247:
163:
420:
229:
8186:
5080:
1296:
3237:
373:, the skewness is defined in terms of this relationship: positive/right nonparametric skew means the mean is greater than (to the right of) the median, while negative/left nonparametric skew means the mean is less than (to the left of) the median. However, the modern definition of skewness and the traditional nonparametric definition do not always have the same sign: while they agree for some families of distributions, they differ in some of the cases, and conflating them is misleading.
8298:
8172:
878:
8210:
8198:
1291:{\displaystyle {\begin{aligned}{\tilde {\mu }}_{3}&=\operatorname {E} \left\\&={\frac {\operatorname {E} -3\mu \operatorname {E} +3\mu ^{2}\operatorname {E} -\mu ^{3}}{\sigma ^{3}}}\\&={\frac {\operatorname {E} -3\mu (\operatorname {E} -\mu \operatorname {E} )-\mu ^{3}}{\sigma ^{3}}}\\&={\frac {\operatorname {E} -3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}.\end{aligned}}}
786:
83:
overall; this is the case for a symmetric distribution but can also be true for an asymmetric distribution where one tail is long and thin, and the other is short but fat. Thus, the judgement on the symmetry of a given distribution by using only its skewness is risky; the distribution shape must be taken into account.
3676:
4480:
2394:
40:
3214:
Many models assume normal distribution; i.e., data are symmetric about the mean. The normal distribution has a skewness of zero. But in reality, data points may not be perfectly symmetric. So, an understanding of the skewness of the dataset indicates whether deviations from the mean are going to be
427:
For example, in the distribution of adult residents across US households, the skew is to the right. However, since the majority of cases is less than or equal to the mode, which is also the median, the mean sits in the heavier left tail. As a result, the rule of thumb that the mean is right of the
4139:
Quantile-based skewness measures are at first glance easy to interpret, but they often show significantly larger sample variations than moment-based methods. This means that often samples from a symmetric distribution (like the uniform distribution) have a large quantile-based skewness, just by
82:
is on the left side of the distribution, and positive skew indicates that the tail is on the right. In cases where one tail is long but the other tail is fat, skewness does not obey a simple rule. For example, a zero value in skewness means that the tails on both sides of the mean balance out
415:. Most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they also contradict the textbook interpretation of the median.
168:
Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the numeric sequence (49, 50, 51), whose values are evenly distributed around a central value of 50. We can transform this sequence into a negatively skewed
517:
2102:
1854:
4702:
1569:
3003:
3367:
5845:
Premaratne, G., Bera, A. K. (2001). Adjusting the Tests for
Skewness and Kurtosis for Distributional Misspecifications. Working Paper Number 01-0116, University of Illinois. Forthcoming in Comm in Statistics, Simulation and Computation. 2016
4241:
3953:
2208:
4282:
A value of skewness equal to zero does not imply that the probability distribution is symmetric. Thus there is a need for another measure of asymmetry that has this property: such a measure was introduced in 2000. It is called
232:
Example of an asymmetric distribution with zero skewness. This figure serves as a counterexample that zero skewness does not imply symmetric distribution necessarily. (Skewness was calculated by
Pearson's moment coefficient of
3120:; their expected values can even have the opposite sign from the true skewness. For instance, a mixed distribution consisting of very thin Gaussians centred at â99, 0.5, and 2 with weights 0.01, 0.66, and 0.33 has a skewness
4869:
220:
the skew is negative. Similarly, we can make the sequence positively skewed by adding a value far above the mean, which is probably a positive outlier, e.g. (49, 50, 51, 60), where the mean is 52.5, and the median is 50.5.
2878:
4346:
224:
As mentioned earlier, a unimodal distribution with zero value of skewness does not imply that this distribution is symmetric necessarily. However, a symmetric unimodal or multimodal distribution always has zero skewness.
4091:
781:{\displaystyle \gamma _{1}:={\tilde {\mu }}_{3}=\operatorname {E} \left={\frac {\mu _{3}}{\sigma ^{3}}}={\frac {\operatorname {E} \left}{(\operatorname {E} \left)^{3/2}}}={\frac {\kappa _{3}}{\kappa _{2}^{3/2}}}}
2720:
91:
Consider the two distributions in the figure just below. Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are called
242:
The skewness is not directly related to the relationship between the mean and median: a distribution with negative skew can have its mean greater than or less than the median, and likewise for positive skew.
396:. This is the case of a coin toss or the series 1,2,3,4,... Note, however, that the converse is not true in general, i.e. zero skewness (defined below) does not imply that the mean is equal to the median.
1408:
1865:
3200:
for a mean will be not only incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level, but they will also result in unequal error probabilities on each side.
403:
Many textbooks teach a rule of thumb stating that the mean is right of the median under right skew, and left of the median under left skew. This rule fails with surprising frequency. It can fail in
1625:
2213:
883:
4549:
5066:
4971:
3693:). The numerator is difference between the average of the upper and lower quartiles (a measure of location) and the median (another measure of location), while the denominator is the
5481:
Bowley, A. L. (1901). Elements of
Statistics, P.S. King & Son, Laondon. Or in a later edition: BOWLEY, AL. "Elements of Statistics, 4th Edn (New York, Charles Scribner)."(1920).
3776:
1419:
299:
218:
509:
153:
instead refers to the right tail being drawn out and, often, the mean being skewed to the right of a typical center of the data. A right-skewed distribution usually appears as a
2916:
123:
instead refers to the left tail being drawn out and, often, the mean being skewed to the left of a typical center of the data. A left-skewed distribution usually appears as a
2132:
8248:
3671:{\displaystyle {\frac {{\frac {{{Q}(3/4)}+{{Q}(1/4)}}{2}}-{{Q}(1/2)}}{\frac {{{Q}(3/4)}-{{Q}(1/4)}}{2}}}={\frac {{{Q}(3/4)}+{{Q}(1/4)}-2{{Q}(1/2)}}{{{Q}(3/4)}-{{Q}(1/4)}}},}
4918:
4533:
2656:
3145:
3118:
2468:
466:
5849:
Premaratne, G., Bera, A. K. (2000). Modeling
Asymmetry and Excess Kurtosis in Stock Return Data. Office of Research Working Paper Number 00-0123, University of Illinois.
4334:
367:
4998:
3172:
3087:
3060:
3033:
2908:
2619:
2592:
2565:
2499:
2424:
2189:
843:, but should not be confused with Pearson's other skewness statistics (see below). The last equality expresses skewness in terms of the ratio of the third cumulant
4154:
343:
319:
2752:
2538:
2389:{\displaystyle {\begin{aligned}G_{1}&={\frac {k_{3}}{k_{2}^{3/2}}}={\frac {n^{2}}{(n-1)(n-2)}}\;b_{1}={\frac {\sqrt {n(n-1)}}{n-2}}\;g_{1},\\\end{aligned}}}
3798:
3174:
has an expected value of about 0.32, since usually all three samples are in the positive-valued part of the distribution, which is skewed the other way.
5269:
4730:
4475:{\displaystyle \operatorname {dSkew} (X):=1-{\frac {\operatorname {E} \|X-X'\|}{\operatorname {E} \|X+X'-2\theta \|}}{\text{ if }}\Pr(X=\theta )\neq 1}
8241:
5748:
Szekely, G. J. and Mori, T. F. (2001) "A characteristic measure of asymmetry and its application for testing diagonal symmetry",
5350:
2764:
7307:
8405:
7812:
8363:
8234:
3964:
4112:) †1 and is well defined without requiring the existence of any moments of the distribution. Bowley's measure of skewness is γ(
7962:
5180:
2661:
7586:
6227:
5382:
2097:{\displaystyle g_{1}={\frac {m_{3}}{m_{2}^{3/2}}}={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{3}}{\left^{3/2}}},}
7360:
137:: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. The distribution is said to be
107:: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. The distribution is said to be
1312:
7799:
1849:{\displaystyle b_{1}={\frac {m_{3}}{s^{3}}}={\frac {{\tfrac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{3}}{\left^{3/2}}}}
6222:
5922:
4697:{\displaystyle \operatorname {dSkew} _{n}(X):=1-{\frac {\sum _{i,j}\|x_{i}-x_{j}\|}{\sum _{i,j}\|x_{i}+x_{j}-2\theta \|}}.}
2143:
173:, e.g. (40, 49, 50, 51). Therefore, the mean of the sequence becomes 47.5, and the median is 49.5. Based on the formula of
380:, then the mean is equal to the median, and the distribution has zero skewness. If the distribution is both symmetric and
6826:
5974:
5147:
5245:
377:
5889:
872:
is finite too, then skewness can be expressed in terms of the non-central moment E by expanding the previous formula:
861:
as the fourth cumulant normalized by the square of the second cumulant. The skewness is also sometimes denoted Skew.
7609:
7501:
5815:
5568:
5337:
5316:
4136: < 1. Another measure can be obtained by integrating the numerator and denominator of this expression.
3218:
3193:
Skewness indicates the direction and relative magnitude of a distribution's deviation from the normal distribution.
8281:
8214:
7787:
7661:
5006:
3690:
8368:
7845:
7506:
7251:
6622:
6212:
3789:
Other names for this measure are Galton's measure of skewness, the YuleâKendall index and the quartile skewness,
246:
7896:
7108:
6915:
6804:
6762:
4923:
2192:
1564:{\displaystyle \Pr=(1-x)^{-3}/2{\mbox{ for negative }}x{\mbox{ and }}\Pr=(1+x)^{-3}/2{\mbox{ for positive }}x.}
6836:
8400:
8139:
7098:
6001:
5876:
5513:
Groeneveld, Richard A (1991). "An influence function approach to describing the skewness of a distribution".
3699:
3958:
A more general formulation of a skewness function was described by
Groeneveld, R. A. and Meeden, G. (1984):
8275:
7690:
7639:
7624:
7614:
7483:
7355:
7322:
7148:
7103:
6933:
5866:
5697:
Hosking, J.R.M. (1992). "Moments or L moments? An example comparing two measures of distributional shape".
3208:
2471:
17:
5357:
4493: = Ξ (with probability 1). Distance skewness is always between 0 and 1, equals 0 if and only if
2998:{\displaystyle \operatorname {var} (b_{1})<\operatorname {var} (g_{1})<\operatorname {var} (G_{1}).}
261:
180:
8202:
8034:
7835:
7759:
7060:
6814:
6483:
5947:
5871:
3264:(not to be confused with Pearson's moment coefficient of skewness, see above). These other measures are:
478:
8357:
7919:
7891:
7886:
7634:
7393:
7299:
7279:
7187:
6898:
6716:
6199:
6071:
4128: = 9/10. This definition leads to a corresponding overall measure of skewness defined as the
3779:
96:, and they provide a visual means to determine which of the two kinds of skewness a distribution has:
8269:
7651:
7419:
7140:
7065:
6994:
6923:
6843:
6831:
6701:
6689:
6682:
6390:
6111:
2110:
412:
8258:
8134:
7901:
7764:
7449:
7414:
7378:
7163:
6605:
6514:
6473:
6385:
6076:
5915:
5113:
4877:
4512:
3694:
2628:
1596:
1589:
60:
43:
Example distribution with positive skewness. These data are from experiments on wheat grass growth.
5425:
5206:
8043:
7656:
7596:
7533:
7171:
7155:
6893:
6755:
6745:
6595:
6509:
3253:
3203:
Skewness can be used to obtain approximate probabilities and quantiles of distributions (such as
3123:
3096:
2433:
444:
404:
5883:
4313:
162:
8081:
8011:
7804:
7741:
7496:
7383:
6380:
6277:
6184:
6063:
5962:
3783:
1603:
419:
5192:
8106:
8048:
7991:
7817:
7710:
7619:
7345:
7229:
7088:
7080:
6970:
6962:
6777:
6673:
6651:
6610:
6575:
6542:
6488:
6463:
6418:
6357:
6317:
6119:
5942:
4236:{\displaystyle \operatorname {skew} (X)={\frac {(\mu -\nu )}{\operatorname {E} (|X-\nu |)}},}
381:
352:
228:
78:
distribution (a distribution with a single peak), negative skew commonly indicates that the
8029:
7604:
7553:
7529:
7491:
7409:
7388:
7340:
7219:
7197:
7166:
7075:
6952:
6903:
6821:
6794:
6750:
6706:
6468:
6244:
6124:
4976:
3150:
3090:
3065:
3038:
3011:
2886:
2622:
2597:
2570:
2543:
2477:
2402:
2167:
2154:
5503:
Yule, George Udny. An introduction to the theory of statistics. C. Griffin, limited, 1912.
