
160:
157:
152:
149:
144:
135:
119:
118:
117:
112:
111:
110:
81:Basic example
73:
71:
67:
62:
60:
56:
52:
48:
41:
37:
33:
19:
18:Weighted mean
20290:
20285:David Terr.
20259:
20242:
20215:
20209:
20194:Inequalities
20193:
20188:
20169:
20163:
20147:
20114:
20110:
20082:. Retrieved
20078:
20066:
20054:. Retrieved
20050:
20041:
20014:
20010:
20000:
19971:
19967:
19961:
19945:
19926:
19840:
19479:
19474:
19429:
19245:
19123:
19109:
18786:
18711:approaching
18710:
18542:
18388:
18083:
18029:
17938:
17602:
17584:
17291:
17048:
16828:
16772:
16704:
16640:
16441:
16438:
15964:
15962:
15845:
15732:
15653:
15651:
15634:standardized
15631:
15439:
15435:
15433:
15425:
15267:
15145:is given by
15111:
15073:
15042:
15034:
15031:
15026:
15024:
14957:
14693:
14322:
13614:
13612:
13595:standardized
13592:
13494:
13275:
13140:
13137:
12980:
12978:
12968:
12964:
12956:
12952:
12946:
12867:
12811:
12503:
12453:
12451:
12413:
12312:
12134:
11975:
11970:
11934:
11796:
11780:
11771:
11545:
11152:
11143:
11135:
11131:
10665:
10371:
10335:
10224:
10220:
10066:
9629:
9626:
9360:
9301:
9292:
9126:
8912:
8907:
8901:
7496:
6906:
6683:
6293:
6095:
6088:
6084:
5361:
5357:
5116:
5114:
4832:
4810:
4750:
4472:
4285:pps sampling
4283:(such as in
4280:
4274:
4130:
4128:
4122:
4040:design based
4039:
4037:
3857:
3770:
3748:
3418:
3226:
3187:
3134:
3130:
3126:
3122:
3116:
3072:
3070:
3060:
3058:
2782:
2639:uncorrelated
2634:
2632:
2396:
2364:
2343:
2230:
1895:
1752:
1750:
1500:
1373:
1225:
1192:
1126:
1037:
893:
889:
726:
460:
378:
287:
278:
276:
268:
174:
123:
116:
84:
63:
46:
44:
20084:22 December
20056:22 December
19627:(squared),
19563:(squared),
17826:(of length
17778:equal to a
16948:commutative
12310:, squared.
11793:Other notes
4506:known
4330:known
4163:known
3073:model based
2650:expectation
1405:with known
20310:Categories
19856:References
19118:See also:
18597:is simply
18383:See also:
17593:See also:
15658:normalized
15638:normalized
13599:normalized
12408:See also:
3856:-expanded
3769:-expanded
2361:Expectancy
1368:See also:
20292:MathWorld
20251:300283069
19815:Weighting
19718:−
19697:¯
19688:−
19655:∑
19639:σ
19597:σ
19581:¯
19572:σ
19538:¯
19529:σ
19502:σ
19489:σ
19457:¯
19445:^
19442:σ
19399:σ
19381:¯
19372:−
19336:∑
19323:−
19298:ν
19294:χ
19259:ν
19255:χ
19224:ν
19220:χ
19207:¯
19198:σ
19182:¯
19170:^
19167:σ
19134:χ
19041:−
19029:−
19020:≤
18968:−
18960:−
18931:−
18900:−
18879:−
18864:−
18833:−
18822:−
18746:−
18687:−
18669:−
18651:−
18622:∑
18508:−
18442:Δ
18439:−
18404:Δ
18355:−
18311:∑
18056:−
17975:¯
17966:σ
17956:¯
17916:−
17871:¯
17862:σ
17798:…
17737:¯
17635:…
17443:−
17408:−
17383:−
17369:−
17346:−
17315:¯
17220:⊤
17101:⊤
17027:−
16989:∑
16972:¯
16881:∑
16868:¯
16845:¯
16807:−
16719:σ
16610:−
16595:∗
16591:μ
16587:−
16554:∗
16550:μ
16546:−
16498:∑
16383:−
16361:∗
16357:μ
16353:−
16320:∗
16316:μ
16312:−
16264:∑
16240:∗
16236:μ
16232:−
16199:∗
16195:μ
16191:−
16143:∑
16104:∑
16100:−
16059:∑
16020:∑
15906:∑
15896:∗
15892:μ
15864:∗
15860:μ
15846:Then the
15801:∑
15685:∑
15608:−
15586:∗
15582:μ
15578:−
15545:∗
15541:μ
15537:−
15489:∑
15388:∗
15384:μ
15380:−
15347:∗
15343:μ
15339:−
15291:∑
15220:∑
15175:∑
15162:∗
15158:μ
15130:∗
15126:μ
15112:Then the
15094:≥
14999:σ
14969:
14902:−
14875:∗
14871:μ
14867:−
14824:∑
14768:−
14745:^
14742:σ
14639:−
14530:−
14466:−
14406:∑
14345:∑
14293:σ
14255:−
14212:
14206:−
14194:
14159:−
14134:
14128:−
14116:
14072:∗
14068:μ
14064:−
14045:
14012:∑
13979:^
13976:σ
13966:
13945:σ
