5522:
870:
the percent error will be negative. This negative result provides additional information about the experimental result. For example, experimentally calculating the speed of light and coming up with a negative percent error says that the experimental value is a velocity that is less than the speed of light. This is a big difference from getting a positive percent error, which means the experimental value is a velocity that is greater than the speed of light (violating the
4629:. Since the derivative of the natural log at 1 is 1, log points are approximately equal to percent change for small differences â for example an increase of 1% equals an increase of 0.995 cNp, and a 5% increase gives a 4.88 cNp increase. This approximation property does not hold for other choices of logarithm base, which introduce a scaling factor due to the derivative not being 1. Log points can thus be used as a replacement for percent change.
4566:
346:
4365:
4637:
Using log change has the advantages of additivity compared to relative change. Specifically, when using log change, the total change after a series of changes equals the sum of the changes. With percent, summing the changes is only an approximation, with larger error for larger changes. For example:
869:
Although it is common practice to use the absolute value version of relative change when discussing percent error, in some situations, it can be beneficial to remove the absolute values to provide more information about the result. Thus, if an experimental value is less than the theoretical value,
193:
typically is uniform, e.g. across an increasing data series. If the relationship of the value with respect to the reference value (that is, larger or smaller) does not matter in a particular application, the absolute value may be used in place of the actual change in the above formula to produce a
3929:
1178:
If a bank were to raise the interest rate on a savings account from 3% to 4%, the statement that "the interest rate was increased by 1%" would be incorrect and misleading. The absolute change in this situation is 1 percentage point (4% â 3%), but the relative change in the interest rate is:
865:
could be "accepted." Experimental value is what has been derived by use of calculation and/or measurement and is having its accuracy tested against the theoretical value, a value that is accepted by the scientific community or a value that could be seen as a goal for a successful result.
1860:
4108:
853:
243:
945:
2336:
1143:
535:
3737:
when the relative error of a measurement is required. Minimum mean change has been recommended for use in econometrics. Logarithmic change has been recommended as a general-purpose replacement for relative change and is discussed more below.
3462:
3752:
2938:
2721:
433:
The domain restriction of relative change to positive numbers often poses a constraint. To avoid this problem it is common to take the absolute value, so that the relative change formula works correctly for all nonzero values of
4269:
5138:
3362:
3250:
4561:{\displaystyle \ln {\frac {V_{1}}{V_{0}}}=\int _{V_{0}}^{V_{1}}{\frac {{\mathrm {d} }V}{V}}\approx \int _{V_{0}}^{V_{1}}{\frac {{\mathrm {d} }V}{V_{0}}}={\frac {V_{1}-V_{0}}{V_{0}}}={\text{classical relative change}}}
772:
is a special case of the percentage form of relative change calculated from the absolute change between the experimental (measured) and theoretical (accepted) values, and dividing by the theoretical (accepted) value.
1702:
4936:
1254:
5019:
759:
3938:
4847:
778:
3138:
2804:
880:
194:
value for the relative change which is always non-negative. The actual difference is not usually a good way to compare the numbers, in particular because it depends on the unit of measurement. For instance,
100:
occurs in measuring situations where the reference value is the accepted or actual value (perhaps theoretically determined) and the value being compared to it is experimentally determined (by measurement).
4360:
2225:
1547:
2859:
2051:
1439:
1360:
1023:
3716:
5219:
1623:
1047:
1258:
In general, the term "percentage point(s)" indicates an absolute change or difference of percentages, while the percent sign or the word "percentage" refers to the relative change or difference.
216:
is 100, giving the impression of a larger difference. But even with constant units, the relative change helps judge the importance of the respective change. For example, an increase in price of
446:
4615:
5430:
607:
2469:
379:
is negative, hence it is not usually defined for negative reference values either. For example, we might want to calculate the relative change of â10 to â6. The above formula gives
3025:
568:
341:{\displaystyle {\text{relative change}}(v_{\text{ref}},v)={\frac {\text{actual change}}{\text{reference value}}}={\frac {\Delta v}{v_{\text{ref}}}}={\frac {v}{v_{\text{ref}}}}-1.}
2643:
3496:
5402:
2595:
2551:
1970:
3655:
3596:
2754:
2160:
425:. Corresponding values of percent change would be obtained by multiplying these values by 100 (and appending the % sign to indicate that the value is a percentage).
5362:
The log change is the unique two-variable function that is additive, and whose linearization matches relative change. There is a family of additive difference functions
4146:
3373:
3291:
3179:
4139:
3540:
2507:
2414:
2084:
1908:
1693:
5484:
5457:
2188:
2120:
539:
This still does not solve the issue when the reference is zero. It is common to instead use an indicator of relative change, and take the absolute values of both
96:
for repeated measurements where the outcomes are expected to be the same. A special case of percent change (relative change expressed as a percentage) called
2865:
2649:
1182:
612:
6063:
972:
For example, if a house is worth $ 100,000 today and the year after its value goes up to $ 110,000, the percentage change of its value can be expressed as
3924:{\displaystyle H(K,L)={\begin{cases}\int _{1}^{K/L}t^{c-1}dt&{\text{when }}K>L\\-\int _{K/L}^{1}t^{c-1}dt&{\text{when }}K<L\end{cases}}}
5042:
3297:
3185:
1855:{\displaystyle {\begin{cases}R(x,y)>0&{\text{iff }}y>x\\R(x,y)=0&{\text{iff }}y=x\\R(x,y)<0&{\text{iff }}y<x\end{cases}}.}
1479:
1377:
1298:
975:
4855:
1564:
4941:
6092:
4804:
957:. Therefore, it is vital to preserve the order as above: subtract the theoretical value from the experimental value and not vice versa.