2722:, i.e., their distributions converge to a normal distribution with mean 0 and variance 6 (
328:
304:
8:
8176:
8101:
8024:
7705:
7469:
7462:
7424:
7332:
7312:
7284:
7017:
6883:
6878:
6868:
6860:
6678:
6639:
6529:
6519:
6428:
6207:
6163:
6081:
6006:
5908:
5447:
5400:
5267:
Joanes, D. N.; Gill, C. A. (1998). "Comparing measures of sample skewness and kurtosis".
5001:
3197:
2729:
1582:
473:
250:
A general relationship of mean and median under differently skewed unimodal distribution.
8329:
8190:
8001:
7855:
7751:
7700:
7576:
7473:
7457:
7434:
7211:
6945:
6928:
6888:
6799:
6694:
6656:
6627:
6587:
6547:
6493:
6410:
6096:
6091:
5783:
5714:
5652:
5530:
5085:
3948:{\displaystyle {\frac {{{Q}(9/10)}+{{Q}(1/10)}-2{{Q}(1/2)}}{{{Q}(9/10)}-{{Q}(1/10)}}}.}
3346:
3333:
3293:
3260:
Other measures of skewness have been used, including simpler calculations suggested by
2523:
800:
370:
255:
174:
119:, despite the fact that the curve itself appears to be skewed or leaning to the right;
48:
8226:
149:, despite the fact that the curve itself appears to be skewed or leaning to the left;
8287:
8185:
8096:
8066:
8058:
7878:
7869:
7794:
7725:
7581:
7566:
7541:
7429:
7370:
7236:
7224:
6850:
6767:
6711:
6634:
6478:
6400:
6179:
6053:
5811:
5787:
5564:
5422:
5397:
5376:
5333:
5312:
5079:
3686:
3285:
3249:
393:
4505:
have the same probability distribution) and equals 1 if and only if X is a constant
8121:
8076:
7840:
7827:
7720:
7695:
7629:
7561:
7439:
7047:
6940:
6873:
6786:
6733:
6552:
6423:
6217:
6101:
6016:
5983:
5834:
5825:
MacGillivray, HL (1992). "Shape properties of the g- and h- and
Johnson families".
5775:
5706:
5644:
5522:
5467:
5462:
5278:
4536:
4259:
71:
about its mean. The skewness value can be positive, zero, negative, or undefined.
8038:
7782:
7644:
7571:
7246:
7120:
7093:
7070:
7039:
6666:
6661:
6615:
6345:
5996:
5108:
4864:{\displaystyle h(x_{i},x_{j})={\frac {(x_{i}-x_{m})-(x_{m}-x_{j})}{x_{i}-x_{j}}}}
4717:
3222:
2502:
1585:
and any other symmetric distribution with finite third moment has a skewness of 0
68:
7528:
169:
distribution by adding a value far below the mean, which is probably a negative
8319:
8314:
7987:
7982:
6445:
6375:
6021:
5731:
Szekely, G.J. (2000). "Pre-limit and post-limit theorems for statistics", In:
4255:
3225:
2873:{\displaystyle \operatorname {var} (G_{1})={\frac {6n(n-1)}{(n-2)(n+1)(n+3)}}.}
2510:
815:
804:
31:
5838:
3236:
3196:
With pronounced skewness, standard statistical inference procedures such as a
8394:
8144:
8111:
7974:
7935:
7746:
7715:
7179:
7133:
6738:
6440:
6267:
6031:
6026:
5607:
5581:
5330:
The
Advanced Theory of Statistics, Volume 1: Distribution Theory, 3rd Edition
5118:
4148:
Groeneveld and Meeden have suggested, as an alternative measure of skewness,
3204:
3187:
3182:
Skewness is a descriptive statistic that can be used in conjunction with the
2723:
5779:
5420:
5311:
Duncan Cramer (1997) Fundamental
Statistics for Social Research. Routledge.
5282:
4274:
in place of moments provides a measure of skewness known as the L-skewness.
8297:
8086:
8019:
7996:
7911:
7241:
6537:
6435:
6370:
6312:
6297:
6234:
6189:
5068:. It can be seen as the median of all possible quantile skewness measures.
3310:
The
Pearson median skewness, or second skewness coefficient, is defined as
3261:
2754:
for sufficiently large samples. More precisely, in a random sample of size
5896:
Closed-skew
Distributions â Simulation, Inversion and Parameter Estimation
4258:, and E() is the expectation operator. This is closely related in form to
3305:
8129:
8091:
7774:
7675:
7537:
7350:
7317:
6809:
6726:
6721:
6365:
6322:
6302:
6282:
6272:
6041:
2135:
64:
5635:
Groeneveld, R.A.; Meeden, G. (1984). "Measuring Skewness and Kurtosis".
5299:
3272:
The Pearson mode skewness, or first skewness coefficient, is defined as
8309:
6975:
6455:
6155:
6086:
6036:
6011:
5931:
5763:
5718:
5683:
5656:
5534:
5098:
5093:
3267:
52:
5895:
5230:
1606:
can have a skewness of any positive value, depending on its parameters
1577:
Examples of distributions with finite skewness include the following.
8379:
7128:
6980:
6600:
6395:
6307:
6292:
6287:
6252:
5736:
5430:
5405:
4713:
4086:{\displaystyle \gamma (u)={\frac {Q(u)+Q(1-u)-2Q(1/2)}{Q(u)-Q(1-u)}}}
3183:
2470:
is the symmetric unbiased estimator of the second cumulant (i.e. the
408:
39:
5852:
5710:
5648:
5526:
2687:
8374:
8344:
8339:
8324:
6644:
6262:
6134:
6129:
5152:
5103:
4271:
4129:
2726:, 1930). The variance of the sample skewness is thus approximately
2715:{\displaystyle {\sqrt {n}}b_{1}\mathrel {\xrightarrow {d} } N(0,6)}
2427:
858:
832:
75:
8149:
7850:
5805:
5547:
2506:
170:
2474:). This adjusted FisherâPearson standardized moment coefficient
8071:
7052:
7026:
7006:
6257:
6048:
4721:
3324:
3245:
2540:
is normally distributed, it can be shown that all three ratios
389:
346:
5395:
1619:
values, two natural estimators of the population skewness are
5900:
5766:; A. Struyf (November 2004). "A Robust Measure of Skewness".
1403:{\displaystyle \Pr \left=x^{-2}{\mbox{ for }}x>1,\ \Pr=0}
5991:
5302:
Journal of Statistics Education 19.2 (2011): 1-18. (Page 7)
3320:
3281:
3241:
2514:
385:
322:
5681:
Hinkley DV (1975) "On power transformations to symmetry",
5148:"2.6 Skewness and the Mean, Median, and Mode â Statistics"
4535:) with probability one. Thus there is a simple consistent
436:
5761:
2520:
Under the assumption that the underlying random variable
4716:
is a scale-invariant robust measure of skewness, with a
4120: = 3/4 while Kelly's measure of skewness is Îł(
2426:
is the unique symmetric unbiased estimator of the third
8256:
5884:
An Asymmetry Coefficient for Multivariate Distributions
5608:"Applied Statistics I: Chapter 5: Measures of skewness"
4336:
denotes the norm in the Euclidean space, then a simple
3306:
Pearson's second skewness coefficient (median skewness)
2910:
has the smaller variance of the three estimators, with
4143:
3357:
Bowley's measure of skewness (from 1901), also called
2002:
1926:
1749:
1673:
1549:
1488:
1478:
1355:
5009:
4979:
4926:
4880:
4733:
4552:
4515:
4349:
4316:
4157:
3967:
3801:
3792:
Similarly, Kelly's measure of skewness is defined as
3703:
3370:
3153:
3126:
3099:
3068:
3041:
3014:
2919:
2889:
2767:
2732:
2664:
2631:
2600:
2573:
2546:
2526:
2480:
2436:
2405:
2211:
2170:
2113:
1868:
1628:
1422:
1315:
881:
520:
481:
447:
355:
331:
307:
264:
183:
7813:
Autoregressive conditional heteroskedasticity (ARCH)
5181:"Mean, Median, and Skew: Correcting a Textbook Rule"
5075:
3778:, which for symmetric distributions is equal to the
3268:
Pearson's first skewness coefficient (mode skewness)
423:
Distribution of adult residents across US households
5668:
5666:
1413:where the third cumulants are infinite, or as when
7275:
5890:On More Robust Estimation of Skewness and Kurtosis
5739:and G. J. Szekely), Dekker, New York, pp. 411â422.
5270:Journal of the Royal Statistical Society, Series D
5060:
4992:
4965:
4912:
4863:
4696:
4527:
4474:
4328:
4306:is an independent identically distributed copy of
4235:
4085:
3947:
3770:
3670:
3166:
3139:
3112:
3081:
3054:
3027:
2997:
2902:
2872:
2746:
2714:
2650:
2613:
2586:
2559:
2532:
2493:
2462:
2418:
2388:
2183:
2126:
2096:
1848:
1563:
1402:
1290:
780:
503:
460:
361:
337:
313:
293:
237:
212:
5827:Communications in Statistics â Theory and Methods
5768:Journal of Computational and Graphical Statistics
5750:Communications in Statistics â Theory and Methods
5630:
5628:
5626:
5624:
5548:Johnson, NL, Kotz, S & Balakrishnan, N (1994)
5174:
5172:
5170:
8392:
5663:
5634:
5605:
5146:Illowsky, Barbara; Dean, Susan (27 March 2020).
4448:
1494:
1423:
1376:
1316:
7361:Multivariate adaptive regression splines (MARS)
5892:Comparison of skew estimators by Kim and White.
5561:Statistical Methods in the Atmospheric Sciences
3256:with the same medians and different skewnesses.
30:For the planarity measure in graph theory, see
5853:Skewness Measures for the Weibull Distribution
5806:Johnson, NL; Kotz, S; Balakrishnan, N (1994).
5621:
5167:
3228:based on sample skewness and sample kurtosis.
8242:
5916:
5207:"1.3.5.11. Measures of Skewness and Kurtosis"
5061:{\displaystyle \{x_{1},x_{2},\ldots ,x_{n}\}}
3231:
2164:is the (biased) sample third central moment.
5824:
5448:"Measuring Skewness: A Forgotten Statistic?"
5300:"Measuring skewness: a forgotten statistic."
5226:
5224:
5145:
5055:
5010:
4685:
4650:
4629:
4603:
4437:
4411:
4400:
4383:
4323:
4317:
27:Measure of the asymmetry of random variables
5675:
5541:
5445:
5343:
5294:
5292:
5241:
5239:
5233:, 2008â2016 by Stan Brown, Oak Road Systems
4497:is diagonally symmetric with respect to Ξ (
2505:and several statistical packages including
8249:
8235:
5961:
5923:
5909:
5512:
5484:
5439:
5266:
5231:"Measures of Shape: Skewness and Kurtosis"
5178:
4291:is a random variable taking values in the
3352:
3190:to characterize the data or distribution.
2368:
2319:
850:to the 1.5th power of the second cumulant
6574:
5553:
5492:Mathematics of Statistics, Pt. 1, 3rd ed.
5466:
5446:Doane, David P.; Seward, Lori E. (2011).