13928:−
13883:
13877:−
13865:
13849:−
13824:
13818:−
13806:
13771:μ
13768:−
13749:
13726:∑
13700:^
13697:σ
13687:
13646:σ
13625:μ
13504:∑
13467:∗
13463:μ
13459:−
13413:∑
13403:−
13373:∑
13340:∑
13117:−
13087:∑
13068:∗
13064:μ
13060:−
13014:∑
12922:σ
12885:^
12882:σ
12821:∑
12763:∑
12744:∗
12740:μ
12736:−
12690:∑
12660:^
12657:σ
12629:μ
12626:−
12590:∑
12564:^
12561:σ
12521:^
12518:σ
12474:^
12471:σ
12456:weighted
12435:∗
12431:μ
12351:∑
12276:σ
12265:¯
12256:σ
12224:σ
12200:≤
12158:∑
12154:≤
12078:∑
12063:σ
12047:¯
12038:σ
12003:σ
11985:σ
11953:¯
11944:σ
11908:σ
11866:∑
11850:¯
11841:σ
11806:σ
11783:Jackknife
11738:¯
11728:−
11697:∑
11678:¯
11657:−
11636:^
11624:¯
11615:σ
11572:∑
11560:¯
11513:¯
11504:−
11488:∑
11471:¯
11446:¯
11433:¯
11424:−
11392:¯
11383:−
11367:∑
11355:¯
11342:−
11320:¯
11307:¯
11298:−
11272:∑
11248:¯
11227:−
11206:^
11187:¯
11176:σ
11096:¯
11086:−
11038:∑
10994:∑
10959:π
10945:¯
10935:−
10910:π
10906:−
10874:∑
10859:^
10838:^
10822:¯
10782:≈
10770:π
10766:−
10690:Δ
10680:≠
10674:∀
10636:π
10622:¯
10612:−
10588:π
10574:¯
10564:−
10536:ˇ
10533:Δ
10504:∑
10483:∑
10468:^
10447:^
10431:¯
10405:^
10395:^
10157:Δ
10147:≠
10141:∀
10111:^
10096:^
10081:^
10067:The term
10041:^
10026:^
10011:^
9999:^
9987:−
9981:^
9971:^
9949:^
9934:^
9924:^
9889:^
9858:π
9840:^
9831:−
9807:π
9789:^
9780:−
9752:ˇ
9749:Δ
9720:∑
9699:∑
9684:^
9663:^
9653:^
9600:π
9578:−
9567:π
9526:∑
9506:≈
9460:∑
9414:∑
9400:^
9390:^
9374:^
9342:^
9332:^
9316:^
9197:π
9145:≈
9136:π
9092:¯
9082:−
9034:∑
8990:∑
8971:^
8955:¯
8844:∑
8798:∑
8779:¯
8718:∑
8675:∑
8656:¯
8620:¯
8564:∑
8518:∑
8465:∑
8419:∑
8394:ˇ
8367:∑
8349:ˇ
8322:∑
8301:π
8265:∑
8251:π
8208:∑
8195:^
8190:¯
8173:^
8117:∑
8074:∑
8052:ˇ
8025:∑
8010:ˇ
7983:∑
7962:π
7936:∑
7922:π
7889:∑
7876:¯
7805:ˇ
7778:∑
7763:π
7730:∑
7686:∑
7676:^
7642:π
7580:^
7511:^
7415:∑
7371:π
7347:π
7322:π
7318:−
7287:∑
7242:ˇ
7223:ˇ
7201:ˇ
7198:Δ
7169:∑
7124:ˇ
7105:ˇ
7083:ˇ
7080:Δ
7051:∑
7030:∑
6980:^
6967:
6952:and that
6937:≈
6925:π
6921:−
6834:∑
6788:^
6783:¯
6768:
6698:≠
6692:∀
6663:π
6659:−
6642:π
6631:π
6621:π
6614:−
6593:ˇ
6590:Δ
6555:π
6544:π
6534:π
6527:−
6506:ˇ
6503:Δ
6470:π
6440:Δ
6427:π
6417:π
6413:−
6401:π
6337:π
6309:ˇ
6266:ˇ
6247:ˇ
6225:ˇ
6222:Δ
6193:∑
6172:∑
6126:^
6121:¯
6106:
6054:¯
6041:−
5995:∑
5985:−
5950:¯
5937:−
5891:∑
5881:−
5788:∑
5778:−
5767:π
5725:∑
5715:−
5642:∑
5628:−
5568:∑
5558:−
5500:^
5490:−
5440:∑
5430:−
5389:^
5376:
5297:∑
5288:¯
5245:¯
5232:−
5186:∑
5176:−
5143:^
5130:
5060:∑
5045:π
5009:∑
5005:≈
4953:∑
4902:∑
4870:^
4818:π
4766:∑
4687:∑
4663:π
4627:∑
4620:≈
4567:∑
4527:^
4498:^
4493:¯
4441:¯
4428:−
4382:∑
4372:−
4322:^
4317:¯
4301:
4214:∑
4207:≈
4184:^
4155:^
4150:¯
4092:×
4080:≈
4069:π
4016:π
3976:ˇ
3941:ˇ
3905:ˇ
3816:π
3788:ˇ
3757:π
3725:π
3721:−
3706:π
3602:π
3416:design).
3361:π
3355:≈
3281:π
3263:∣
3196:Bernoulli
3171:^
3007:∑
2995:¯
2930:∑
2917:¯
2885:−
2864:¯
2855:−
2822:∑
2798:^
2795:σ
2757:¯
2746:¯
2716:^
2713:σ
2691:¯
2678:
2606:μ
2594:¯
2562:μ
2550:μ
2519:μ
2484:∑
2471:¯
2429:μ
2379:¯
2313:σ
2280:∑
2267:¯
2258:σ
2248:¯
2195:−
2186:σ
2165:∑
2142:σ
2133:−
2124:σ
2102:∑
2080:σ
2038:∑
2022:¯
2013:σ
1983:σ
1970:σ
1961:when all
1930:σ
1914:¯
1905:σ
1847:∑
1824:−
1815:σ
1794:∑
1776:¯
1767:σ
1703:∑
1682:⋅
1647:∑
1620:σ
1593:∑
1569:σ
1531:∑
1518:¯
1470:σ
1417:σ
1303:∑
1297:σ
1287:¯
1278:σ
1246:¯
1237:σ
1206:σ
1149:∑
1079:∑
980:∑
970:¯
903:∑
856:⋯
802:⋯
744:¯
679:∑
636:∑
623:¯
576:…
504:…
429:×
411:×
396:¯
228:×
210:×
192:¯
139:¯
20155:, 2011.