3031:
2760:
609:, which can usually be addressed by appropriately extending the indicator. For example, for arithmetic mean this formula may be used:
857:
The terms "Experimental" and "Theoretical" used in the equation above are commonly replaced with similar terms. Other terms used for
4295:
2815:
1978:
4103:{\displaystyle H(K,L)={\begin{cases}{\frac {1}{c}}\cdot ((K/L)^{c}-1)&c\neq 0\\\ln(K/L)&c=0,K>0,L>0\end{cases}}}
3676:
5965:
5870:
5819:
5673:
5143:
231:
We can adjust the comparison to take into account the "size" of the quantities involved, by defining, for positive values of
848:{\displaystyle \%{\text{ Error}}={\frac {|{\text{Experimental}}-{\text{Theoretical}}|}{|{\text{Theoretical}}|}}\times 100.}
940:{\displaystyle \%{\text{ Error}}={\frac {{\text{Experimental}}-{\text{Theoretical}}}{|{\text{Theoretical}}|}}\times 100.}
17:
5982:
5500:
4625:(cNp) when measured for root-power quantities. This quantity has also been referred to as a log percentage and denoted
2331:{\displaystyle {\text{Relative change}}(x,y)={\frac {{\text{Actual change}}\,\Delta }{f(x,y)}}={\frac {y-x}{f(x,y)}}.}
6223:
6177:
6054:
5565:
5543:
969:
is a way to express a change in a variable. It represents the relative change between the old value and the new one.
5536:
4801:), it does not correspond to the same variation. The conversion between relative and log changes may be computed as
2197:
Usually the indicator of relative change is presented as the actual change Î scaled by some function of the values
1460:
was considered the reference value, but we could have made the choice the other way and considered the cost of car
2086:
still satisfies the other conditions besides normalization. Furthermore, due to the independence condition, every
6070:
5947:
5592:
1640:
It is the use of the words "of" and "less/more than" that distinguish between ratios and relative differences.
6074:
4573:
5407:
1695:
is a binary real-valued function defined for the domain of interest which satisfies the following properties:
1138:{\displaystyle {\text{Percentage change}}={\frac {\Delta V}{V_{1}}}={\frac {V_{2}-V_{1}}{V_{1}}}\times 100\%.}
573:
350:
The relative change is independent of the unit of measurement employed; for example, the relative change from
5896:
3598:, which means that all such indicators have a "symmetry" property that the classical relative change lacks:
2420:
1648:
The (classical) relative change above is but one of the possible measures/indicators of relative change. An
104:
The relative change formula is not well-behaved under many conditions. Various alternative formulas, called
2952:
546:
6208:
5505:
2606:
5809:
3468:
530:{\displaystyle {\text{Relative change}}(v_{\text{ref}},v)={\frac {v-v_{\text{ref}}}{|v_{\text{ref}}|}}.}
5365:
2567:
2523:
1913:
1158:
When the variable in question is a percentage itself, it is better to talk about its change by using
6155:
The CES and par production techniques, income distribution and the neoclassical theory of production
4176:
3968:
3782:
3601:
3382:
1711:
5530:
3545:
3457:{\displaystyle {\begin{cases}{\frac {y-x}{\ln {\frac {y}{x}}}}&x\neq y\\x&x=y\end{cases}}}
2732:
6099:
5547:
4279:
3730:
3261:
3149:
422:
5860:
5663:
4115:
2125:
3510:
2475:
2384:
2063:
1887:
1663:
6218:
6203:
5462:
5435:
3726:
871:
31:
5690:
8:
5495:
3734:
2165:
2097:
1167:
1163:
185:
6141:
6023:
6005:
6000:
Brauen, Silvan; Erpf, Philipp; Wasem, Micha (2020). "On
Absolute and Relative Change".
5928:
5908:
2933:{\displaystyle {\frac {\left({\frac {y}{x}}-1\right)\left(1+{\frac {x}{y}}\right)}{2}}}
2716:{\displaystyle {\frac {{\frac {y}{x}}-1}{{\frac {1}{2}}\left(1+{\frac {y}{x}}\right)}}}
85:
are also commonly used. The terms "change" and "difference" are used interchangeably.
46:
while taking into account the "sizes" of the things being compared, i.e. dividing by a
6173:
6050:
6027:
5961:
5866:
5815:
5669:
5510:
4285:
3503:
As can be seen in the table, all but the first two indicators have, as denominator a
877:
The percent error equation, when rewritten by removing the absolute values, becomes:
89:
5932:
1374:. It is also common to express the comparison as a ratio, which in this example is,
6133:
6118:
6015:
5953:
5918:
5895:
Karjus, Andres; Blythe, Richard A.; Kirby, Simon; Smith, Kenny (10 February 2020).
5351:
1159:
4264:{\displaystyle H(K,L)={\begin{cases}(K-L)/K&c=-1\\(K-L)/L&c=1\end{cases}}}
6213:
93:
6164:
5957:
189:
is sometimes also used even though the absolute value is not taken; the sign of
5640:, p. 11: "We suggest that this indicator should be used more extensively."
5022:
3722:
3665:
should have the same magnitude as a relative change in the opposite direction,
2340:
As with classical relative change, the general relative change is undefined if
190:
5923:
5133:{\displaystyle {\frac {V_{1}-V_{0}}{V_{0}}}\neq -{\frac {V_{0}-V_{1}}{V_{1}}}}
3357:{\displaystyle {\frac {{\frac {y}{x}}-1}{\min \left(1,{\frac {y}{x}}\right)}}}
3245:{\displaystyle {\frac {{\frac {y}{x}}-1}{\max \left(1,{\frac {y}{x}}\right)}}}
6197:
5585:
6153:
4570:
In the same way that relative change is scaled by 100 to get percentages,
6019:
4284:
Of these indicators of relative change, the most natural arguably is the
954:
861:
could be "measured," "calculated," or "actual" and another term used for
6098:. Illinois State University, Dept of Physics. 2004-07-20. Archived from
6145:
4938:, and therefore additivity implies a sort of symmetry property, namely
4931:{\displaystyle \ln {\frac {V_{1}}{V_{0}}}+\ln {\frac {V_{0}}{V_{1}}}=0}
3733:
with a certain tolerance. Another application is in the computation of
1249:{\displaystyle {\frac {4\%-3\%}{3\%}}=0.333\ldots =33{\frac {1}{3}}\%.}
204:
116:
to be satisfactory indicators, but these have not seen widespread use.