5305:
5221:
5141:
5139:
4966:{\displaystyle x_{i}\geq x_{m}\geq x_{j}}
857:. This is analogous to the definition of
5755:
5289:
5246:Pearson's moment coefficient of skewness
5236:
4340:with respect to location parameter Ξ is
3235:
837:Pearson's moment coefficient of skewness
418:
407:, or in distributions where one tail is
245:
227:
38:
5696:
4132:of this over the range 1/2 â€
3771:{\displaystyle ({Q}(3/4)}-{{Q}(1/4))/2}
1574:where the third cumulant is undefined.
437:Fisher's moment coefficient of skewness
14:
8393:
7887:KaplanâMeier estimator (product limit)
5690:
5381:: CS1 maint: archived copy as title (
5136:
2625:estimators of the population skewness
2153:is the (biased) sample second central
8230:
7960:
7527:
7274:
6573:
6343:
5960:
5904:
5579:
5421:
5414:
5396:
5298:Doane, David P., and Lori E. Seward.
4724:of the values of the kernel function
4260:Pearson's second skewness coefficient
3147:of about â9.77, but in a sample of 3
59:is a measure of the asymmetry of the
8406:Statistical deviation and dispersion
8197:
7897:Accelerated failure time (AFT) model
5262:
5260:
5258:
5256:
5254:
4277:
294:{\displaystyle (\mu -\nu )/\sigma ,}
213:{\displaystyle (\mu -\nu )/\sigma ,}
8209:
7492:Analysis of variance (ANOVA, anova)
6344:
5810:. Vol. 1 (2 ed.). Wiley.
5808:Continuous Univariate Distributions
4144:Groeneveld and Meeden's coefficient
504:{\displaystyle {\tilde {\mu }}_{3}}
24:
7587:CochranâMantelâHaenszel statistics
6213:Pearson product-moment correlation
4539:of diagonal symmetry based on the
4405:
4377:
4196:
3345:Which is a simple multiple of the
1610:
1306:Skewness can be infinite, as when
1214:
1157:
1129:
1095:
1041:
1003:
972:
912:
679:
636:
556:
399:A 2005 journal article points out:
25:
8417:
5859:
5426:"Pearson's skewness coefficients"
5328:Kendall, M.G.; Stuart, A. (1969)
5251:
2198:Another common definition of the
835:. It is sometimes referred to as
8296:
8282:cumulative distribution function
8208:
8196:
8184:
8171:
8170:
7961:
5490:Kenney JF and Keeping ES (1962)
5078:
3691:cumulative distribution function
428:median under right skew failed.
161:
8369:probability-generating function
7846:Least-squares spectral analysis
5742:
5733:Statistics for the 21st Century
5725:
5599:
5573:
5506:
5497:
5475:
5455:Journal of Statistics Education
5389:
5185:Journal of Statistics Education
3177:
2127:{\displaystyle {\overline {x}}}
238:Relationship of mean and median
86:
6827:Mean-unbiased minimum-variance
5930:
5468:10.1080/10691898.2011.11889611
5322:
5199:
4907:
4881:
4830:
4804:
4798:
4772:
4763:
4737:
4572:
4566:
4463:
4451:
4362:
4356:
4295:-dimensional Euclidean space,
4224:
4220:
4206:
4202:
4191:
4179:
4170:
4164:
4077:
4065:
4056:
4050:
4042:
4028:
4016:
4004:
3995:
3989:
3977:
3971:
3935:
3921:
3908:
3894:
3882:
3868:
3852:
3838:
3825:
3811:
3757:
3754:
3740:
3726:
3712:
3704:
3658:
3644:
3631:
3617:
3605:
3591:
3575:
3561:
3548:
3534:
3511:
3497:
3484:
3470:
3457:
3443:
3424:
3410:
3397:
3383:
3008:For non-normal distributions,
2989:
2976:
2964:
2951:
2939:
2926:
2861:
2849:
2846:
2834:
2831:
2819:
2814:
2802:
2787:
2774:
2709:
2697:
2351:
2339:
2313:
2301:
2298:
2286:
2061:
2034:
1985:
1958:
1816:
1789:
1732:
1705:
1528:
1515:
1509:
1497:
1457:
1444:
1438:
1426:
1391:
1379:
1233:
1220:
1172:
1169:
1163:
1148:
1135:
1126:
1114:
1101:
1053:
1047:
1022:
1009:
991:
978:
893:
841:moment coefficient of skewness
718:
703:
690:
676:
660:
647:
541:
489:
277:
265:
196:
184:
13:
1:
8140:Geographic information system
7356:Simultaneous equations models
5124:
4913:{\displaystyle (x_{i},x_{j})}
4528:{\displaystyle c\neq \theta }
2651:{\displaystyle \gamma _{1}=0}
431:
8276:probability density function
7323:Coefficient of determination
6934:Uniformly most powerful test
5179:von Hippel, Paul T. (2005).
5129:
4707:
4265:
4254:is the median, |...| is the
3361:(from 1912) is defined as:
2758:from a normal distribution,
2119:
2055:
1979:
1810:
1726:
7:
7892:Proportional hazards models
7836:Spectral density estimation
7818:Vector autoregression (VAR)
7252:Maximum posterior estimator
6484:Randomized controlled trial
5872:Encyclopedia of Mathematics
5494:, Van Nostrand, (page 102).
5071:
4104:) satisfies â1 â€
3219:D'Agostino's K-squared test
3140:{\displaystyle \gamma _{1}}
3113:{\displaystyle \gamma _{1}}
3093:of the population skewness
2463:{\displaystyle k_{2}=s^{2}}
1592:has a skewness just below 1
1301:
461:{\displaystyle \gamma _{1}}
10:
8422:
8358:moment-generating function
7652:Multivariate distributions
6072:Average absolute deviation
5798:
4329:{\displaystyle \|\cdot \|}
3689:(i.e., the inverse of the
3232:Other measures of skewness
29:
8353:
8305:
8294:
8270:probability mass function
8265:
8259:probability distributions
8166:
8120:
8057:
8010:
7973:
7969:
7956:
7928:
7910:
7877:
7868:
7826:
7773:
7734:
7683:
7674:
7640:Structural equation model
7595:
7552:
7548:
7523:
7482:
7448:
7402:
7369:
7331:
7298:
7294:
7270:
7210:
7119:
7038:
7002:
6993:
6976:Score/Lagrange multiplier
6961:
6914:
6859:
6785:
6776:
6586:
6582:
6569:
6528:
6502:
6454:
6409:
6391:Sample size determination
6356:
6352:
6339:
6243:
6198:
6172:
6154:
6110:
6062:
5982:
5973:
5969:
5956:
5938:
5839:10.1080/03610929208830842
5699:The American Statistician
5515:The American Statistician
4287:and denoted by dSkew. If
2144:sample standard deviation
8135:Environmental statistics
7657:Elliptical distributions
7450:Generalized linear model
7379:Simple linear regression
7149:HodgesâLehmann estimator
6606:Probability distribution
6515:Stochastic approximation
6077:Coefficient of variation
5563:, p 27. Academic Press.
5114:Skew normal distribution
4541:sample distance skewness
4299:has finite expectation,
3695:semi-interquartile range
3254:log-normal distributions
3209:CornishâFisher expansion
2501:is the version found in
1597:exponential distribution
1590:half-normal distribution
1551: for positive
1480: for negative
405:multimodal distributions
61:probability distribution
8364:characteristic function
7795:Cross-correlation (XCF)
7403:Non-standard predictors
6837:LehmannâScheffĂ© theorem
6510:Adaptive clinical trial
5867:"Asymmetry coefficient"
5780:10.1198/106186004X12632
5401:"Pearson Mode Skewness"
5283:10.1111/1467-9884.00122
4874:taken over all couples
3353:Quantile-based measures
376:If the distribution is
362:{\displaystyle \sigma }
254:In the older notion of
8191:Mathematics portal
8012:Engineering statistics
7920:NelsonâAalen estimator
7497:Analysis of covariance
7384:Ordinary least squares
7308:Pearson product-moment
6712:Statistical functional
6623:Empirical distribution
6456:Controlled experiments
6185:Frequency distribution
5963:Descriptive statistics
5752:30/8&9, 1633â1639.
5606:A.W.L. Pubudu Thilan.
5062:
4994:
4967:
4914:
4865:
4698:
4529:
4476:
4330:
4237:
4087:
3949:
3772:
3672:
3257:
3215:positive or negative.
3168:
3141:
3114:
3083:
3056:
3029:
2999:
2904:
2874:
2748:
2716:
2652:
2615:
2588:
2561:
2534:
2495:
2464:
2420:
2390:
2185:
2128:
2098:
2033:
1957:
1850:
1788:
1704:
1604:lognormal distribution
1565:
1404:
1292:
782:
505:
462:
424:
417:
363:
339:
315:
295:
251:
234:
214:
44:
8107:Population statistics
8049:System identification
7783:Autocorrelation (ACF)
7711:Exponential smoothing
7625:Discriminant analysis
7620:Canonical correlation
7484:Partition of variance
7346:Regression validation
7190:(JonckheereâTerpstra)
7089:Likelihood-ratio test
6778:Frequentist inference
6690:Locationâscale family
6611:Sampling distribution
6576:Statistical inference
6543:Cross-sectional study
6530:Observational studies
6489:Randomized experiment
6318:Stem-and-leaf display
6120:Central limit theorem
5586:mathworld.wolfram.com
5063:
5000:is the median of the
4995:
4993:{\displaystyle x_{m}}
4968:
4915:
4866:
4699:
4530:
4489:) := 0 for
4477:
4331:
4238:
4088:
3950:
3773:
3673:
3239:
3169:
3167:{\displaystyle G_{1}}
3142:
3115:
3084:
3082:{\displaystyle G_{1}}
3057:
3055:{\displaystyle g_{1}}
3030:
3028:{\displaystyle b_{1}}
3000:
2905:
2903:{\displaystyle b_{1}}
2875:
2749:
2717:
2653:
2616:
2614:{\displaystyle G_{1}}
2589:
2587:{\displaystyle g_{1}}
2562:
2560:{\displaystyle b_{1}}
2535:
2496:
2494:{\displaystyle G_{1}}
2465:
2421:
2419:{\displaystyle k_{3}}
2391:
2186:
2184:{\displaystyle g_{1}}
2129:
2099:
2013:
1937:
1851:
1768:
1684:
1566:
1405:
1293:
783:
506:
468:of a random variable
463:
422:
401:
364:
340:
316:
296:
249:
231:
215:
42:
8401:Moment (mathematics)
8030:Probabilistic design
7615:Principal components
7458:Exponential families
7410:Nonlinear regression
7389:General linear model
7351:Mixed effects models
7341:Errors and residuals
7318:Confounding variable
7220:Bayesian probability
7198:Van der Waerden test
7188:Ordered alternative
6953:Multiple comparisons
6832:RaoâBlackwellization
6795:Estimating equations
6751:Statistical distance
6469:Factorial experiment
6002:Arithmetic-Geometric
5615:University of Ruhuna
5195:on 20 February 2016.
5007:
4977:
4924:
4878:
4731:
4550:
4513:
4347:
4338:measure of asymmetry
4314:
4155:
3965:
3799:
3700:
3368:
3207:in finance) via the
3151:
3124:
3097:
3066:
3039:
3012:
2917:
2887:
2765:
2730:
2662:
2629:
2598:
2571:
2544:
2524:
2478:
2434:
2403:
2209:
2168:
2111:
1866:
1626:
1420:
1313:
879:
805:expectation operator
518:
479:
445:
353:
338:{\displaystyle \nu }
329:
314:{\displaystyle \mu }
305:
262:
181:
8102:Official statistics
8025:Methods engineering
7706:Seasonal adjustment
7474:Poisson regressions
7394:Bayesian regression
7333:Regression analysis
7313:Partial correlation
7285:Regression analysis
6884:Prediction interval
6879:Likelihood interval
6869:Confidence interval
6861:Interval estimation
6822:Unbiased estimators
6640:Model specification
6520:Up-and-down designs
6208:Partial correlation
6164:Index of dispersion
6082:Interquartile range
5886:by Michel Petitjean
5672:MacGillivray (1992)
5580:Weisstein, Eric W.