20139:37828617
19994:pdf link
19743:See also
18191:at time
16762:and the
15965:unbiased
15963:and the
15780:′
15440:unbiased
12416:variance
12330:′
12189:′
12109:′
11897:′
10297:′
10244:′
10227:between
9595:′
9562:′
9499:′
9453:′
8883:′
8837:′
8762:estimand
8603:′
8557:′
8504:′
8458:′
8405:′
8360:′
7816:′
6357:. Also,
5099:′
5040:′
4984:′
4933:′
4726:′
4658:′
4598:′
4253:′
4038:In this
3952:′
3860:values:
3643:′
3549:′
3490:′
3102:are not
2646:variance
2624:Variance
2514:′
2069:′
1751:and the
1407:variance
1333:′
1058:′
1010:′
933:′
465:of data
279:relative
99:students
76:Examples
20227:, 1980.
20204:, 1988.
20131:5073694
20051:Gnu.org
20019:Bibcode
19976:Bibcode
19750:Average
19278:is the
18124:, with
18030:where:
17846:). The
17692:is the
16946:is not
16766:by the
16737:by the
15850:vector
15116:vector
14624:is the
7662:we get
4275:If the
3071:From a
1266:to be:
536:weights
105:
101:
93:
89:
87:classes
55:average
20266:
20249:
20223:
20200:
20176:
20137:
20129:
19933:
19879:
19246:where
18543:where
17558:0.9901
17551:0.9901
17503:0.9901
17486:0.9901
15438:, the
15003:actual
14958:where
14726:
14563:
14323:where
14297:actual
13949:actual
13650:actual
13330:
13004:
12812:where
12576:
12504:biased
12454:biased
12135:where
11546:where
6461:where
5270:where
2975:, and
1224:, the
20316:Means
20135:S2CID
20107:(PDF)
20075:(PDF)
19832:Notes
17774:is a
13543:(not
9299:Proof
8915:ratio
6904:Proof
6294:With
4470:Proof
3104:i.i.d
2783:With
2662:proof
2660:(see
2654:i.i.d
463:tuple
38:, or
20264:ISBN
20247:OCLC
20221:ISBN
20198:ISBN
20174:ISBN
20127:PMID
20086:2017
20058:2017
19931:ISBN
19877:ISBN
19760:Mean
18980:0.61
18891:0.39
18407:<
18401:<
17939:and
17597:and
15989:is:
15636:nor
14384:and
13597:nor
12940:for
12452:The
12418:and
11785:and
10280:and
2648:and
1755:is:
1127:The
150:4300
45:The
20119:doi
20027:doi
19984:doi
19012:is
17252:100
17150:100
12942:iid
9217:or
9127:If
6964:Var
6765:Var
6103:Var
5396:pwr
5373:Var
5127:Var
5117:pwr
4833:pwr
4298:Var
4123:pwr
3117:In
2675:Var
2664:):
606:is
441:86.
426:0.6
408:0.4
364:0.6
320:0.4
254:86.
161:86.
107:and
95:one
20312::
20289:.
20219:,
20133:.
20125:.
20115:35
20113:.
20109:.
20094:^
20077:.
20049:.
20025:.
20015:27
20013:.
20009:.
19992:-
19982:.
19972:29
19970:.
19929:.
19887:^
19864:^
19477:.
19282::
18784:.
18286:.
17723:,
17670:,
17241::=
17191::=
17122::=
17072::=
15760::
15718:1.
15660::
15109:.
15022:.
12971:.
12542::
12395:.
11973:.
11789:.
10064:.
8910:.
8899:.
6681:.
5355:.
5112:.
4272:.
4117:.
3746:.
2903:,
1228:,
432:90
414:80
355:30
349:20
345:30
311:30
305:20
301:20
285:.
245:30
239:20
231:90
225:30
213:80
207:20
153:50
72:.
34:,
20295:.
20272:.
20253:.
20182:.
20141:.
20121::
20088:.
20060:.
20035:.
20029::
20021::
19990:.
19986::
19978::
19939:.
19848:.
19727:.
19721:1
19715:n
19708:2
19704:)
19694:x
19683:i
19679:x
19675:(
19670:n
19665:1
19662:=
19659:i
19648:=
19643:2
19611:n
19607:/
19601:2
19593:=
19588:2
19578:x
19545:2
19535:x
19506:0
19498:=
19493:i
19454:x
19415:;
19408:2
19403:i
19392:2
19388:)
19378:x
19367:i
19363:x
19359:(
19351:n
19346:1
19343:=
19340:i
19329:)
19326:1
19320:n
19317:(
19313:1
19308:=
19303:2
19264:2
19229:2
19214:2
19204:x
19194:=
19189:2
19179:x
19138:2
19090:w
19070:n
19047:)
19044:w
19038:1
19035:(
19032:n
19025:e
19000:n
18977:=
18971:1
18964:e
18957:1
18934:1
18927:e
18906:)
18903:w
18897:1
18894:(
18888:=
18885:)
18882:w
18876:1
18873:(
18867:1
18860:e
18836:1
18829:)
18825:w
18819:1
18816:(
18795:w
18772:m
18752:)
18749:w
18743:1
18740:(
18736:/
18732:1
18729:=
18724:1
18720:V
18696:,
18690:w
18684:1
18677:m
18673:w
18666:1
18660:=
18654:1
18648:i
18644:w
18637:m
18632:1
18629:=
18626:i
18618:=
18613:1
18609:V
18583:1
18579:V
18556:1
18552:V
18528:,
18521:1
18517:V
18511:1
18505:i
18501:w
18495:=
18490:i
18486:w
18462:m
18436:1
18433:=
18430:w
18410:1
18398:0
18363:.