70:
43:
6047:
Using and
Understanding Mathematics: A Quantitative Reasoning Approach
5014:{\displaystyle \ln {\frac {V_{1}}{V_{0}}}=-\ln {\frac {V_{0}}{V_{1}}}}
754:{\displaystyle d_{r}(x,y)={\frac {|x-y|}{(|x|+|y|)/2}},\ d_{r}(0,0)=0}
5776:
5774:
5772:
950:
6137:
1279:
costs $ 40,000. We wish to compare these costs. With respect to car
6010:
5913:
5792:
5621:
5619:
5617:
5615:
5613:
4842:{\displaystyle {\text{log change}}=\ln(1+{\text{relative change}})}
418:
108:, have been proposed in the literature. Several authors have found
63:
5769:
6172:. ETLA A 4. Helsinki: Research Institute of the Finnish Economy.
4288:(ln) of the ratio of the two numbers (final and initial), called
2162:. This implies all indicators behave like the classical one when
2056:
The normalization condition is motivated by the observation that
5610:
3133:{\displaystyle {\frac {{\frac {y}{x}}-1}{\left^{\frac {1}{k}}}}}
2799:{\displaystyle {\frac {{\frac {y}{x}}-1}{\sqrt {\frac {y}{x}}}}}
1027:
It can then be said that the worth of the house went up by 10%.
372:) is zero, and gives negative values for positive increases if
66:
5897:"Quantifying the dynamics of topical fluctuations in language"
5039:
is chosen as the reference. In contrast, for relative change,
4355:{\displaystyle \left|{\frac {V_{1}-V_{0}}{V_{0}}}\right|\ll 1}
69:. By multiplying these ratios by 100 they can be expressed as
4622:
3741:
Tenhunen defines a general relative difference function from
365:. The relative change is not defined if the reference value (
353:
197:
88:
Relative change is often used as a quantitative indicator of
59:
414:, indicating a decrease, yet in fact the reading increased.
5946:
Roe, John; deForest, Russ; Jamshidi, Sara (26 April 2018).
4257:
4096:
3917:
3450:
1984:
1845:
1542:{\displaystyle {\frac {-\$ 10,000}{\$ 50,000}}=-0.20=-20\%}
5593:"IEC 60050 â Details for IEV number 112-03-07: "relative""
2854:{\displaystyle {\frac {2}{{\frac {1}{x}}+{\frac {1}{y}}}}}
2046:{\displaystyle \left.{\frac {d}{dy}}R(1,y)\right|_{y=1}=1}
1434:{\displaystyle {\frac {\$ 50,000}{\$ 40,000}}=1.25=125\%,}
5235:
approaches 0 while the other remains fixed. For example:
3657:. This agrees with intuition that a relative change from
1355:{\displaystyle {\frac {\$ 10,000}{\$ 40,000}}=0.25=25\%,}
1018:{\displaystyle {\frac {110000-100000}{100000}}=0.1=10\%.}
5025:
of a change expressed in log change is the same whether
3721:
Maximum mean change has been recommended when comparing
3711:{\displaystyle {\frac {y}{x}}={\frac {1}{\frac {x}{y}}}}
1173:
5828:
5214:{\displaystyle {\frac {(V_{1}-V_{0})^{2}}{V_{0}V_{1}}}}
1618:{\displaystyle {\frac {\$ 40,000}{\$ 50,000}}=0.8=80\%}
1464:
as the reference value. The absolute difference is now
5894:
5840:
2425:
6117:
Törnqvist, Leo; Vartia, Pentti; Vartia, Yrjö (1985),
6116:
5780:
5637:
5625:
5465:
5438:
5410:
5368:
5146:
5045:
4944:
4858:
4807:
4576:
4368:
4298:
4149:
4118:
3941:
3755:
3679:
3604:
3548:
3513:
3471:
3376:
3300:
3264:
3188:
3152:
3034:
2955:
2868:
2818:
2763:
2735:
2652:
2609:
2570:
2526:
2478:
2423:
2387:
2228:
2168:
2128:
2100:
2066:
1981:
1916:
1890:
1705:
1666:
1567:
1482:
1380:
1301:
1185:
1050:
978:
883:
781:
615:
576:
549:
449:
246:
5757:
5745:
4617:
can be scaled by 100 to get what is commonly called
5945:
949:It is important to note that the two values in the
5814:. Belmont, Calif., Wadsworth Pub. Co. p. 17.
5643:
5478:
5451:
5424:
5396:
5213:
5132:
5013:
4930:
4841:
4609:
4560:
4354:
4263:
4133:
4102:
3923:
3710:
3649:
3590:
3534:
3490:
3456:
3356:
3285:
3244:
3173:
3132:
3019:
2932:
2853:
2798:
2748:
2715:
2637:
2589:
2545:
2501:
2463:
2408:
2330:
2182:
2154:
2114:
2078:
2045:
1964:
1902:
1854:
1687:
1617:
1541:
1433:
1354:
1248:
1137:
1017:
939:
847:
753:
601:
562:
529:
340:
5862:Data Analysis for Business, Economics, and Policy
220:of a valuable is considered big if changing from
6195:
3322:
3265:
3210:
3153:
1884:Independent of the unit of measurement: for all
1643:
6158:(PhD). A. Vol. 290. University of Tampere.
5999:
5983:"The Case for a Logarithmic Performance Metric"
5695:Smadent - Best Educational Website of Pakistan
6044:
5890:
5888:
5886:
5884:
5882:
5739:
5727:
5715:
2090:can be written as a single argument function
1147:Some calculators directly support this via a
5357:
5865:. Cambridge University Press. p. 203.
3507:. One of the properties of a mean function
2122:. The normalization condition is then that
212:is 1 while the absolute difference between
6119:"How Should Relative Changes Be Measured?"