5191:(2). Archived from
3198:confidence interval
2883:In normal samples,
2747:{\displaystyle 6/n}
2691:
2267:
1916:
1599:has a skewness of 2
1583:normal distribution
775:
474:standardized moment
147:skewed to the right
8330:standard deviation
8122:Spatial statistics
8002:Medical statistics
7902:First hitting time
7856:Whittle likelihood
7507:Degrees of freedom
7502:Multivariate ANOVA
7435:Heteroscedasticity
7247:Bayesian estimator
7212:Bayesian inference
7061:KolmogorovâSmirnov
6946:Randomization test
6916:Testing hypotheses
6889:Tolerance interval
6800:Maximum likelihood
6695:Exponential family
6628:Density estimation
6588:Statistical theory
6548:Natural experiment
6494:Scientific control
6411:Survey methodology
6097:Standard deviation
5423:Weisstein, Eric W.
5398:Weisstein, Eric W.
5086:Mathematics portal
5058:
4990:
4963:
4910:
4861:
4720:of 25%. It is the
4694:
4649:
4602:
4525:
4472:
4326:
4233:
4083:
3945:
3729:
3668:
3359:Yule's coefficient
3347:nonparametric skew
3334:standard deviation
3294:standard deviation
3258:
3164:
3137:
3110:
3079:
3052:
3025:
2995:
2900:
2870:
2744:
2712:
2648:
2611:
2584:
2557:
2530:
2491:
2460:
2416:
2386:
2384:
2245:
2181:
2124:
2094:
2011:
1935:
1894:
1846:
1766:
1682:
1561:
1553:
1492:
1482:
1400:
1359:
1288:
1286:
801:standard deviation
778:
753:
501:
458:
425:
371:standard deviation
359:
335:
311:
291:
256:nonparametric skew
252:
235:
210:
175:nonparametric skew
117:skewed to the left
49:probability theory
45:
8388:
8387:
8288:quantile function
8224:
8223:
8162:
8161:
8158:
8157:
8097:National accounts
8067:Actuarial science
8059:Social statistics
7952:
7951:
7948:
7947:
7944:
7943:
7879:Survival function
7864:
7863:
7726:Granger causality
7567:Contingency table
7542:Survival analysis
7519:
7518:
7515:
7514:
7371:Linear regression
7266:
7265:
7262:
7261:
7237:Credible interval
7206:
7205:
6989:
6988:
6805:Method of moments
6674:Parametric family
6635:Statistical model
6565:
6564:
6561:
6560:
6479:Random assignment
6401:Statistical power
6335:
6334:
6331:
6330:
6180:Contingency table
6150:
6149:
6017:Generalized/power
4859:
4689:
4634:
4587:
4446:
4441:
4285:distance skewness
4278:Distance skewness
4228:
4081:
3940:
3687:quantile function
3663:
3520:
3519:
3432:
3091:biased estimators
2865:
2692:
2670:
2621:are unbiased and
2533:{\displaystyle X}
2366:
2354:
2317:
2268:
2193:method of moments
2122:
2089:
2058:
2010:
1982:
1934:
1917:
1844:
1813:
1765:
1729:
1681:
1664:
1552:
1491:
1481:
1375:
1358:
1279:
1199:
1080:
943:
896:
776:
736:
628:
587:
544:
492:
411:but the other is
16:(Redirected from
8413:
8300:
8251:
8244:
8237:
8228:
8227:
8212:
8211:
8200:
8199:
8189:
8188:
8174:
8173:
8077:Crime statistics
7971:
7970:
7958:
7957:
7875:
7874:
7841:Fourier analysis
7828:Frequency domain
7808:
7755:
7721:Structural break
7681:
7680:
7630:Cluster analysis
7577:Log-linear model
7550:
7549:
7525:
7524:
7466:
7440:Homoscedasticity
7296:
7295:
7272:
7271:
7191:
7183:
7175:
7174:(KruskalâWallis)
7159:
7144:
7099:Cross validation
7084:
7066:AndersonâDarling
7013:
7000:
6999:
6971:Likelihood-ratio
6963:Parametric tests
6941:Permutation test
6924:1- & 2-tails
6815:Minimum distance
6787:Point estimation
6783:
6782:
6734:Optimal decision
6685:
6584:
6583:
6571:
6570:
6553:Quasi-experiment
6503:Adaptive designs
6354:
6353:
6341:
6340:
6218:Rank correlation
5980:
5979:
5971:
5970:
5958:
5957:
5925:
5918:
5911:
5902:
5901:
5880:
5842:
5833:(5): 1244â1250.
5821:
5792:
5791:
5759:
5753:
5746:
5740:
5729:
5723:
5722:
5694:
5688:
5679:
5673:
5670:
5661:
5660:
5637:The Statistician
5632:
5619:
5618:
5612:
5603:
5597:
5596:
5594:
5592:
5577:
5571:
5559:Wilks DS (1995)
5557:
5551:
5545:
5539:
5538:
5510:
5504:
5501:
5495:
5488:
5482:
5479:
5473:
5472:
5470:
5452:
5443:
5437:
5436:
5435:
5418:
5412:
5411:
5410:
5393:
5387:
5386:
5380:
5372:
5370:
5368:
5362:
5356:. Archived from
5355:
5347:
5341:
5326:
5320:
5309:
5303:
5296:
5287:
5286:
5264:
5249:
5243:
5234:
5228:
5219:
5218:
5216:
5214:
5203:
5197:
5196:
5176:
5165:
5164:
5162:
5160:
5143:
5109:Shape parameters
5088:
5083:
5082:
5067:
5065:
5064:
5059:
5054:
5053:
5035:
5034:
5022:
5021:
4999:
4997:
4996:
4991:
4989:
4988:
4972:
4970:
4969:
4964:
4962:
4961:
4949:
4948:
4936:
4935:
4919:
4917:
4916:
4911:
4906:
4905:
4893:
4892:
4870:
4868:
4867:
4862:
4860:
4858:
4857:
4856:
4844:
4843:
4833:
4829:
4828:
4816:
4815:
4797:
4796:
4784:
4783:
4770:
4762:
4761:
4749:
4748:
4703:
4701:
4700:
4695:
4690:
4688:
4675:
4674:
4662:
4661:
4648:
4632:
4628:
4627:
4615:
4614:
4601:
4585:
4562:
4561:
4537:statistical test
4534:
4532:
4531:
4526:
4481:
4479:
4478:
4473:
4447:
4444:
4442:
4440:
4427:
4403:
4399:
4375:
4335:
4333:
4332:
4327:
4305:
4242:
4240:
4239:
4234:
4229:
4227:
4223:
4209:
4194:
4177:
4092:
4090:
4089:
4084:
4082:
4080:
4045:
4038:
3984:
3954:
3952:
3951:
3946:
3941:
3939:
3938:
3931:
3920:
3911:
3904:
3893:
3886:
3885:
3878:
3867:
3855:
3848:
3837:
3828:
3821:
3810:
3803:
3777:
3775:
3774:
3769:
3768:
3764:
3750:
3739:
3730:
3722:
3711:
3677:
3675:
3674:
3669:
3664:
3662:
3661:
3654:
3643:
3634:
3627:
3616:
3609:
3608:
3601:
3590:
3578:
3571:
3560:
3551:
3544:
3533:
3526:
3521:
3515:
3514:
3507:
3496:
3487:
3480:
3469:
3462:
3461:
3460:
3453:
3442:
3433:
3428:
3427:
3420:
3409:
3400:
3393:
3382:
3375:
3372:
3340:
3339:
3337:
3336:
3331:
3328:
3300:
3299:
3297:
3296:
3291:
3288:
3173:
3171:
3170:
3165:
3163:
3162:
3146:
3144:
3143:
3138:
3136:
3135:
3119:
3117:
3116:
3111:
3109:
3108:
3088:
3086:
3085:
3080:
3078:
3077:
3061:
3059:
3058:
3053:
3051:
3050:
3034:
3032:
3031:
3026:
3024:
3023:
3004:
3002:
3001:
2996:
2988:
2987:
2963:
2962:
2938:
2937:
2909:
2907:
2906:
2901:
2899:
2898:
2879:
2877:
2876:
2871:
2866:
2864:
2817:
2794:
2786:
2785:
2753:
2751:
2750:
2745:
2740:
2721:
2719:
2718:
2713:
2693:
2683:
2681:
2680:
2671:
2666:
2657:
2655:
2654:
2649:
2641:
2640:
2620:
2618:
2617:
2612:
2610:
2609:
2593:
2591:
2590:
2585:
2583:
2582:
2566:
2564:
2563:
2558:
2556:
2555:
2539:
2537:
2536:
2531:
2500:
2498:
2497:
2492:
2490:
2489:
2469:
2467:
2466:
2461:
2459:
2458:
2446:
2445:
2425:
2423:
2422:
2417:
2415:
2414:
2395:
2393:
2392:
2387:
2385:
2378:
2377:
2367:
2365:
2335:
2334:
2329:
2328:
2318:
2316:
2284:
2283:
2274:
2269:
2266:
2262:
2253:
2244:
2243:
2234:
2225:
2224:
2190:
2188:
2187:
2182:
2180:
2179:
2133:
2131:
2130:
2125:
2123:
2115:
2103:
2101:
2100:
2095:
2090:
2088:
2087:
2083:
2074:
2070:
2069:
2068:
2059:
2051:
2046:
2045:
2032:
2027:
2012:
2003:
1994:
1993:
1992:
1983:
1975:
1970:
1969:
1956:
1951:
1936:
1927:
1923:
1918:
1915:
1911:
1902:
1893:
1892:
1883:
1878:
1877:
1855:
1853:
1852:
1847:
1845:
1843:
1842:
1838:
1829:
1825:
1824:
1823:
1814:
1806:
1801:
1800:
1787:
1782:
1767:
1764:
1750:
1741:
1740:
1739:
1730:
1722:
1717:
1716:
1703:
1698:
1683:
1674:
1670:
1665:
1663:
1662:
1653:
1652:
1643:
1638:
1637:
1615:For a sample of
1570:
1568:
1567:
1562:
1554:
1550:
1544:
1539:
1538:
1493:
1489:
1483:
1479:
1473:
1468:
1467:
1409:
1407:
1406:
1401:
1373:
1360:
1356:
1353:
1352:
1337:
1333:
1297:
1295:
1294:
1289:
1287:
1280:
1278:
1277:
1268:
1267:
1266:
1254:
1253:
1232:
1231:
1212:
1204:
1200:
1198:
1197:
1188:
1187:
1186:
1147:
1146:
1113:
1112:
1093:
1085:
1081:
1079:
1078:
1069:
1068:
1067:
1040:
1039:
1021:
1020:
990:
989:
970:
962:
958:
954:
953:
948:
944:
939:
928:
904:
903:
898:
897:
889:
839:, or simply the
787:
785:
784:
779:
777:
774:
770:
761:
752:
751:
742:
737:
735:
734:
733:
729:
716:
712:
711:
710:
674:
673:
669:
668:
667:
634:
629:
627:
626:
617:
616:
607:
602:
598:
597:
592:
588:
583:
572:
552:
551:
546:
545:
537:
530:
529:
510:
508:
507:
502:
500:
499:
494:
493:
485:
467:
465:
464:
459:
457:
456:
368:
366:
365:
360:
344:
342:
341:
336:
320:
318:
317:
312:
300:
298:
297:
292:
284:
219:
217:
216:
211:
203:
165:
135:
134:
105:
104:
21:
8421:
8420:
8416:
8415:
8414:
8412:
8411:
8410:
8391:
8390:
8389:
8384:
8349:
8301:
8292:
8261:
8255:
8225:
8220:
8183:
8154:
8116:
8053:
8039:quality control
8006:
7988:Clinical trials
7965:
7940:
7924:
7912:Hazard function
7906:
7860:
7822:
7806:
7769:
7765:BreuschâGodfrey
7753:
7730:
7670:
7645:Factor analysis
7591:
7572:Graphical model
7544:
7511:
7478:
7464:
7444:
7398:
7365:
7327:
7290:
7289:
7258:
7202:
7189:
7181:
7173:
7157:
7142:
7121:Rank statistics
7115:
7094:Model selection
7082:
7040:Goodness of fit
7034:
7011:
6985:
6957:
6910:
6855:
6844:Median unbiased
6772:
6683:
6616:Order statistic
6578:
6557:
6524:
6498:
6450:
6405:
6348:
6346:Data collection
6327:
6239:
6194:
6168:
6146:
6106:
6058:
5975:Continuous data
5965:
5952:
5934:
5929:
5865:
5862:
5857:
5818:
5801:
5796:
5795:
5774:(4): 996â1017.