18358:i
18352:1
18349:+
18346:k
18342:x
18336:i
18332:w
18326:m
18321:1
18318:=
18315:i
18307:=
18302:k
18298:z
18274:m
18254:z
18231:i
18227:x
18204:i
18200:t
18179:y
18157:i
18153:t
18132:n
18112:y
18092:x
18064:.
18059:1
18051:C
18046:=
18042:W
18015:,
18012:)
18008:X
18003:W
17997:T
17992:J
17987:(
17982:2
17972:x
17962:=
17953:x
17924:,
17919:1
17912:)
17907:J
17902:W
17896:T
17891:J
17886:(
17883:=
17878:2
17868:x
17834:n
17812:T
17808:]
17804:1
17801:,
17795:,
17792:1
17789:[
17761:J
17734:x
17709:i
17705:x
17679:C
17656:T
17652:]
17646:n
17642:x
17638:,
17632:,
17627:1
17623:x
17619:[
17616:=
17612:X
17564:]
17545:[
17540:=
17535:]
17529:1
17522:1
17516:[
17509:]
17498:0
17491:0
17480:[
17475:=
17464:)
17458:2
17453:x
17446:1
17438:2
17433:C
17428:+
17423:1
17418:x
17411:1
17403:1
17398:C
17392:(
17386:1
17378:)
17372:1
17364:2
17359:C
17354:+
17349:1
17341:1
17336:C
17330:(
17325:=
17311:x
17275:]
17269:1
17264:0
17257:0
17246:[
17236:2
17231:C
17225:,
17214:]
17208:1
17203:0
17197:[
17186:2
17181:x
17156:]
17145:0
17138:0
17133:1
17127:[
17117:1
17112:C
17106:,
17095:]
17089:0
17084:1
17078:[
17067:1
17062:x
17035:,
17030:1
17022:)
17016:i
17011:W
17004:n
16999:1
16996:=
16993:i
16984:(
16979:=
16968:x
16960:C
16930:,
16926:)
16920:i
16915:x
16908:i
16903:W
16896:n
16891:1
16888:=
16885:i
16876:(
16864:x
16856:C
16851:=
16841:x
16815:.
16810:1
16802:i
16797:C
16792:=
16787:i
16782:W
16749:C
16723:2
16685:N
16681:/
16677:1
16674:=
16669:1
16665:V
16660:/
16654:i
16650:w
16626:.
16618:2
16614:V
16607:1
16601:)
16582:i
16577:x
16571:(
16565:T
16560:)
16541:i
16536:x
16530:(
16523:i
16519:w
16513:N
16508:1
16505:=
16502:i
16491:=
16487:C
16463:1
16460:=
16455:1
16451:V
16420:.
16414:)
16409:1
16405:V
16400:/
16394:2
16390:V
16386:(
16378:1
16374:V
16367:)
16348:i
16343:x
16337:(
16331:T
16326:)
16307:i
16302:x
16296:(
16289:i
16285:w
16279:N
16274:1
16271:=
16268:i
16257:=
16246:)
16227:i
16222:x
16216:(
16210:T
16205:)
16186:i
16181:x
16175:(
16168:i
16164:w
16158:N
16153:1
16150:=
16147:i
16134:2
16129:i
16125:w
16119:N
16114:1
16111:=
16108:i
16095:2
16090:)
16084:i
16080:w
16074:N
16069:1
16066:=
16063:i
16054:(
16045:i
16041:w
16035:N
16030:1
16027:=
16024:i
16013:=
16005:C
15976:C
15948:.
15943:i
15938:x
15931:i
15927:w
15921:N
15916:1
15913:=
15910:i
15902:=
15826:i
15822:w
15816:N
15811:1
15808:=
15805:i
15794:i
15790:w
15784:=
15776:i
15772:w
15746:1
15742:V
15715:=
15710:i
15706:w
15700:N
15695:1
15692:=
15689:i
15681:=
15676:1
15672:V
15617:.
15611:1
15603:1
15599:V
15592:)
15573:i
15568:x
15562:(
15556:T
15551:)
15532:i
15527:x
15521:(
15514:i
15510:w
15504:N
15499:1
15496:=
15493:i
15482:=
15478:C
15452:C
15411:.
15404:1
15400:V
15394:)
15375:i
15370:x
15364:(
15358:T
15353:)
15334:i
15329:x
15323:(
15316:i
15312:w
15306:N
15301:1
15298:=
15295:i
15284:=
15280:C
15253:.