6093:"Percent Difference – Percent Error"
6045:Bennett, Jeffrey; Briggs, William (2005),
5879:
5858:
5808:Rao, Potluri; Miller, Roger LeRoy (1971).
5665:Optimization: Foundations and Applications
4610:{\displaystyle \ln {\frac {V_{1}}{V_{0}}}}
2355:is zero. Various choices for the function
6009:
5922:
5912:
5859:Békés, Gåbor; Kézdi, Gåbor (6 May 2021).
5807:
5597:International Electrotechnical Vocabulary
5566:Learn how and when to remove this message
5418:
2260:
6151:
6064:"Understanding Measurement and Graphing"
5846:
5529:This article includes a list of general
5425:{\displaystyle \lambda \in \mathbb {R} }
4621:. Log points are equivalent to the unit
602:{\displaystyle v=v_{\text{reference}}=0}
58:value. The comparison is expressed as a
14:
6196:
6162:
5834:
5763:
5751:
5661:
5649:
2464:{\displaystyle {\tfrac {y-x}{f(x,y)}}}
208:, but the absolute difference between
5980:
5688:
4362:, the following approximation holds:
4273:
1174:Example of percentages of percentages
5515:
4785:Note that in the above table, since
4112:In particular for the special cases
3020:{\displaystyle \left^{\frac {1}{k}}}
1561:. The ratio form of the comparison,
960:
570:. Then the only problematic case is
563:{\displaystyle v_{\text{reference}}}
224:but rather small when changing from
27:Comparisons in quantitative sciences
5781:Törnqvist, Vartia & Vartia 1985
5668:. New York: John Wiley & Sons.
5662:Miller, H. Ronald (29 March 2011).
5638:Törnqvist, Vartia & Vartia 1985
5626:Törnqvist, Vartia & Vartia 1985
2638:{\displaystyle {\frac {1}{2}}(x+y)}
24:
6166:Relative changes and index numbers
5535:it lacks sufficient corresponding
5501:Errors and residuals in statistics
4793:) has the same numerical value as
4487:
4436:
3491:{\displaystyle \ln {\frac {y}{x}}}
2261:
1612:
1585:
1571:
1536:
1503:
1489:
1425:
1398:
1384:
1346:
1319:
1305:
1240:
1209:
1201:
1192:
1129:
1062:
1009:
884:
782:
293:
25:
6235:
6049:(3rd ed.), Boston: Pearson,
5397:{\displaystyle F_{\lambda }(x,y)}
5691:"Percentage Increase Calculator"
5689:Kazmi, Kumail (March 26, 2021).
5520:
2590:{\displaystyle 1-{\frac {x}{y}}}
2546:{\displaystyle {\frac {y}{x}}-1}
1656:(initial or reference value) to
1456:In this example the cost of car
763:
417:Measures of relative change are
124:Given two numerical quantities,
6071:North Carolina State University
5993:
5974:
5939:
5852:
5801:
5793:What's a good way to check for
5786:
5432:, such that absolute change is
2417:where the indicator's value is
1965:{\displaystyle R(ax,ay)=R(x,y)}
1466:â$ 10,000 = $ 40,000 â $ 50,000
5949:Mathematics for Sustainability
5733:
5721:
5709:
5682:
5655:
5631:
5391:
5379:
5177:
5150:
4836:
4822:
4232:
4220:
4191:
4179:
4165:
4153:
4055:
4041:
4017:
4002:
3987:
3984:
3957:
3945:
3771:
3759:
3650:{\displaystyle R(x,y)=-R(y,x)}
3644:
3632:
3620:
3608:
3585:
3573:
3564:
3552:
3529:
3517:
3280:
3268:
3168:
3156:
2998:
2972:
2632:
2620:
2496:
2482:
2454:
2442:
2403:
2391:
2374:Indicators of relative change
2319:
2307:
2281:
2269:
2246:
2234:
2143:
2137:
2017:
2005:
1959:
1947:
1938:
1920:
1817:
1805:
1773:
1761:
1729:
1717:
1682:
1670:
1295:. The relative difference is,
1285:$ 10,000 = $ 50,000 â $ 40,000
1266:
924:
914:
874:) and is a newsworthy result.
832:
822:
815:
797:
742:
730:
700:
696:
688:
680:
672:
668:
662:
648:
638:
626:
517:
502:
474:
455:
271:
252:
83:relative percentage difference
13:
1:
6038:
5981:Doyle, Patrick (2016-08-24).
4632:
3591:{\displaystyle m(x,y)=m(y,x)}
1867:is an increasing function of
1644:Indicators of relative change
1472:costs $ 10,000 less than car
1291:costs $ 10,000 more than car
1283:, the absolute difference is
1162:, to avoid confusion between
1037:represents the old value and
119:
106:indicators of relative change
6073:. 2008-08-20. Archived from
5901:Language Dynamics and Change
2944:Moment mean change of order
2749:{\displaystyle {\sqrt {xy}}}
1650:indicator of relative change
7:
5958:10.1007/978-3-319-76660-7_4
5506:Relative standard deviation
5489:
2513:(Classical) Relative change
1549:is also negative since car
1476:. The relative difference,
1261:
10:
6240:
4659:Total relative change (%)
4277:
3368:Logarithmic (mean) change
6126:The American Statistician
5952:. Springer. p. 190.
5924:10.1163/22105832-01001200
5740:Bennett & Briggs 2005
5728:Bennett & Briggs 2005
5716:Bennett & Briggs 2005
5358:Uniqueness and extensions
4555:classical relative change
3673:, just like the relation
3286:{\displaystyle \min(x,y)}
3174:{\displaystyle \max(x,y)}
428:
6224:Dimensionless quantities
6163:Vartia, Yrjö O. (1976).
6152:Tenhunen, Lauri (1990).
5578:
5350:Here 0 means taking the
2557:Reversed relative change
42:are used to compare two
6002:SSRN Electronic Journal
5797:floating-point equality
5550:more precise citations.