5760:
5756:
5747:
5743:
5730:
5726:
5711:10.2307/2685210
5695:
5691:
5680:
5676:
5671:
5664:
5649:10.2307/2987742
5633:
5622:
5610:
5604:
5600:
5590:
5588:
5578:
5574:
5558:
5554:
5546:
5542:
5527:10.2307/2684367
5511:
5507:
5502:
5498:
5489:
5485:
5480:
5476:
5450:
5444:
5440:
5419:
5415:
5394:
5390:
5374:
5373:
5366:
5364:
5360:
5353:
5351:"Archived copy"
5349:
5348:
5344:
5327:
5323:
5310:
5306:
5297:
5290:
5265:
5252:
5244:
5237:
5229:
5222:
5212:
5210:
5205:
5204:
5200:
5177:
5168:
5158:
5156:
5144:
5137:
5132:
5127:
5084:
5077:
5074:
5049:
5045:
5030:
5026:
5017:
5013:
5008:
5005:
5004:
4984:
4980:
4978:
4975:
4974:
4957:
4953:
4944:
4940:
4931:
4927:
4925:
4922:
4921:
4901:
4897:
4888:
4884:
4879:
4876:
4875:
4852:
4848:
4839:
4835:
4834:
4824:
4820:
4811:
4807:
4792:
4788:
4779:
4775:
4771:
4769:
4757:
4753:
4744:
4740:
4732:
4729:
4728:
4718:breakdown point
4710:
4670:
4666:
4657:
4653:
4638:
4633:
4623:
4619:
4610:
4606:
4591:
4586:
4584:
4557:
4553:
4551:
4548:
4547:
4514:
4511:
4510:
4443:
4420:
4404:
4392:
4376:
4374:
4348:
4345:
4344:
4315:
4312:
4311:
4303:
4280:
4268:
4219:
4205:
4195:
4178:
4176:
4156:
4153:
4152:
4146:
4124:) evaluated at
4116:) evaluated at
4046:
4034:
3985:
3983:
3966:
3963:
3962:
3927:
3916:
3915:
3900:
3889:
3888:
3887:
3874:
3863:
3862:
3844:
3833:
3832:
3817:
3806:
3805:
3804:
3802:
3800:
3797:
3796:
3760:
3746:
3735:
3734:
3718:
3707:
3702:
3701:
3698:
3697:
3650:
3639:
3638:
3623:
3612:
3611:
3610:
3597:
3586:
3585:
3567:
3556:
3555:
3540:
3529:
3528:
3527:
3525:
3503:
3492:
3491:
3476:
3465:
3464:
3463:
3449:
3438:
3437:
3416:
3405:
3404:
3389:
3378:
3377:
3376:
3374:
3373:
3371:
3369:
3366:
3365:
3355:
3332:
3329:
3318:
3317:
3315:
3314:
3308:
3292:
3289:
3280:
3279:
3277:
3276:
3270:
3234:
3223:goodness-of-fit
3186:and the normal
3180:
3158:
3154:
3152:
3149:
3148:
3131:
3127:
3125:
3122:
3121:
3104:
3100:
3098:
3095:
3094:
3073:
3069:
3067:
3064:
3063:
3046:
3042:
3040:
3037:
3036:
3019:
3015:
3013:
3010:
3009:
2983:
2979:
2958:
2954:
2933:
2929:
2918:
2915:
2914:
2894:
2890:
2888:
2885:
2884:
2818:
2795:
2793:
2781:
2777:
2766:
2763:
2762:
2736:
2731:
2728:
2727:
2682:
2676:
2672:
2665:
2663:
2660:
2659:
2636:
2632:
2630:
2627:
2626:
2605:
2601:
2599:
2596:
2595:
2578:
2574:
2572:
2569:
2568:
2551:
2547:
2545:
2542:
2541:
2525:
2522:
2521:
2485:
2481:
2479:
2476:
2475:
2472:sample variance
2454:
2450:
2441:
2437:
2435:
2432:
2431:
2410:
2406:
2404:
2401:
2400:
2383:
2382:
2373:
2369:
2355:
2333:
2324:
2320:
2285:
2279:
2275:
2273:
2258:
2254:
2249:
2239:
2235:
2233:
2226:
2220:
2216:
2212:
2210:
2207:
2206:
2200:sample skewness
2175:
2171:
2169:
2166:
2165:
2163:
2152:
2114:
2112:
2109:
2108:
2079:
2075:
2064:
2060:
2050:
2041:
2037:
2028:
2017:
2001:
2000:
1996:
1995:
1988:
1984:
1974:
1965:
1961:
1952:
1941:
1925:
1924:
1922:
1907:
1903:
1898:
1888:
1884:
1882:
1873:
1869:
1867:
1864:
1863:
1834:
1830:
1819:
1815:
1805:
1796:
1792:
1783:
1772:
1754:
1748:
1747:
1743:
1742:
1735:
1731:
1721:
1712:
1708:
1699:
1688:
1672:
1671:
1669:
1658:
1654:
1648:
1644:
1642:
1633:
1629:
1627:
1624:
1623:
1613:
1611:Sample skewness
1548:
1540:
1531:
1527:
1490: and
1487:
1477:
1469:
1460:
1456:
1421:
1418:
1417:
1357: for
1354:
1345:
1341:
1323:
1319:
1314:
1311:
1310:
1304:
1285:
1284:
1273:
1269:
1262:
1258:
1249:
1245:
1227:
1223:
1213:
1211:
1202:
1201:
1193:
1189:
1182:
1178:
1142:
1138:
1108:
1104:
1094:
1092:
1083:
1082:
1074:
1070:
1063:
1059:
1035:
1031:
1016:
1012:
985:
981:
971:
969:
960:
959:
949:
929:
927:
923:
922:
918:
905:
899:
888:
887:
886:
882:
880:
877:
876:
856:
849:
826:
813:
766:
762:
757:
747:
743:
741:
725:
721:
717:
706:
702:
689:
685:
675:
663:
659:
646:
642:
635:
633:
622:
618:
612:
608:
606:
593:
573:
571:
567:
566:
562:
547:
536:
535:
534:
525:
521:
519:
516:
515:
495:
484:
483:
482:
480:
477:
476:
452:
448:
446:
443:
442:
439:
434:
354:
351:
350:
330:
327:
326:
306:
303:
302:
280:
263:
260:
259:
240:
199:
182:
179:
178:
132:
131:
102:
101:
89:
69:random variable
35:
28:
23:
22:
15:
12:
11:
5:
8419:
8409:
8408:
8403:
8386:
8385:
8383:
8382:
8377:
8372:
8366:
8361:
8354:
8351:
8350:
8348:
8347:
8342:
8337:
8332:
8327:
8322:
8317:
8315:central moment
8312:
8306:
8303:
8302:
8295:
8293:
8291:
8290:
8285:
8279:
8273:
8266:
8263:
8262:
8254:
8253:
8246:
8239:
8231:
8222:
8221:
8219:
8218:
8206:
8194:
8180:
8167:
8164:
8163:
8160:
8159:
8156:
8155:
8153:
8152:
8147:
8142:
8137:
8132:
8126:
8124:
8118:
8117:
8115:
8114:
8109:
8104:
8099:
8094:
8089:
8084:
8079:
8074:
8069:
8063:
8061:
8055:
8054:
8052:
8051:
8046:
8041:
8032:
8027:
8022:
8016:
8014:
8008:
8007:
8005:
8004:
7999:
7994:
7985:
7983:Bioinformatics
7979:
7977:
7967:
7966:
7954:
7953:
7950:
7949:
7946:
7945:
7942:
7941:
7939:
7938:
7932:
7930:
7926:
7925:
7923:
7922:
7916:
7914:
7908:
7907:
7905:
7904:
7899:
7894:
7889:
7883:
7881:
7872:
7866:
7865:
7862:
7861:
7859:
7858:
7853:
7848:
7843:
7838:
7832:
7830:
7824:
7823:
7821:
7820:
7815:
7810:
7802:
7797:
7792:
7791:
7790:
7788:partial (PACF)
7779:
7777:
7771:
7770:
7768:
7767:
7762:
7757:
7749:
7744:
7738:
7736:
7735:Specific tests
7732:
7731:
7729:
7728:
7723:
7718:
7713:
7708:
7703:
7698:
7693:
7687:
7685:
7678:
7672:
7671:
7669:
7668:
7667:
7666:
7665:
7664:
7649:
7648:
7647:
7637:
7635:Classification
7632:
7627:
7622:
7617:
7612:
7607:
7601:
7599:
7593:
7592:
7590:
7589:
7584:
7582:McNemar's test
7579:
7574:
7569:
7564:
7558:
7556:
7546:
7545:
7521:
7520:
7517:
7516:
7513:
7512:
7510:
7509:
7504:
7499:
7494:
7488:
7486:
7480:
7479:
7477:
7476:
7460:
7454:
7452:
7446:
7445:
7443:
7442:
7437:
7432:
7427:
7422:
7420:Semiparametric
7417:
7412:
7406:
7404:
7400:
7399:
7397:
7396:
7391:
7386:
7381:
7375:
7373:
7367:
7366:
7364:
7363:
7358:
7353:
7348:
7343:
7337:
7335:
7329:
7328:
7326:
7325:
7320:
7315:
7310:
7304:
7302:
7292:
7291:
7288:
7287:
7282:
7276:
7268:
7267:
7264:
7263:
7260:
7259:
7257:
7256:
7255:
7254:
7244:
7239:
7234:
7233:
7232:
7227:
7216:
7214:
7208:
7207:
7204:
7203:
7201:
7200:
7195:
7194:
7193:
7185:
7177:
7161:
7158:(MannâWhitney)
7153:
7152:
7151:
7138:
7137:
7136:
7125:
7123:
7117:
7116:
7114:
7113:
7112:
7111:
7106:
7101:
7091:
7086:
7083:(ShapiroâWilk)
7078:
7073:
7068:
7063:
7058:
7050:
7044:
7042:
7036:
7035:
7033:
7032:
7024:
7015:
7003:
6997:
6995:Specific tests
6991:
6990:
6987:
6986:
6984:
6983:
6978:
6973:
6967:
6965:
6959:
6958:
6956:
6955:
6950:
6949:
6948:
6938:
6937:
6936:
6926:
6920:
6918:
6912:
6911:
6909:
6908:
6907:
6906:
6901:
6891:
6886:
6881:
6876:
6871:
6865:
6863:
6857:
6856:
6854:
6853:
6848:
6847:
6846:
6841:
6840:
6839:
6834:
6819:
6818:
6817:
6812:
6807:
6802:
6791:
6789:
6780:
6774:
6773:
6771:
6770:
6765:
6760:
6759:
6758:
6748:
6743:
6742:
6741:
6731:
6730:
6729:
6724:
6719:
6709:
6704:
6699:
6698:
6697:
6692:
6687:
6671:
6670:
6669:
6664:
6659:
6649:
6648:
6647:
6642:
6632:
6631:
6630:
6620:
6619:
6618:
6608:
6603:
6598:
6592:
6590:
6580:
6579:
6567:
6566:
6563:
6562:
6559:
6558:
6556:
6555:
6550:
6545:
6540:
6534:
6532:
6526:
6525:
6523:
6522:
6517:
6512:
6506:
6504:
6500:
6499:
6497:
6496:
6491:
6486:
6481:
6476:
6471:
6466:
6460:
6458:
6452:
6451:
6449:
6448:
6446:Standard error
6443:
6438:
6433:
6432:
6431:
6426:
6415:
6413:
6407:
6406:
6404:
6403:
6398:
6393:
6388:
6383:
6378:
6376:Optimal design
6373:
6368:
6362:
6360:
6350:
6349:
6337:
6336:
6333:
6332:
6329:
6328:
6326:
6325:
6320:
6315:
6310:
6305:
6300:
6295:
6290:
6285:
6280:
6275:
6270:
6265:
6260:
6255:
6249:
6247:
6241:
6240:
6238:
6237:
6232:
6231:
6230:
6225:
6215:
6210:
6204:
6202:
6196:
6195:
6193:
6192:
6187:
6182:
6176:
6174:
6173:Summary tables
6170:
6169:
6167:
6166:
6160:
6158:
6152:
6151:
6148:
6147:
6145:
6144:
6143:
6142:
6137:
6132:
6122:
6116:
6114:
6108:
6107:
6105:
6104:
6099:
6094:
6089:
6084:
6079:
6074:
6068:
6066:
6060:
6059:
6057:
6056:
6051:
6046:
6045:
6044:
6039:
6034:
6029:
6024:
6019:
6014:
6009:
6007:Contraharmonic
6004:
5999:
5988:
5986:
5977:
5967:
5966:
5954:
5953:
5951:
5950:
5945:
5939:
5936:
5935:
5928:
5927:
5920:
5913:
5905:
5899:
5898:
5893:
5887:
5881:
5861:
5860:External links
5858:
5856:
5855:
5850:
5847:
5843:
5822:
5816:
5802:
5800:
5797:
5794:
5793:
5754:
5741:
5724:
5705:(3): 186â189.