15245:i
15241:w
15235:N
15230:1
15227:=
15224:i
15212:i
15207:x
15200:i
15196:w
15190:N
15185:1
15182:=
15179:i
15168:=
15097:0
15089:i
15085:w
15074:K
15058:i
15053:x
15027:N
15008:2
14995:=
14992:]
14987:2
14981:w
14976:s
14972:[
14966:E
14939:,
14933:)
14928:1
14924:V
14919:/
14913:2
14909:V
14905:(
14897:1
14893:V
14885:2
14881:)
14862:i
14858:x
14854:(
14849:i
14845:w
14839:N
14834:1
14831:=
14828:i
14817:=
14804:)
14799:2
14794:1
14790:V
14785:/
14779:2
14775:V
14771:(
14765:1
14759:2
14753:w
14733:=
14721:2
14715:w
14710:s
14678:)
14672:2
14667:1
14663:V
14658:/
14652:2
14648:V
14643:(
14636:1
14610:f
14607:f
14604:e
14600:N
14596:=
14591:2
14587:V
14582:/
14576:2
14571:1
14567:V
14542:)
14537:N
14533:1
14527:N
14521:(
14499:)
14491:2
14486:1
14482:V
14476:2
14472:V
14463:1
14459:(
14436:2
14431:i
14427:w
14421:N
14416:1
14413:=
14410:i
14402:=
14397:2
14393:V
14370:i
14366:w
14360:N
14355:1
14352:=
14349:i
14341:=
14336:1
14332:V
14302:2
14288:)
14280:2
14275:1
14271:V
14265:2
14261:V
14252:1
14248:(
14244:=
14234:]
14229:2
14225:)
14221:]
14218:X
14215:[
14209:E
14203:X
14200:(
14197:[
14191:E
14184:2
14179:1
14175:V
14169:2
14165:V
14156:]
14151:2
14147:)
14143:]
14140:X
14137:[
14131:E
14125:X
14122:(
14119:[
14113:E
14110:=
14096:1
14092:V
14087:]
14082:2
14078:)
14059:i
14055:x
14051:(
14048:[
14042:E
14037:i
14033:w
14027:N
14022:1
14019:=
14016:i
14005:=
13998:]
13993:2
13987:w
13969:[
13963:E
13954:2
13940:)
13935:N
13931:1
13925:N
13919:(
13915:=
13905:]
13900:2
13896:)
13892:]
13889:X
13886:[
13880:E
13874:X
13871:(
13868:[
13862:E
13857:N
13854:1
13846:]
13841:2
13837:)
13833:]
13830:X
13827:[
13821:E
13815:X
13812:(
13809:[
13803:E
13800:=
13788:N
13784:]
13779:2
13775:)
13763:i
13759:x
13755:(
13752:[
13746:E
13741:N
13736:1
13733:=
13730:i
13719:=
13712:]
13707:2
13690:[
13684:E
13655:2
13576:i
13572:w
13551:N
13529:i
13525:w
13519:N
13514:1
13511:=
13508:i
13478:2
13473:)
13454:i
13450:x
13445:(
13438:i
13434:w
13428:N
13423:1
13420:=
13417:i
13406:1
13398:i
13394:w
13388:N
13383:1
13380:=
13377:i
13365:i
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13355:N
13350:1
13347:=
13344:i
13333:=
13325:2
13321:s
13297:}
13292:i
13288:w
13284:{
13261:}
13258:3
13255:,
13252:1
13249:,
13246:2
13243:{
13223:}
13220:5
13217:,
13214:4
13211:,
13208:2
13205:{
13185:}
13182:5
13179:,
13176:5
13173:,
13170:5
13167:,
13164:4
13161:,
13158:2
13155:,
13152:2
13149:{
13120:1
13112:i
13108:w
13102:N
13097:1
13094:=
13091:i
13079:2
13074:)
13055:i
13051:x
13046:(
13039:i
13035:w
13029:N
13024:1
13021:=
13018:i
13007:=
12999:2
12995:s
12957:N
12953:N
12926:2
12899:2
12893:w
12854:1
12851:=
12846:i
12842:w
12836:N
12831:1
12828:=
12825:i
12788:i
12784:w
12778:N
12773:1
12770:=
12767:i
12755:2
12750:)
12731:i
12727:x
12722:(
12715:i
12711:w
12705:N
12700:1
12697:=
12694:i
12683:=
12674:2
12668:w
12644:N
12638:2
12633:)
12621:i
12617:x
12612:(
12605:N
12600:1
12597:=
12594:i
12583:=
12571:2
12528:2
12488:2
12482:w
12377:i
12373:w
12366:n
12361:1
12358:=
12355:i
12344:i
12340:w
12334:=
12326:i
12322:w
12292:n
12286:/
12280:0
12272:=
12262:x
12233:2
12228:0
12203:1
12193:2
12184:i
12180:w
12173:n
12168:1
12165:=
12162:i
12151:n
12147:/
12143:1
12120:,
12113:2
12104:i
12100:w
12093:n
12088:1
12085:=
12082:i
12072:2
12067:0
12059:=
12054:2
12044:x
12012:2
12007:0
11999:=
11994:2
11989:i
11950:x
11917:2
11912:i
11901:2
11892:i
11888:w
11881:n
11876:1
11873:=
11870:i
11862:=
11857:2
11847:x
11815:2
11810:i
11755:2
11751:)
11745:w
11735:x
11723:i
11719:x
11715:(
11710:2
11705:i
11701:w
11689:2
11685:)
11675:w
11669:n
11666:(
11663:)
11660:1
11654:n
11651:(
11647:n
11642:=
11631:2
11621:x
11586:n
11580:i
11576:w
11566:=
11557:w
11530:]
11524:2
11520:)
11510:w
11499:i
11495:w
11491:(
11483:2
11478:w
11468:x
11461:+
11458:)
11453:w
11443:x
11430:w
11419:i
11415:x
11409:i
11405:w
11401:(
11398:)
11389:w
11378:i
11374:w
11370:(
11362:w
11352:x
11345:2
11337:2
11333:)
11327:w
11317:x
11304:w
11293:i
11289:x
11283:i
11279:w
11275:(
11268:[
11259:2
11255:)
11245:w
11239:n
11236:(
11233:)
11230:1
11224:n
11221:(
11217:n
11212:=
11201:2
11194:w
11184:x
11118:.