4134:{\displaystyle c=\pm 1}
2155:{\displaystyle H'(1)=1}
1275:costs $ 50,000 and car
421:numbers expressed as a
79:percent(age) difference
5480:
5453:
5426:
5398:
5215:
5140:, with the difference
5134:
5015:
4932:
4843:
4650:Total log change (cNp)
4611:
4562:
4356:
4280:Logarithmic derivative
4265:
4135:
4104:
3925:
3712:
3651:
3592:
3536:
3535:{\displaystyle m(x,y)}
3492:
3458:
3358:
3287:
3246:
3175:
3134:
3021:
2934:
2855:
2800:
2750:
2717:
2639:
2601:Arithmetic mean change
2591:
2547:
2503:
2502:{\displaystyle H(y/x)}
2465:
2410:
2409:{\displaystyle f(x,y)}
2332:
2184:
2156:
2116:
2080:
2079:{\displaystyle c>0}
2047:
1966:
1904:
1903:{\displaystyle a>0}
1856:
1689:
1688:{\displaystyle R(x,y)}
1619:
1543:
1435:
1356:
1250:
1139:
1019:
941:
849:
755:
603:
564:
531:
342:
5481:
5479:{\displaystyle F_{1}}
5454:
5452:{\displaystyle F_{0}}
5427:
5399:
5216:
5135:
5016:
4933:
4844:
4656:Relative change 1 (%)
4653:Relative change 0 (%)
4612:
4563:
4357:
4266:
4136:
4105:
3926:
3745:(reference value) to
3727:programming languages
3713:
3652:
3593:
3537:
3493:
3459:
3359:
3288:
3247:
3176:
3135:
3022:
2935:
2856:
2801:
2751:
2727:Geometric mean change
2718:
2640:
2592:
2548:
2504:
2466:
2411:
2333:
2185:
2157:
2117:
2081:
2060:scaled by a constant
2048:
1967:
1905:
1857:
1690:
1620:
1544:
1436:
1357:
1251:
1140:
1020:
942:
850:
756:
604:
565:
532:
343:
6020:10.2139/ssrn.3739890
5907:(1). Section A.3.1.
5811:Applied econometrics
5697:. Smadent Publishing
5463:
5436:
5408:
5366:
5260:Relative change (%)
5144:
5043:
4942:
4856:
4805:
4574:
4366:
4296:
4147:
4116:
3939:
3753:
3735:approximation errors
3677:
3602:
3546:
3511:
3469:
3374:
3298:
3262:
3186:
3150:
3032:
2953:
2866:
2816:
2810:Harmonic mean change
2761:
2733:
2650:
2607:
2568:
2524:
2476:
2421:
2385:
2370:have been proposed:
2226:
2166:
2126:
2098:
2064:
1979:
1914:
1888:
1703:
1664:
1565:
1480:
1441:and we say that car
1378:
1362:and we say that car
1299:
1183:
1048:
976:
881:
872:theory of relativity
779:
613:
574:
547:
447:
244:
32:quantitative science
5730:, pp. 137–139
5496:Approximation error
5221:becoming larger as
4481:
4430:
3875:
3807:
3256:Minimum mean change
3144:Maximum mean change
2375:
2183:{\displaystyle y/x}
2115:{\displaystyle y/x}
1168:absolute difference
1164:relative difference
1030:More generally, if
214:200 and 100 cm
186:absolute difference
40:relative difference
18:Relative difference
6209:Numerical analysis
5476:
5459:and log change is
5449:
5422:
5394:
5211:
5130:
5011:
4928:
4839:
4647:Log change 1 (cNp)
4644:Log change 0 (cNp)
4607:
4558:
4453:
4402:
4352:
4274:Logarithmic change
4261:
4256:
4131:
4100:
4095:
3921:
3916:
3853:
3785:
3708:
3647:
3588:
3532:
3488:
3454:
3449:
3354:
3283:
3242:
3171:
3130:
3017:
2930:
2851:
2796:
2746:
2713:
2635:
2587:
2543:
2499:
2461:
2459:
2406:
2373:
2328:
2180:
2152:
2112:
2076:
2043:
1962:
1900:
1852:
1844:
1699:Appropriate sign:
1685:
1615:
1539:
1431:
1352:
1246:
1135:
1015:
937:
845:
751:
599:
560:
527:
363:200 to 100 cm
361:, the same as for
338:
226:$ 10,000 to 10,100
5967:978-3-319-76660-7
5872:978-1-108-48301-8
5837:, pp. 17â18.