5689:
5674:
5662:
5643:(4): 391â399.
5620:
5598:
5572:
5552:
5550:p. 3 and p. 40
5540:
5505:
5496:
5483:
5474:
5438:
5413:
5388:
5363:on 5 July 2010
5342:
5321:
5304:
5288:
5277:(1): 183â189.
5250:
5248:, FXSolver.com
5235:
5220:
5198:
5166:
5134:
5133:
5131:
5128:
5126:
5123:
5122:
5121:
5116:
5111:
5106:
5101:
5096:
5090:
5089:
5073:
5070:
5057:
5052:
5048:
5044:
5041:
5038:
5033:
5029:
5025:
5020:
5016:
5012:
4987:
4983:
4960:
4956:
4952:
4947:
4943:
4939:
4934:
4930:
4909:
4904:
4900:
4896:
4891:
4887:
4883:
4872:
4871:
4855:
4851:
4847:
4842:
4838:
4832:
4827:
4823:
4819:
4814:
4810:
4806:
4803:
4800:
4795:
4791:
4787:
4782:
4778:
4774:
4768:
4765:
4760:
4756:
4752:
4747:
4743:
4739:
4736:
4709:
4706:
4705:
4704:
4693:
4687:
4684:
4681:
4678:
4673:
4669:
4665:
4660:
4656:
4652:
4647:
4644:
4641:
4637:
4631:
4626:
4622:
4618:
4613:
4609:
4605:
4600:
4597:
4594:
4590:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4560:
4556:
4524:
4521:
4518:
4483:
4482:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4445: if
4439:
4436:
4433:
4430:
4426:
4423:
4419:
4416:
4413:
4410:
4407:
4402:
4398:
4395:
4391:
4388:
4385:
4382:
4379:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4325:
4322:
4319:
4279:
4276:
4267:
4264:
4256:absolute value
4244:
4243:
4232:
4226:
4222:
4218:
4215:
4212:
4208:
4204:
4201:
4198:
4193:
4190:
4187:
4184:
4181:
4175:
4172:
4169:
4166:
4163:
4160:
4145:
4142:
4094:
4093:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4044:
4041:
4037:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3982:
3979:
3976:
3973:
3970:
3956:
3955:
3944:
3937:
3934:
3930:
3926:
3923:
3919:
3914:
3910:
3907:
3903:
3899:
3896:
3892:
3884:
3881:
3877:
3873:
3870:
3866:
3861:
3858:
3854:
3851:
3847:
3843:
3840:
3836:
3831:
3827:
3824:
3820:
3816:
3813:
3809:
3767:
3763:
3759:
3756:
3753:
3749:
3745:
3742:
3738:
3733:
3728:
3725:
3721:
3717:
3714:
3710:
3706:
3679:
3678:
3667:
3660:
3657:
3653:
3649:
3646:
3642:
3637:
3633:
3630:
3626:
3622:
3619:
3615:
3607:
3604:
3600:
3596:
3593:
3589:
3584:
3581:
3577:
3574:
3570:
3566:
3563:
3559:
3554:
3550:
3547:
3543:
3539:
3536:
3532:
3524:
3518:
3513:
3510:
3506:
3502:
3499:
3495:
3490:
3486:
3483:
3479:
3475:
3472:
3468:
3459:
3456:
3452:
3448:
3445:
3441:
3436:
3431:
3426:
3423:
3419:
3415:
3412:
3408:
3403:
3399:
3396:
3392:
3388:
3385:
3381:
3354:
3351:
3343:
3342:
3307:
3304:
3303:
3302:
3269:
3266:
3240:Comparison of
3233:
3230:
3226:normality test
3179:
3176:
3161:
3157:
3134:
3130:
3107:
3103:
3089:are generally
3076:
3072:
3049:
3045:
3022:
3018:
3006:
3005:
2994:
2991:
2986:
2982:
2978:
2975:
2972:
2969:
2966:
2961:
2957:
2953:
2950:
2947:
2944:
2941:
2936:
2932:
2928:
2925:
2922:
2897:
2893:
2881:
2880:
2869:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2792:
2789:
2784:
2780:
2776:
2773:
2770:
2743:
2739:
2735:
2711:
2708:
2705:
2702:
2699:
2696:
2690:
2686:
2679:
2675:
2669:
2647:
2644:
2639:
2635:
2608:
2604:
2581:
2577:
2554:
2550:
2529:
2488:
2484:
2457:
2453:
2449:
2444:
2440:
2413:
2409:
2397:
2396:
2381:
2376:
2372:
2364:
2361:
2358:
2353:
2350:
2347:
2344:
2341:
2338:
2332:
2327:
2323:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2282:
2278:
2272:
2265:
2261:
2257:
2252:
2248:
2242:
2238:
2232:
2229:
2227:
2223:
2219:
2215:
2214:
2178:
2174:
2161:
2150:
2121:
2118:
2105:
2104:
2093:
2086:
2082:
2078:
2073:
2067:
2063:
2057:
2054:
2049:
2044:
2040:
2036:
2031:
2026:
2023:
2020:
2016:
2009:
2006:
1999:
1991:
1987:
1981:
1978:
1973:
1968:
1964:
1960:
1955:
1950:
1947:
1944:
1940:
1933:
1930:
1921:
1914:
1910:
1906:
1901:
1897:
1891:
1887:
1881:
1876:
1872:
1857:
1856:
1841:
1837:
1833:
1828:
1822:
1818:
1812:
1809:
1804:
1799:
1795:
1791:
1786:
1781:
1778:
1775:
1771:
1763:
1760:
1757:
1753:
1746:
1738:
1734:
1728:
1725:
1720:
1715:
1711:
1707:
1702:
1697:
1694:
1691:
1687:
1680:
1677:
1668:
1661:
1657:
1651:
1647:
1641:
1636:
1632:
1612:
1609:
1608:
1607:
1600:
1593:
1586:
1572:
1571:
1560:
1557:
1547:
1543:
1537:
1534:
1530:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1486:
1476:
1472:
1466:
1463:
1459:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1411:
1410:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1372:
1369:
1366:
1363:
1351:
1348:
1344:
1340:
1336:
1332:
1329:
1326:
1322:
1318:
1303:
1300:
1299:
1298:
1283:
1276:
1272:
1265:
1261:
1257:
1252:
1248:
1244:
1241:
1238:
1235:
1230:
1226:
1222:
1219:
1216:
1210:
1207:
1205:
1203:
1196:
1192:
1185:
1181:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1145:
1141:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1111:
1107:
1103:
1100:
1097:
1091:
1088:
1086:
1084:
1077:
1073:
1066:
1062:
1058:
1055:
1052:
1049:
1046:
1043:
1038:
1034:
1030:
1027:
1024:
1019:
1015:
1011:
1008:
1005:
1002:
999:
996:
993:
988:
984:
980:
977:
974:
968:
965:
963:
961:
957:
952:
947:
942:
938:
935:
932:
926:
921:
917:
914:
911:
908:
906:
902:
895:
892:
885:
884:
868:is finite and
854:
847:
822:
816:central moment
811:
789:
788:
773:
769:
765:
760:
756:
750:
746:
740:
732:
728:
724:
720:
715:
709:
705:
701:
698:
695:
692:
688:
684:
681:
678:
672:
666:
662:
658:
655:
652:
649:
645:
641:
638:
632:
625:
621:
615:
611:
605:
601:
596:
591:
586:
582:
579:
576:
570:
565:
561:
558:
555:
550:
543:
540:
533:
528:
524:
511:, defined as:
498:
491:
488:
455:
451:
438:
435:
433:
430:
358:
334:
310:
290:
287:
283:
279:
276:
273:
270:
267:
239:
236:
209:
206:
202:
198:
195:
192:
189:
186:
159:
158:
128:
88:
85:
32:Graph skewness
26:
9:
6:
4:
3:
2:
8418:
8407:
8404:
8402:
8399:
8398:
8396:
8381:
8378:
8376:
8373:
8370:
8367:
8365:
8362:
8359:
8356:
8355:
8352:
8346:
8343:
8341:
8338:
8336:
8333:
8331:
8328:
8326:
8323:
8321:
8318:
8316:
8313:
8311:
8308:
8307:
8304:
8299:
8289:
8286:
8283:
8280:
8277:
8274:
8271:
8268:
8267:
8264:
8260:
8252:
8247:
8245:
8240:
8238:
8233:
8232:
8229:
8217:
8216:
8207:
8205:
8204:
8195:
8193:
8192:
8187:
8181:
8179:
8178:
8169:
8168:
8165:
8151:
8148:
8146:
8145:Geostatistics
8143:
8141:
8138:
8136:
8133:
8131:
8128:
8127:
8125:
8123:
8119:
8113:
8112:Psychometrics
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8090:
8088:
8085:
8083:
8080:
8078:
8075:
8073:
8070:
8068:
8065:
8064:
8062:
8060:
8056:
8050:
8047:
8045:
8042:
8040:
8036:
8033:
8031:
8028:
8026:
8023:
8021:
8018:
8017:
8015:
8013:
8009:
8003:
8000:
7998:
7995:
7993:
7989:
7986:
7984:
7981:
7980:
7978:
7976:
7975:Biostatistics
7972:
7968:
7964:
7959:
7955:
7937:
7936:Log-rank test
7934:
7933:
7931:
7927:
7921:
7918:
7917:
7915:
7913:
7909:
7903:
7900:
7898:
7895:
7893:
7890:
7888:
7885:
7884:
7882:
7880:
7876:
7873:
7871:
7867:
7857:
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7833:
7831:
7829:
7825:
7819:
7816:
7814:
7811:
7809:
7807:(BoxâJenkins)
7803:
7801:
7798:
7796:
7793:
7789:
7786:
7785:
7784:
7781:
7780:
7778:
7776:
7772:
7766:
7763:
7761:
7760:DurbinâWatson
7758:
7756:
7750:
7748:
7745:
7743:
7742:DickeyâFuller
7740:
7739:
7737:
7733:
7727:
7724:
7722:
7719:
7717:
7716:Cointegration
7714:
7712:
7709:
7707:
7704:
7702:
7699:
7697:
7694:
7692:
7691:Decomposition
7689:
7688:
7686:
7682:
7679:
7677:
7673:
7663:
7660:
7659:
7658:
7655:
7654:
7653:
7650:
7646:
7643:
7642:
7641:
7638:
7636:
7633:
7631:
7628:
7626:
7623:
7621:
7618:
7616:
7613:
7611:
7608:
7606:
7603:
7602:
7600:
7598:
7594:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7565:
7563:
7562:Cohen's kappa
7560:
7559:
7557:
7555:
7551:
7547:
7543:
7539:
7535:
7531:
7526:
7522:
7508:
7505:
7503:
7500:
7498:
7495:
7493:
7490:
7489:
7487:
7485:
7481:
7475:
7471:
7467:
7461:
7459:
7456:
7455:
7453:
7451:
7447:
7441:
7438:
7436:
7433:
7431:
7428:
7426:
7423:
7421:
7418:
7416:
7415:Nonparametric
7413:
7411:
7408:
7407:
7405:
7401:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7376:
7374:
7372:
7368:
7362:
7359:
7357:
7354:
7352:
7349:
7347:
7344:
7342:
7339:
7338:
7336:
7334:
7330:
7324:
7321:
7319:
7316:
7314:
7311:
7309:
7306:
7305:
7303:
7301:
7297:
7293:
7286:
7283:
7281:
7278:
7277:
7273:
7269:
7253:
7250:
7249:
7248:
7245:
7243:
7240:
7238:
7235:
7231:
7228:
7226:
7223:
7222:
7221:
7218:
7217:
7215:
7213:
7209:
7199:
7196:
7192:
7186:
7184:
7178:
7176:
7170:
7169:
7168:
7165:
7164:Nonparametric
7162:
7160:
7154:
7150:
7147:
7146:
7145:
7139:
7135:
7134:Sample median
7132:
7131:
7130:
7127:
7126:
7124:
7122:
7118:
7110:
7107:
7105:
7102:
7100:
7097:
7096:
7095:
7092:
7090:
7087:
7085:
7079:
7077:
7074:
7072:
7069:
7067:
7064:
7062:
7059:
7057:
7055:
7051:
7049:
7046:
7045:
7043:
7041:
7037:
7031:
7029:
7025:
7023:
7021:
7016:
7014:
7009:
7005:
7004:
7001:
6998:
6996:
6992:
6982:
6979:
6977:
6974:
6972:
6969:
6968:
6966:
6964:
6960:
6954:
6951:
6947:
6944:
6943:
6942:
6939:
6935:
6932:
6931:
6930:
6927:
6925:
6922:
6921:
6919:
6917:
6913:
6905:
6902:
6900:
6897:
6896:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6870:
6867:
6866:
6864:
6862:
6858:
6852:
6849:
6845:
6842:
6838:
6835:
6833:
6830:
6829:
6828:
6825:
6824:
6823:
6820:
6816:
6813:
6811:
6808:
6806:
6803:
6801:
6798:
6797:
6796:
6793:
6792:
6790:
6788:
6784:
6781:
6779:
6775:
6769:
6766:
6764:
6761:
6757:
6754:
6753:
6752:
6749:
6747:
6744:
6740:
6739:loss function
6737:
6736:
6735:
6732:
6728:
6725:
6723:
6720:
6718:
6715:
6714:
6713:
6710:
6708:
6705:
6703:
6700:
6696:
6693:
6691:
6688:
6686:
6680:
6677:
6676:
6675:
6672:
6668:
6665:
6663:
6660:
6658:
6655:
6654:
6653:
6650:
6646:
6643:
6641:
6638:
6637:
6636:
6633:
6629:
6626:
6625:
6624:
6621:
6617:
6614:
6613:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6594:
6593:
6591:
6589:
6585:
6581:
6577:
6572:
6568:
6554:
6551:
6549:
6546:
6544:
6541:
6539:
6536:
6535:
6533:
6531:
6527:
6521:
6518:
6516:
6513:
6511:
6508:
6507:
6505:
6501:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6475:
6472:
6470:
6467:
6465:
6462:
6461:
6459:
6457:
6453:
6447:
6444:
6442:
6441:Questionnaire
6439:
6437:
6434:
6430:
6427:
6425:
6422:
6421:
6420:
6417:
6416:
6414:
6412:
6408:
6402:
6399:
6397:
6394:
6392:
6389:
6387:
6384:
6382:
6379:
6377:
6374:
6372:
6369:
6367:
6364:
6363:
6361:
6359:
6355:
6351:
6347:
6342:
6338:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6281:
6279:
6276:
6274:
6271:
6269:
6268:Control chart
6266:
6264:
6261:
6259:
6256:
6254:
6251:
6250:
6248:
6246:
6242:
6236:
6233:
6229:
6226:
6224:
6221:
6220:
6219:
6216:
6214:
6211:
6209:
6206:
6205:
6203:
6201:
6197:
6191:
6188:
6186:
6183:
6181:
6178:
6177:
6175:
6171:
6165:
6162:
6161:
6159:
6157:
6153:
6141:
6138:
6136:
6133:
6131:
6128:
6127:
6126:
6123:
6121:
6118:
6117:
6115:
6113:
6109:
6103:
6100:
6098:
6095:
6093:
6090:
6088:
6085:
6083:
6080:
6078:
6075:
6073:
6070:
6069:
6067:
6065:
6061:
6055:
6052:
6050:
6047:
6043:
6040:
6038:
6035:
6033:
6030:
6028:
6025:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6003:
6000:
5998:
5995:
5994:
5993:
5990:
5989:
5987:
5985:
5981:
5978:
5976:
5972:
5968:
5964:
5959:
5955:
5949:
5946:
5944:
5941:
5940:
5937:
5933:
5926:
5921:
5919:
5914:
5912:
5907:
5906:
5903:
5897:
5894:
5891:
5888:
5885:
5882:
5878:
5874:
5873:
5868:
5864:
5863:
5854:
5851:
5848:
5844:
5840:
5836:
5832:
5828:
5823:
5819:
5817:0-471-58495-9
5813:
5809:
5804:
5803:
5789:
5785:
5781:
5777:
5773:
5769:
5765:
5758:
5751:
5745:
5738:
5734:
5728:
5720:
5716:
5712:
5708:
5704:
5700:
5693:
5687:
5686:, 62, 101â111
5685:
5678:
5669:
5667:
5658:
5654:
5650:
5646:
5642:
5638:
5631:
5629:
5627:
5625:
5617:. p. 21.
5616:
5609:
5602:
5587:
5583:
5576:
5570:
5569:0-12-751965-3
5566:
5562:
5556:
5549:
5544:
5536:
5532:
5528:
5524:
5521:(2): 97â102.
5520:
5516:
5509:
5500:
5493:
5487:
5478:
5469:
5464:
5460:
5456:
5449:
5442:
5433:
5432:
5427:
5424:
5417:
5408:
5407:
5402:
5399:
5392:
5384:
5378:
5359:
5352:
5346:
5339:
5338:0-85264-141-9
5335:
5331:
5325:
5318:
5317:9780415172042
5314:
5308:
5301:
5295:
5293:
5284:
5280:
5276:
5272:
5271:
5263:
5261:
5259:
5257:
5255:
5247:
5242:
5240:
5232:
5227:
5225:
5208:
5202:
5194:
5190:
5186:
5182:
5175:
5173:
5171:
5155:
5154:
5149:
5142:
5140:
5135:
5120:
5119:Skewness risk
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5100:
5097:
5095:
5092:
5091:
5087:
5081:
5076:
5069:
5050:
5046:
5042:
5039:
5036:
5031:
5027:
5023:
5018:
5014:
5003:
4985:
4981:
4958:
4954:
4950:
4945:
4941:
4937:
4932:
4928:
4902:
4898:
4894:
4889:
4885:
4853:
4849:
4845:
4840:
4836:
4825:
4821:
4817:
4812:
4808:
4801:
4793:
4789:
4785:
4780:
4776:
4766:
4758:
4754:
4750:
4745:
4741:
4734:
4727:
4726:
4725:
4723:
4719:
4715:
4691:
4682:
4679:
4676:
4671:
4667:
4663:
4658:
4654:
4645:
4642:
4639:
4635:
4624:
4620:
4616:
4611:
4607:
4598:
4595:
4592:
4588:
4581:
4578:
4575:
4569:
4563:
4558:
4554:
4546:
4545:
4544:
4542:
4538:
4522:
4519:
4516:
4508:
4504:
4500:
4496:
4492:
4488:
4469:
4466:
4460:
4457:
4454:
4434:
4431:
4428:
4424:
4421:
4417:
4414:
4408:
4396:
4393:
4389:
4386:
4380:
4371:
4368:
4365:
4359:
4353:
4350:
4343:
4342:
4341:
4339:
4320:
4309:
4302:
4298:
4294:
4290:
4286:
4275:
4273:
4263:
4261:
4257:
4253:
4250:is the mean,
4249:
4230:
4216:
4213:
4210:
4199:
4188:
4185:
4182:
4173:
4167:
4161:
4158:
4151:
4150:
4149:
4141:
4137:
4135:
4131:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4096:The function
4074:
4071:
4068:
4062:
4059:
4053:
4047:
4039:
4035:
4031:
4025:
4022:
4019:
4013:
4010:
4007:
4001:
3998:
3992:
3986:
3980:
3974:
3968:
3961:
3960:
3959:
3942:
3932:
3928:
3924:
3917:
3912:
3905:
3901:
3897:
3890:
3879:
3875:
3871:
3864:
3859:
3856:
3849:
3845:
3841:
3834:
3829:
3822:
3818:
3814:
3807:
3795:
3794:
3793:
3790:
3787:
3785:
3781:
3765:
3761:
3751:
3747:
3743:
3736:
3731:
3723:
3719:
3715:
3708:
3696:
3692:
3688:
3684:
3665:
3655:
3651:
3647:
3640:
3635:
3628:
3624:
3620:
3613:
3602:
3598:
3594:
3587:
3582:
3579:
3572:
3568:
3564:
3557:
3552:
3545:
3541:
3537:
3530:
3522:
3516:
3508:
3504:
3500:
3493:
3488:
3481:
3477:
3473:
3466:
3454:
3450:
3446:
3439:
3434:
3429:
3421:
3417:
3413:
3406:
3401:
3394:
3390:
3386:
3379:
3364:
3363:
3362:
3360:
3350:
3348:
3335:
3326:
3322:
3313:
3312:
3311:
3295:
3287:
3283:
3275:
3274:
3273:
3265:
3263:
3255:
3251:
3247:
3243:
3238:
3229:
3227:
3224:
3220:
3216:
3212:
3210:
3206:
3205:value at risk
3201:
3199:
3194:
3191:
3189:
3188:quantile plot
3185:
3175:
3159:
3155:
3132:
3128:
3105:
3101:
3092:
3074:
3070:
3047:
3043:
3020:
3016:
2992:
2984:
2980:
2973:
2970:
2967:
2959:
2955:
2948:
2945:
2942:
2934:
2930:
2923:
2920:
2913:
2912:
2911:
2895:
2891:
2867:
2858:
2855:
2852:
2843:
2840:
2837:
2828:
2825:
2822:
2811:
2808:
2805:
2799:
2796:
2790:
2782:
2778:
2771:
2768:
2761:
2760:
2759:
2757:
2741:
2737:
2733:
2725:
2706:
2703:
2700:
2694:
2688:
2684:
2677:
2673:
2667:
2645:
2642:
2637:
2633:
2624:
2606:
2602:
2579:
2575:
2552:
2548:
2527:
2518:
2516:
2512:
2508:
2504:
2486:
2482:
2473:
2455:
2451:
2447:
2442:
2438:
2429:
2411:
2407:
2379:
2374:
2370:
2362:
2359:
2356:
2348:
2345:
2342:
2336:
2330:
2325:
2321:
2310:
2307:
2304:
2295:
2292:
2289:
2280:
2276:
2270:
2263:
2259:
2255:
2250:
2246:
2240:
2236:
2230:
2228:
2221:
2217:
2205:
2204:
2203:
2201:
2196:
2195:estimator.