11113:2
11109:)
11103:w
11093:y
11081:i
11077:y
11073:(
11068:2
11063:i
11059:w
11053:n
11048:1
11045:=
11042:i
11029:2
11025:)
11019:i
11015:w
11009:n
11004:1
11001:=
10998:i
10990:(
10986:1
10981:=
10976:2
10971:)
10963:i
10952:w
10942:y
10930:i
10926:y
10919:)
10914:i
10903:1
10900:(
10896:(
10889:n
10884:1
10881:=
10878:i
10866:2
10856:N
10849:1
10844:=
10834:)
10829:w
10819:y
10812:(
10809:V
10785:1
10779:)
10774:i
10763:1
10760:(
10740:0
10737:=
10734:)
10729:j
10725:I
10721:,
10716:i
10712:I
10708:(
10705:C
10702:=
10697:j
10694:i
10686::
10683:j
10677:i
10652:.
10648:)
10640:j
10629:w
10619:y
10607:j
10603:y
10592:i
10581:w
10571:y
10559:i
10555:y
10546:j
10543:i
10525:(
10519:n
10514:1
10511:=
10508:j
10498:n
10493:1
10490:=
10487:i
10475:2
10465:N
10458:1
10453:=
10443:)
10438:w
10428:y
10421:(
10418:V
10411:=
10401:)
10392:R
10386:(
10383:V
10357:1
10354:=
10349:i
10345:z
10319:i
10315:z
10309:i
10305:I
10301:=
10293:i
10289:z
10266:i
10262:y
10256:i
10252:I
10248:=
10240:i
10236:y
10225:i
10221:n
10207:0
10204:=
10201:)
10196:j
10192:I
10188:,
10183:i
10179:I
10175:(
10172:C
10169:=
10164:j
10161:i
10153::
10150:j
10144:i
10117:)
10108:Z
10102:,
10093:Y
10087:(
10078:C
10051:]
10047:)
10038:Z
10032:,
10023:Y
10017:(
10008:C
9996:R
9990:2
9977:)
9968:Z
9962:(
9959:V
9946:R
9940:+
9930:)
9921:Y
9915:(
9912:V
9904:[
9896:2
9886:Z
9879:1
9874:=
9870:)
9862:j
9851:j
9847:z
9837:R
9826:j
9822:y
9811:i
9800:i
9796:z
9786:R
9775:i
9771:y
9762:j
9759:i
9741:(
9735:n
9730:1
9727:=
9724:j
9714:n
9709:1
9706:=
9703:i
9691:2
9681:Z
9674:1
9669:=
9659:)
9650:R
9644:(
9641:V
9612:)
9604:i
9591:i
9587:z
9581:R
9571:i
9558:i
9554:y
9547:(
9541:n
9536:1
9533:=
9530:i
9520:Z
9517:1
9512:+
9509:R
9495:i
9491:z
9485:i
9481:w
9475:n
9470:1
9467:=
9464:i
9449:i
9445:y
9439:i
9435:w
9429:n
9424:1
9421:=
9418:i
9407:=
9397:Z
9387:Y
9380:=
9371:R
9339:Z
9329:Y
9322:=
9313:R
9293:N
9277:i
9273:w
9248:i
9244:p
9240:1
9235:=
9230:i
9226:w
9201:i
9193:1
9188:=
9183:i
9179:w
9158:n
9153:i
9149:p
9140:i
9123:.
9109:2
9105:)
9099:w
9089:y
9077:i
9073:y
9069:(
9064:2
9059:i
9055:w
9049:n
9044:1
9041:=
9038:i
9025:2
9021:)
9015:i
9011:w
9005:n
9000:1
8997:=
8994:i
8986:(
8982:1
8977:=
8967:)
8962:w
8952:y
8945:(
8942:V
8908:R
8879:i
8875:1
8869:i
8865:w
8859:n
8854:1
8851:=
8848:i
8833:i
8829:y
8823:i
8819:w
8813:n
8808:1
8805:=
8802:i
8791:=
8786:w
8776:y
8743:i
8739:w
8733:n
8728:1
8725:=
8722:i
8710:i
8706:y
8700:i
8696:w
8690:n
8685:1
8682:=
8679:i
8668:=
8663:w
8653:y
8627:w
8617:y
8610:=
8599:i
8595:1
8589:i
8585:w
8579:n
8574:1
8571:=
8568:i
8553:i
8549:y
8543:i
8539:w
8533:n
8528:1
8525:=
8522:i
8511:=
8500:i
8496:1
8490:i
8486:w
8480:N
8475:1
8472:=
8469:i
8454:i
8450:y
8444:i
8440:w
8434:N
8429:1
8426:=
8423:i
8412:=
8401:i
8391:1
8382:N
8377:1
8374:=
8371:i
8356:i
8346:y
8337:N
8332:1
8329:=
8326:i
8315:=
8305:i
8297:1
8290:i
8286:I
8280:N
8275:1
8272:=
8269:i
8255:i
8245:i
8241:y
8233:i
8229:I
8223:N
8218:1
8215:=
8212:i
8201:=
8187:Y
8179:=
8170:R
8142:i
8138:w
8132:N
8127:1
8124:=
8121:i
8109:i
8105:y
8099:i
8095:w
8089:N
8084:1
8081:=
8078:i
8067:=
8059:i
8049:1
8040:N
8035:1
8032:=
8029:i
8017:i
8007:y
7998:N
7993:1
7990:=
7987:i
7976:=
7966:i
7958:1
7951:N
7946:1
7943:=
7940:i
7926:i
7916:i
7912:y
7904:N
7899:1
7896:=
7893:i
7882:=
7873:Y
7867:=
7864:R
7842:i
7838:y
7812:i
7802:1
7793:n
7788:1
7785:=
7782:i
7774:=
7767:i
7757:i
7753:I
7745:n
7740:1
7737:=
7734:i
7726:=
7721:i
7717:I
7711:i
7707:w
7701:n
7696:1
7693:=
7690:i
7682:=
7673:N
7646:i
7638:1
7633:=
7628:i
7624:w
7603:N
7577:N
7554:N
7534:N
7508:Y
7491:π
7467:2
7462:)
7456:i
7452:y
7446:i
7442:w
7437:(
7430:n
7425:1
7422:=
7419:i
7407:2
7403:N
7399:1
7394:=
7383:)
7375:i
7365:i
7361:y
7351:i
7341:i
7337:y
7331:)
7326:i
7315:1
7312:(
7308:(
7302:n
7297:1
7294:=
7291:i
7279:2
7275:N
7271:1
7266:=
7255:)
7249:i
7239:y
7230:i
7220:y
7211:i
7208:i
7190:(
7184:n
7179:1
7176:=
7173:i
7161:2
7157:N
7153:1
7148:=
7137:)
7131:j
7121:y
7112:i
7102:y
7093:j
7090:i
7072:(
7066:n
7061:1
7058:=
7055:j
7045:n
7040:1
7037:=
7034:i
7022:2
7018:N
7014:1
7009:=
7002:)
6996:)
6992:N
6977:Y
6970:(
6940:1
6934:)
6929:i
6918:1
6915:(
6886:2
6881:)
6875:i
6871:y
6865:i
6861:w
6856:(
6849:n
6844:1
6841:=
6838:i
6826:2
6822:N
6818:1
6813:=
6810:)
6804:)
6800:N
6780:Y
6771:(
6742:0
6739:=
6736:)
6731:j
6727:I
6723:,
6718:i
6714:I
6710:(
6707:C
6704::
6701:j
6695:i
6667:i
6656:1
6653:=
6646:i
6635:i
6625:i
6611:1
6608:=
6603:i
6600:i
6562:j
6559:i
6548:j
6538:i
6524:1
6521:=
6516:j
6513:i
6477:j
6474:i
6447:j
6444:i
6436:=
6431:j
6421:i
6408:j
6405:i
6397:=
6394:)
6389:j
6385:I
6381:,
6376:i
6372:I
6368:(
6365:C
6341:i
6331:i
6327:y
6321:=
6316:i
6306:y
6279:)
6273:j
6263:y
6254:i
6244:y
6235:j
6232:i
6214:(
6208:n
6203:1
6200:=
6197:j
6187:n
6182:1
6179:=
6176:i
6164:2
6160:N
6156:1
6151:=
6148:)
6142:)
6138:N
6118:Y
6109:(
6065:2
6060:)
6050:y
6047:w
6036:i
6032:y
6026:i
6022:w
6017:(
6010:n
6005:1
6002:=
5999:i
5988:1
5982:n
5978:n
5973:=
5961:2
5956:)
5946:y
5943:w
5932:i
5928:y
5922:i
5918:w
5913:(
5906:n
5901:1
5898:=
5895:i
5884:1
5878:n
5874:1
5867:n
5862:2
5858:n
5852:=
5840:2
5835:)
5829:n
5823:i
5819:y
5813:i
5809:w
5803:n
5798:1
5795:=
5792:i
5781:n
5771:i
5761:i
5757:y
5751:n
5747:(
5740:n
5735:1
5732:=
5729:i
5718:1
5712:n
5708:1
5701:n
5698:1
5693:=
5688:2
5683:)
5677:i
5673:y
5667:i
5663:w
5657:n
5652:1
5649:=
5646:i
5636:n
5633:n
5621:i
5617:p
5611:i
5607:y
5599:n
5596:n
5590:(
5583:n
5578:1
5575:=
5572:i
5561:1
5555:n
5551:1
5544:n
5541:1
5536:=
5524:2
5519:)
5513:r
5510:w
5507:p
5497:Y
5483:i
5479:p
5473:i
5469:y
5462:(
5455:n
5450:1
5447:=
5444:i
5433:1
5427:n
5423:1
5416:n
5413:1
5408:=
5401:)
5386:Y
5379:(
5341:n
5335:i
5331:y
5325:i
5321:w
5312:n
5307:1
5304:=
5301:i
5293:=
5284:y
5281:w
5256:2
5251:)
5241:y
5238:w
5227:i
5223:y
5217:i
5213:w
5208:(
5201:n
5196:1
5193:=
5190:i
5179:1
5173:n
5169:n
5164:=
5161:)
5156:r
5153:w
5150:p
5140:Y
5133:(
5095:i
5091:y
5085:i
5081:w
5075:n
5070:1
5067:=
5064:i
5056:=
5049:i
5036:i
5032:y
5024:n
5019:1
5016:=
5013:i
4997:i
4993:p
4989:n
4980:i
4976:y
4968:n
4963:1
4960:=
4957:i
4949:=
4942:i
4938:p
4929:i
4925:y
4917:n
4912:1
4909:=
4906:i
4896:n
4893:1
4888:=
4883:r
4880:w
4877:p
4867:Y
4843:p
4791:i
4787:y
4781:N
4776:1
4773:=
4770:i
4762:=
4759:Y
4736:.