2194:
2176:
2172:
2160:
2156:
2149:
2145:
2141:
2137:
2116:
2091:
2084:
2080:
2076:
2071:
2065:
2052:
2047:
2042:
2038:
2029:
2024:
2021:
2018:
2014:
2007:
2004:
1997:
1989:
1976:
1971:
1966:
1962:
1953:
1948:
1945:
1942:
1938:
1931:
1928:
1919:
1912:
1908:
1904:
1899:
1895:
1889:
1885:
1879:
1874:
1870:
1862:
1861:
1860:
1839:
1835:
1831:
1826:
1820:
1807:
1802:
1797:
1793:
1784:
1779:
1776:
1773:
1769:
1761:
1758:
1755:
1751:
1744:
1736:
1723:
1718:
1713:
1709:
1700:
1695:
1692:
1689:
1685:
1678:
1675:
1666:
1659:
1655:
1649:
1645:
1639:
1634:
1630:
1622:
1621:
1620:
1618:
1605:
1601:
1598:
1594:
1591:
1587:
1584:
1580:
1579:
1578:
1575:
1558:
1555:
1545:
1541:
1535:
1532:
1524:
1521:
1518:
1512:
1506:
1503:
1500:
1484:
1474:
1470:
1464:
1461:
1453:
1450:
1447:
1441:
1435:
1432:
1429:
1416:
1415:
1414:
1397:
1394:
1388:
1385:
1382:
1370:
1367:
1364:
1361:
1349:
1346:
1342:
1338:
1334:
1330:
1327:
1324:
1320:
1309:
1308:
1307:
1281:
1274:
1270:
1263:
1259:
1255:
1250:
1246:
1242:
1239:
1236:
1228:
1224:
1217:
1208:
1206:
1194:
1190:
1183:
1179:
1175:
1166:
1160:
1154:
1151:
1143:
1139:
1132:
1123:
1120:
1117:
1109:
1105:
1098:
1089:
1087:
1075:
1071:
1064:
1060:
1056:
1050:
1044:
1036:
1032:
1028:
1025:
1017:
1013:
1006:
1000:
997:
994:
986:
982:
975:
966:
964:
955:
950:
945:
940:
936:
933:
930:
924:
919:
915:
909:
907:
900:
890:
875:
874:
873:
871:
867:
862:
860:
853:
846:
842:
838:
834:
830:
825:
821:
817:
814:is the third
810:
806:
802:
798:
795:is the mean,
794:
771:
767:
763:
758:
754:
748:
744:
738:
730:
726:
722:
713:
707:
699:
696:
693:
686:
682:
670:
664:
656:
653:
650:
643:
639:
630:
623:
619:
613:
609:
603:
599:
594:
589:
584:
580:
577:
574:
568:
563:
559:
553:
548:
538:
531:
526:
522:
514:
513:
512:
496:
486:
475:
472:is the third
471:
453:
449:
441:The skewness
429:
421:
416:
414:
410:
406:
400:
397:
395:
391:
387:
383:
379:
374:
372:
356:
348:
332:
324:
308:
288:
285:
281:
274:
271:
268:
258:, defined as
257:
248:
244:
230:
226:
222:
207:
204:
200:
193:
190:
187:
177:, defined as
176:
172:
166:
164:
156:
152:
148:
144:
140:
136:
133:positive skew
129:
126:
125:right-leaning
122:
118:
114:
110:
106:
103:negative skew
99:
98:
97:
95:
84:
81:
77:
72:
70:
66:
62:
58:
54:
50:
41:
37:
33:
19:
8334:
8213:
8201:
8182:
8175:
8087:Econometrics
8037: /
8020:Chemometrics
7997:Epidemiology
7990: /
7963:Applications
7805:ARIMA model
7752:Q-statistic
7701:Stationarity
7597:Multivariate
7540: /
7536: /
7534:Multivariate
7532: /
7472: /
7468: /
7242:Bayes factor
7141:Signed rank
7053:
7027:
7019:
7007:
6702:Completeness
6538:Cohort study
6436:Opinion poll
6371:Missing data
6358:Study design
6313:Scatter plot
6235:Scatter plot
6228:Spearman's Ï
6190:Grouped data
6139:
5870:
5830:
5826:
5807:
5771:
5767:
5757:
5749:
5744:
5732:
5727:
5702:
5698:
5692:
5682:
5677:
5640:
5636:
5614:
5601:
5589:. Retrieved
5585:
5575:
5560:
5555:
5543:
5518:
5514:
5508:
5499:
5491:
5486:
5477:
5458:
5454:
5441:
5429:
5416:
5404:
5391:
5365:. Retrieved
5358:the original
5345:
5329:
5324:
5307:
5274:
5268:
5211:. Retrieved
5201:
5193:the original
5188:
5184:
5157:. Retrieved
5151:
4873:
4711:
4540:
4506:
4502:
4498:
4494:
4490:
4486:
4484:
4337:
4307:
4300:
4296:
4292:
4288:
4284:
4281:
4269:
4251:
4247:
4245:
4147:
4138:
4133:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4097:
4095:
3957:
3791:
3788:
3682:
3680:
3358:
3356:
3344:
3309:
3271:
3262:Karl Pearson
3259:
3217:
3213:
3202:
3195:
3192:
3181:
3178:Applications
3007:
2882:
2755:
2519:
2398:
2199:
2197:
2158:
2147:
2139:
2106:
1858:
1616:
1614:
1576:
1573:
1412:
1305:
869:
865:
863:
851:
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840:
836:
828:
823:
819:
808:
796:
792:
790:
469:
440:
426:
402:
398:
375:
253:
241:
223:
167:
160:
155:left-leaning
154:
150:
146:
143:right-tailed
142:
139:right-skewed
138:
130:
124:
120:
116:
112:
108:
100:
93:
90:
87:Introduction
79:
73:
56:
46:
36:
18:Right-skewed
8215:WikiProject
8130:Cartography
8092:Jurimetrics
8044:Reliability
7775:Time domain
7754:(LjungâBox)
7676:Time-series
7554:Categorical
7538:Time-series
7530:Categorical
7465:(Bernoulli)
7300:Correlation
7280:Correlation
7076:JarqueâBera
7048:Chi-squared
6810:M-estimator
6763:Asymptotics
6707:Sufficiency
6474:Interaction
6386:Replication
6366:Effect size
6323:Violin plot
6303:Radar chart
6283:Forest plot
6273:Correlogram
6223:Kendall's Ï
5591:21 November
5461:(2): 1â18.
5332:, Griffin.
5159:21 December
3782:measure of
2136:sample mean
803:, E is the
384:, then the
113:left-tailed
109:left-skewed
8395:Categories
8310:raw moment
8257:Theory of
8082:Demography
7800:ARMA model
7605:Regression
7182:(Friedman)
7143:(Wilcoxon)
7081:Normality
7071:Lilliefors
7018:Student's
6894:Resampling
6768:Robustness
6756:divergence
6746:Efficiency
6684:(monotone)
6679:Likelihood
6596:Population
6429:Stratified
6381:Population
6200:Dependence
6156:Count data
6087:Percentile
6064:Dispersion
5997:Arithmetic
5932:Statistics
5684:Biometrika
5582:"Skewness"
5125:References
5099:Coskewness
5094:Bragg peak
4920:such that
4485:and dSkew(
3784:dispersion
2623:consistent
432:Definition
233:skewness.)
53:statistics
8380:combinant
7463:Logistic
7230:posterior
7156:Rank sum
6904:Jackknife
6899:Bootstrap
6717:Bootstrap
6652:Parameter
6601:Statistic
6396:Statistic
6308:Run chart
6293:Pie chart
6288:Histogram
6278:Fan chart
6253:Bar chart
6135:L-moments
6022:Geometric
5877:EMS Press
5788:120919149
5764:M. Hubert
5762:G. Brys;
5737:C. R. Rao
5431:MathWorld
5406:MathWorld
5340:(Ex 12.9)
5130:Citations
5040:…
4951:≥
4938:≥
4846:−
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4802:−
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4708:Medcouple
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4683:θ
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4272:L-moments
4266:L-moments
4217:ν
4214:−
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4072:−
4060:−
4020:−
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3857:−
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3489:−
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3184:histogram
3129:γ
3102:γ
2974:
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2924:
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1980:¯
1972:−
1939:∑
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1759:−
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1237:−
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1176:−
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1155:μ
1152:−
1133:
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1118:−
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1072:σ
1061:μ
1057:−
1045:
1033:μ
1007:
1001:μ
995:−
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941:σ
937:μ
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916:
894:~
891:μ
833:cumulants
755:κ
745:κ
700:μ
697:−
683:
657:μ
654:−
640:
620:σ
610:μ
585:σ
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578:−
560:
542:~
539:μ
523:γ
490:~
487:μ
450:γ
378:symmetric
357:σ
333:ν
309:μ
286:σ
275:ν
272:−
269:μ
205:σ
194:ν
191:−
188:μ
8375:cumulant
8345:L-moment
8340:kurtosis
8335:skewness
8325:variance
8177:Category
7870:Survival
7747:Johansen
7470:Binomial
7425:Isotonic
7012:(normal)
6657:location
6464:Blocking
6419:Sampling
6298:QâQ plot
6263:Box plot
6245:Graphics
6140:Skewness
6130:Kurtosis
6102:Variance
6032:Heronian
6027:Harmonic
5377:cite web
5213:18 March
5153:OpenStax
5104:Kurtosis
5072:See also
4973:, where
4425:′
4397:′
4140:chance.
4130:supremum
2685:→
2428:cumulant
1302:Examples
859:kurtosis
827:are the
382:unimodal
76:unimodal
67:-valued
57:skewness
8203:Commons
8150:Kriging
8035:Process
7992:studies
7851:Wavelet
7684:General
6851:Plug-in
6645:L space
6424:Cluster
6125:Moments
5943:Outline
5879:, 2001
5799:Sources
5719:2685210
5657:2987742
5535:2684367
5367:9 April
4501:and 2Ξâ
4270:Use of
3685:is the
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2658:, with
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2142:is the
2134:is the
799:is the
369:is the
345:is the
321:is the
171:outlier
8072:Census
7662:Normal
7610:Manova
7430:Robust
7180:2-way
7172:1-way
7010:-test
6681:
6258:Biplot
6049:Median
6042:Lehmer
5984:Center
5814:
5786:
5735:(eds.
5717:
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5567:
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5319:(p 85)
5315:
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4722:median
4310:, and
4246:where
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2724:Fisher
2399:where
2191:is a
2157:, and
2155:moment
2107:where
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818:, and
791:where
390:median
349:, and
347:median
301:where
157:curve.
127:curve.
74:For a
8371:(pgf)
8360:(mgf)
8284:(cdf)
8278:(pdf)
8272:(pmf)
7696:Trend
7225:prior
7167:anova
7056:-test
7030:-test
7022:-test
6929:Power
6874:Pivot
6667:shape
6662:scale
6112:Shape
6092:Range
6037:Heinz
6012:Cubic
5948:Index
5784:S2CID
5715:JSTOR
5653:JSTOR
5611:(PDF)
5531:JSTOR
5451:(PDF)
5361:(PDF)
5354:(PDF)
4555:dSkew
4351:dSkew
4304:'
3221:is a
2503:Excel
413:heavy
151:right
145:, or
115:, or
94:tails
63:of a
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7929:Test
7129:Sign
6981:Wald
6054:Mode
5992:Mean
5846:1â15
5812:ISBN
5593:2019
5565:ISBN
5383:link
5369:2010
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5313:ISBN
5215:2012
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