4731:N
4722:i
4718:y
4712:i
4708:w
4702:n
4697:1
4694:=
4691:i
4680:=
4675:N
4667:i
4654:i
4650:y
4642:n
4637:1
4634:=
4631:i
4615:N
4607:i
4603:p
4594:i
4590:y
4582:n
4577:1
4574:=
4571:i
4561:n
4558:1
4550:=
4545:N
4540:r
4537:w
4534:p
4524:Y
4515:=
4510:N
4490:Y
4452:2
4447:)
4437:y
4434:w
4423:i
4419:y
4413:i
4409:w
4404:(
4397:n
4392:1
4389:=
4386:i
4375:1
4369:n
4365:n
4356:2
4352:N
4348:1
4343:=
4339:)
4334:N
4314:Y
4305:(
4281:n
4258:N
4249:i
4245:y
4239:i
4235:w
4229:n
4224:1
4221:=
4218:i
4202:N
4197:r
4194:w
4191:p
4181:Y
4172:=
4167:N
4147:Y
4131:N
4100:i
4096:p
4089:n
4085:1
4073:i
4065:1
4060:=
4055:i
4051:w
4020:i
4009:i
4005:y
3999:i
3995:I
3988:=
3983:i
3973:y
3964:i
3960:I
3956:=
3948:i
3938:y
3912:i
3902:y
3895:n
3892:=
3885:i
3881:p
3875:i
3871:y
3858:y
3844:p
3820:i
3810:i
3806:y
3800:=
3795:i
3785:y
3771:y
3734:)
3729:i
3718:1
3715:(
3710:i
3700:2
3695:i
3691:y
3687:=
3684:]
3679:i
3675:I
3671:[
3668:V
3663:2
3658:i
3654:y
3650:=
3647:]
3639:i
3635:y
3631:[
3628:V
3606:i
3596:i
3592:y
3588:=
3585:]
3580:i
3576:I
3572:[
3569:E
3564:i
3560:y
3556:=
3553:]
3545:i
3541:y
3537:[
3534:E
3512:i
3508:I
3502:i
3498:y
3494:=
3486:i
3482:y
3459:i
3455:I
3432:i
3428:y
3397:i
3393:p
3370:n
3365:i
3350:i
3346:p
3342:=
3339:)
3330:|
3326:1
3323:=
3318:i
3314:I
3310:(
3307:P
3285:i
3277:=
3274:)
3271:n
3260:1
3257:=
3252:i
3248:I
3244:(
3241:P
3227:i
3211:i
3207:I
3188:y
3168:N
3145:N
3135:T
3131:Y
3127:y
3123:y
3088:i
3084:y
3061:y
3043:n
3037:2
3032:i
3028:w
3022:n
3017:1
3014:=
3011:i
3000:=
2990:2
2986:w
2961:n
2955:i
2951:w
2945:n
2940:1
2937:=
2934:i
2923:=
2914:w
2888:1
2882:n
2875:2
2871:)
2861:y
2850:i
2846:y
2842:(
2837:n
2832:1
2829:=
2826:i
2815:=
2810:2
2805:y
2764:2
2754:w
2741:2
2737:w
2728:2
2723:y
2706:=
2703:)
2698:w
2688:y
2681:(
2635:n
2609:.
2603:=
2600:)
2591:x
2585:(
2582:E
2559:=
2554:i
2529:.
2523:i
2510:i
2506:w
2499:n
2494:1
2491:=
2488:i
2480:=
2477:)
2468:x
2462:(
2459:E
2439:,
2433:i
2424:=
2421:)
2416:i
2412:x
2408:(
2405:E
2376:x
2329:.
2322:2
2317:i
2307:i
2303:x
2295:n
2290:1
2287:=
2284:i
2274:2
2264:x
2254:=
2245:x
2216:.
2209:2
2204:)
2198:2
2190:i
2180:n
2175:1
2172:=
2169:i
2160:(
2151:2
2146:i
2136:4
2128:i
2117:n
2112:1
2109:=
2106:i
2095:=
2089:2
2084:i
2073:2
2064:i
2060:w
2053:n
2048:1
2045:=
2042:i
2034:=
2029:2
2019:x
1987:0
1979:=
1974:i
1949:n
1945:/
1939:2
1934:0
1926:=
1921:2
1911:x
1881:,
1872:i
1868:w
1862:n
1857:1
1854:=
1851:i
1842:1
1836:=
1827:2
1819:i
1809:n
1804:1
1801:=
1798:i
1789:1
1783:=
1773:x
1736:,
1728:i
1724:w
1718:n
1713:1
1710:=
1707:i
1696:)
1690:i
1686:w
1677:i
1673:x
1668:(
1662:n
1657:1
1654:=
1651:i
1640:=
1629:2
1624:i
1616:1
1608:n
1603:1
1600:=
1597:i
1586:)
1578:2
1573:i
1563:i
1559:x
1552:(
1546:n
1541:1
1538:=
1535:i
1524:=
1515:x
1486:.
1479:2
1474:i
1466:1
1461:=
1456:i
1452:w
1426:2
1421:i
1387:i
1383:x
1337:2
1328:i
1324:w
1318:n
1313:1
1310:=
1307:i
1294:=
1284:x
1243:x
1210:2
1175:i
1171:x
1164:n
1159:1
1156:=
1153:i
1143:n
1140:1
1123:.
1105:j
1101:w
1094:n
1089:1
1086:=
1083:j
1072:i
1068:w
1062:=
1054:i
1050:w
1034:.
1019:i
1015:x
1006:i
1002:w
995:n
990:1
987:=
984:i
976:=
967:x
941:1
938:=
929:i
925:w
918:n
913:1
910:=
907:i
875:.
867:n
863:w
859:+
853:+
848:2
844:w
840:+
835:1
831:w
823:n
819:x
813:n
809:w
805:+
799:+
794:2
790:x
784:2
780:w
776:+
771:1
767:x
761:1
757:w
750:=
741:x
712:,
704:i
700:w
694:n
689:1
686:=
683:i
671:i
667:x
661:i
657:w
651:n
646:1
643:=
640:i
629:=
620:x
593:)
587:n
583:w
579:,
573:,
568:2
564:w
560:,
555:1
551:w
546:(
521:)
515:n
511:x
507:,
501:,
496:2
492:x
488:,
483:1
479:x
474:(
438:=
435:)
423:(
420:+
417:)
405:(
402:=
393:x
361:=
352:+
317:=
308:+
251:=
242:+
234:)
222:(
219:+
216:)
204:(
198:=
189:x
158:=
145:=
136:x
103:—
91:—
42:.
20:)
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