5821:978-0-534-00031-8
5675:978-1-118-03118-6
5576:
5575:
5568:
5511:Logarithmic scale
5348:
5347:
5209:
5128:
5083:
5009:
4973:
4920:
4887:
4834:
4811:
4791:relative change 1
4787:relative change 0
4783:
4782:
4605:
4556:
4548:
4506:
4448:
4397:
4340:
4286:natural logarithm
3979:
3903:
3835:
3706:
3705:
3688:
3501:
3500:
3486:
3416:
3413:
3352:
3344:
3312:
3240:
3232:
3200:
3128:
3125:
3095:
3069:
3046:
3014:
2970:
2928:
2917:
2885:
2849:
2846:
2833:
2794:
2793:
2792:
2775:
2744:
2711:
2703:
2682:
2664:
2618:
2585:
2535:
2458:
2323:
2285:
2258:
2232:
2000:
1831:
1787:
1743:
1598:
1516:
1411:
1332:
1238:
1213:
1160:percentage points
1121:
1079:
1054:
1053:Percentage change
995:
967:percentage change
961:Percentage change
929:
921:
909:
901:
890:
837:
829:
812:
804:
788:
719:
712:
590:
557:
522:
513:
496:
465:
453:
330:
327:
310:
307:
285:
284:
281:
262:
250:
152:actual difference
90:quality assurance
75:percentage change
16:(Redirected from
6231:
6190:
6188:
6186:
6171:
6159:
6148:
6123:
6113:
6111:
6110:
6104:
6097:
6088:
6086:
6085:
6079:
6068:
6059:
6032:
6031:
6013:
5997:
5991:
5990:
5978:
5972:
5971:
5943:
5937:
5936:
5926:
5916:
5892:
5877:
5876:
5856:
5850:
5844:
5838:
5832:
5826:
5825:
5805:
5799:
5790:
5784:
5778:
5767:
5761:
5755:
5749:
5743:
5737:
5731:
5725:
5719:
5713:
5707:
5706:
5704:
5702:
5686:
5680:
5679:
5659:
5653:
5647:
5641:
5635:
5629:
5623:
5608:
5607:
5605:
5604:
5589:
5571:
5564:
5560:
5557:
5551:
5546:this article by
5537:inline citations
5524:
5523:
5516:
5485:
5483:
5482:
5477:
5475:
5474:
5458:
5456:
5455:
5450:
5448:
5447:
5431:
5429:
5428:
5423:
5421:
5403:
5401:
5400:
5395:
5378:
5377:
5352:limit from above
5257:Log change (cNp)
5238:
5237:
5220:
5218:
5217:
5212:
5210:
5208:
5207:
5206:
5197:
5196:
5186:
5185:
5184:
5175:
5174:
5162:
5161:
5148:
5139:
5137:
5136:
5131:
5129:
5127:
5126:
5117:
5116:
5115:
5103:
5102:
5092:
5084:
5082:
5081:
5072:
5071:
5070:
5058:
5057:
5047:
5020:
5018:
5017:
5012:
5010:
5008:
5007:
4998:
4997:
4988:
4974:
4972:
4971:
4962:
4961:
4952:
4937:
4935:
4934:
4929:
4921:
4919:
4918:
4909:
4908:
4899:
4888:
4886:
4885:
4876:
4875:
4866:
4848:
4846:
4845:
4840:
4835:
4832:
4812:
4809:
4641:
4640:
4616:
4614:
4613:
4608:
4606:
4604:
4603:
4594:
4593:
4584:
4567:
4565:
4564:
4559:
4557:
4554:
4549:
4547:
4546:
4537:
4536:
4535:
4523:
4522:
4512:
4507:
4505:
4504:
4495:
4491:
4490:
4483:
4480:
4479:
4478:
4468:
4467:
4466:
4449:
4444:
4440:
4439:
4432:
4429:
4428:
4427:
4417:
4416:
4415:
4398:
4396:
4395:
4386:
4385:
4376:
4361:
4359:
4358:
4353:
4345:
4341:
4339:
4338:
4329:
4328:
4327:
4315:
4314:
4304:
4270:
4268:
4267:
4262:
4260:
4259:
4239:
4198:
4140:
4138:
4137:
4132:
4109:
4107:
4106:
4101:
4099:
4098:
4051:
4010:
4009:
3997:
3980:
3972:
3930:
3928:
3927:
3922:
3920:
3919:
3904:
3901:
3891:
3890:
3874:
3869:
3865:
3836:
3833:
3823:
3822:
3806:
3802:
3793:
3717:
3715:
3714:
3709:
3707:
3698:
3694:
3689:
3681:
3656:
3654:
3653:
3648:
3597:
3595:
3594:
3589:
3541:
3539:
3538:
3533:
3497:
3495:
3494:
3489:
3487:
3479:
3463:
3461:
3460:
3455:
3453:
3452:
3417:
3415:
3414:
3406:
3397:
3386:
3363:
3361:
3360:
3355:
3353:
3351:
3350:
3346:
3345:
3337:
3320:
3313:
3305:
3302:
3292:
3290:
3289:
3284:
3251:
3249:
3248:
3243:
3241:
3239:
3238:
3234:
3233:
3225:
3208:
3201:
3193:
3190:
3180:
3178:
3177:
3172:
3139:
3137:
3136:
3131:
3129:
3127:
3126:
3118:
3116:
3112:
3111:
3107:
3106:
3105:
3100:
3096:
3088:
3070:
3062:
3054:
3047:
3039:
3036:
3026:
3024:
3023:
3018:
3016:
3015:
3007:
3005:
3001:
2997:
2996:
2984:
2983:
2971:
2963:
2947:
2939:
2937:
2936:
2931:
2929:
2924:
2923:
2919:
2918:
2910:
2897:
2893:
2886:
2878:
2870:
2860:
2858:
2857:
2852:
2850:
2848:
2847:
2839:
2834:
2826:
2820:
2805:
2803:
2802:
2797:
2795:
2785:
2784:
2783:
2776:
2768:
2765:
2755:
2753:
2752:
2747:
2745:
2737:
2722:
2720:
2719:
2714:
2712:
2710:
2709:
2705:
2704:
2696:
2683:
2675:
2672:
2665:
2657:
2654:
2644:
2642:
2641:
2636:
2619:
2611:
2596:
2594:
2593:
2588:
2586:
2578:
2562:
2552:
2550:
2549:
2544:
2536:
2528:
2518:
2508:
2506:
2505:
2500:
2492:
2470:
2468:
2467:
2462:
2460:
2457:
2437:
2426:
2415:
2413:
2412:
2407:
2376:
2372:
2369:
2354:
2337:
2335:
2334:
2329:
2324:
2322:
2302:
2291:
2286:
2284:
2264:
2259:
2256:
2253:
2233:
2230:
2219:
2193:
2189:
2187:
2186:
2181:
2176:
2161:
2159:
2158:
2153:
2136:
2121:
2119:
2118:
2113:
2108:
2093:
2089:
2085:
2083:
2082:
2077:
2059:
2052:
2050:
2049:
2044:
2036:
2035:
2024:
2020:
2001:
1999:
1988:
1971:
1969:
1968:
1963:
1909:
1907:
1906:
1901:
1880:
1874:
1870:
1866:
1861:
1859:
1858:
1853:
1848:
1847:
1832:
1829:
1788:
1785:
1744:
1741:
1694:
1692:
1691:
1686:
1624:
1622:
1621:
1616:
1599:
1597:
1583:
1569:
1548:
1546:
1545:
1540:
1517:
1515:
1501:
1484:
1467:
1449:the cost of car
1440:
1438:
1437:
1432:
1412:
1410:
1396:
1382:
1361:
1359:
1358:
1353:
1333:
1331:
1317:
1303:
1286:
1255:
1253:
1252:
1247:
1239:
1231:
1214:
1212:
1204:
1187:
1144:
1142:
1141:
1136:
1122:
1120:
1119:
1110:
1109:
1108:
1096:
1095:
1085:
1080:
1078:
1077:
1068:
1060:
1055:
1052:
1024:
1022:
1021:
1016:
996:
991:
980:
946:
944:
943:
938:
930:
928:
927:
922:
919:
917:
911:
910:
907:
902:
899:
896:
891:
888:
854:
852:
851:
846:
838:
836:
835:
830:
827:
825:
819:
818:
813:
810:
805:
802:
800:
794:
789:
786:
760:
758:
757:
752:
729:
728:
717:
713:
711:
707:
699:
691:
683:
675:
666:
665:
651:
645:
625:
624:
608:
606:
605:
600:
592:
591:
588:
569:
567:
566:
561:
559:
558:
555:
542:
536:
534:
533:
528:
523:
521:
520:
515:
514:
511:
505:
499:
498:
497:
494:
481:
467:
466:
463:
454:
451:
413:
411:
409:
408:
405:
402:
395:
393:
392:
389:
386:
364:
360:
356:
347:
345:
344:
339:
331:
329:
328:
325:
316:
311:
309:
308:
305:
299:
291:
286:
282:
279:
278:
264:
263:
260:
251:
248:
227:
223:
219:
215:
211:
207:
200:
178:
144:reference value,
21:
6239:
6238:
6234:
6233:
6232:
6230:
6229:
6228:
6194:
6193:
6184:
6182:
6180:
6169:
6138:10.2307/2683905
6121:
6108:
6106:
6102:
6095:
6091:
6083:
6081:
6077:
6066:
6062:
6057:
6041:
6036:
6035:
5998:
5994:
5979:
5975:
5968:
5944:
5940:
5893:
5880:
5873:
5857:
5853:
5845:
5841:
5833:
5829:
5822:
5806:
5802:
5791:
5787:
5779:
5770:
5762:
5758:
5750:
5746:
5738:
5734:
5726:
5722:
5714:
5710:
5700:
5698:
5687:
5683:
5676:
5660:
5656:
5648:
5644:
5636:
5632:
5624:
5611:
5602:
5600:
5591:
5590:
5586:
5581:
5572:
5561:
5555:
5552:
5542:Please help to
5541:
5525:
5521:
5492:
5470:
5466:
5464:
5461:
5460:
5443:
5439:
5437:
5434:
5433:
5417:
5409:
5406:
5405:
5373:
5369:
5367:
5364:
5363:
5360:
5254:
5246:
5234:
5227:
5202:
5198:
5192:
5188:
5187:
5180:
5176:
5170:
5166:
5157:
5153:
5149:
5147:
5145:
5142:
5141:
5122:
5118:
5111:
5107:
5098:
5094:
5093:
5091:
5077:
5073:
5066:
5062:
5053:
5049:
5048:
5046:
5044:
5041:
5040:
5038:
5031:
5003:
4999:
4993:
4989:
4987:
4967:
4963:
4957:
4953:
4951:
4943:
4940:
4939:
4914:
4910:
4904:
4900:
4898:
4881:
4877:
4871:
4867:
4865:
4857:
4854:
4853:
4852:By additivity,
4833:relative change
4831:
4808:
4806:
4803:
4802:
4635:
4599:
4595:
4589:
4585:
4583:
4575:
4572:
4571:
4553:
4542:
4538:
4531:
4527:
4518:
4514:
4513:
4511:
4500:
4496:
4486:
4485:
4484:
4482:
4474:
4470:
4469:
4462:
4458:
4457:
4435:
4434:
4433:
4431:
4423:
4419:
4418:
4411:
4407:
4406:
4391:
4387:
4381:
4377:
4375:
4367:
4364:
4363:
4334:
4330:
4323:
4319:
4310:
4306:
4305:
4303:
4299:
4297:
4294:
4293:
4292:. Indeed, when
4282:
4276:
4255:
4254:
4243:
4235:
4217:
4216:
4202:
4194:
4172:
4171:
4148:
4145:
4144:
4117:
4114:
4113:
4094:
4093:
4058:
4047:
4032:
4031:
4020:
4005:
4001:
3993:
3971:
3964:
3963:
3940:
3937:
3936:
3933:which leads to
3915:
3914:
3900:
3898:
3880:
3876:
3870:
3861:
3857:
3847:
3846:
3832:
3830:
3812:
3808:
3798:
3794:
3789:
3778:
3777:
3754:
3751:
3750:
3693:
3680:
3678:
3675:
3674:
3603:
3600:
3599:
3547:
3544:
3543:
3512:
3509:
3508:
3478:
3470:
3467:
3466:
3448:
3447:
3436:
3430:
3429:
3418:
3405:
3398:
3387:
3385:
3378:
3377:
3375:
3372:
3371:
3336:
3329:
3325:
3321:
3304:
3303:
3301:
3299:
3296:
3295:
3263:
3260:
3259:
3224:
3217:
3213:
3209:
3192:
3191:
3189:
3187:
3184:
3183:
3151:
3148:
3147:
3117:
3101:
3087:
3083:
3082:
3075:
3071:
3061:
3060:
3056:
3055:
3038:
3037:
3035:
3033:
3030:
3029:
3006:
2992:
2988:
2979:
2975:
2962:
2961:
2957:
2956:
2954:
2951:
2950:
2945:
2909:
2902:
2898:
2877:
2876:
2872:
2871:
2869:
2867:
2864:
2863:
2838:
2825:
2824:
2819:
2817:
2814:
2813:
2767:
2766:
2764:
2762:
2759:
2758:
2736:
2734:
2731:
2730:
2695:
2688:
2684:
2674:
2673:
2656:
2655:
2653:
2651:
2648:
2647:
2610:
2608:
2605:
2604:
2577:
2569:
2566:
2565:
2560:
2527:
2525:
2522:
2521:
2516:
2488:
2477:
2474:
2473:
2438:
2427:
2424:
2422:
2419:
2418:
2416:
2386:
2383:
2382:
2356:
2341:
2303:
2292:
2290:
2265:
2255:
2254:
2252:
2231:Relative change
2229:
2227:
2224:
2223:
2206:
2191:
2172:
2167:
2164:
2163:
2129:
2127:
2124:
2123:
2104:
2099:
2096:
2095:
2091:
2087:
2065:
2062:
2061:
2057:
2025:
1992:
1987:
1986:
1983:
1982:
1980:
1977:
1976:
1915:
1912:
1911:
1889:
1886:
1885:
1878:
1872:
1868:
1864:
1843:
1842:
1828:
1826:
1799:
1798:
1784:
1782:
1755:
1754:
1740:
1738:
1707:
1706:
1704:
1701:
1700:
1665:
1662:
1661:
1646:
1584:
1570:
1568:
1566:
1563:
1562:
1502:
1485:
1483:
1481:
1478:
1477:
1465:
1397:
1383:
1381:
1379:
1376:
1375:
1318:
1304:
1302:
1300:
1297:
1296:
1287:. That is, car
1284:
1269:
1264:
1230:
1205:
1188:
1186:
1184:
1181:
1180:
1176:
1154:
1150:
1115:
1111:
1104:
1100:
1091:
1087:
1086:
1084:
1073:
1069:
1061:
1059:
1051:
1049:
1046:
1045:
1043:
1036:
981:
979:
977:
974:
973:
963:
923:
918:
913:
912:
906:
898:
897:
895:
887:
882:
879:
878:
831:
826:
821:
820:
814:
809:
801:
796:
795:
793:
785:
780:
777:
776:
766:
724:
720:
703:
695:
687:
679:
671:
667:
661:
647:
646:
644:
620:
616:
614:
611:
610:
587:
583:
575:
572:
571:
554:
550:
548:
545:
544:
540:
516:
510:
506:
501:
500:
493:
489:
482:
480:
462:
458:
452:Relative change
450:
448:
445:
444:
440:
431:
406:
403:
400:
399:
397:
390:
387:
384:
383:
381:
380:
377:
370:
362:
358:
351:
324:
320:
315:
304:
300:
292:
290:
283:reference value
277:
259:
255:
249:relative change
247:
245:
242:
241:
236:
225:
221:
217:
213:
209:
202:
201:is the same as
195:
176:
162:
156:absolute change
140:
129:
122:
94:quality control
36:relative change
28:
23:
22:
15:
12:
11:
5:
6237:
6227:
6226:
6221:
6216:
6211:
6206:
6192:
6191:
6178:
6160:
6149:
6114:
6089:
6060:
6055:
6040:
6037:
6034:
6033:
5992:
5987:Vena Solutions
5973:
5966:
5938:
5878:
5871:
5851:
5839:
5827:
5820:
5800:
5785:
5768:
5756:
5744:
5732:
5720:
5708:
5681:
5674:
5654:
5642:
5630:
5609:
5583:
5582:
5580:
5577:
5574:
5573:
5528:
5526:
5519:
5514:
5513:
5508:
5503:
5498:
5491:
5488:
5473:
5469:
5446:
5442:
5420:
5416:
5413:
5393:
5390:
5387:
5384:
5381:
5376:
5372:
5359:
5356:
5346:
5345:
5342:
5339:
5336:
5332:
5331:
5328:
5325:
5322:
5318:
5317:
5314:
5311:
5308:
5304:
5303:
5300:
5297:
5294:
5290:
5289:
5286:
5283:
5280:
5276:
5275:
5272:
5269:
5266:
5262:
5261:
5258:
5255:
5252:
5247:
5244:
5232:
5225:
5205:
5201:
5195:
5191:
5183:
5179:
5173:
5169:
5165:
5160:
5156:
5152:
5125:
5121:
5114:
5110:
5106:
5101:
5097:
5090:
5087:
5080:
5076:
5069:
5065:
5061:
5056:
5052:
5036:
5029:
5006:
5002:
4996:
4992:
4986:
4983:
4980:
4977:
4970:
4966:
4960:
4956:
4950:
4947:
4927:
4924:
4917:
4913:
4907:
4903:
4897:
4894:
4891:
4884:
4880:
4874:
4870:
4864:
4861:
4838:
4830:
4827:
4824:
4821:
4818:
4815:
4797:(respectively
4789:(respectively
4781:
4780:
4777:
4774:
4771:
4768:
4765:
4761:
4760:
4757:
4754:
4751:
4748:
4745:
4741:
4740:
4737:
4734:
4731:
4728:
4725:
4721:
4720:
4717:
4714:
4711:
4708:
4705:
4701:
4700:
4697:
4694:
4691:
4688:
4685:
4681:
4680:
4677:
4674:
4671:
4668:
4665:
4661:
4660:
4657:
4654:
4651:
4648:
4645:
4634:
4631:
4602:
4598:
4592:
4588:
4582:
4579:
4552:
4545:
4541:
4534:
4530:
4526:
4521:
4517:
4510:
4503:
4499:
4494:
4489:
4477:
4473:
4465:
4461:
4456:
4452:
4447:
4443:
4438:
4426:
4422:
4414:
4410:
4405:
4401:
4394:
4390:
4384:
4380:
4374:
4371:
4351:
4348:
4344:
4337:
4333:
4326:
4322:
4318:
4313:
4309:
4302:
4275:
4272:
4258:
4253:
4250:
4247:
4244:
4242:
4238:
4234:
4231:
4228:
4225:
4222:
4219:
4218:
4215:
4212:
4209:
4206:
4203:
4201:
4197:
4193:
4190:
4187:
4184:
4181:
4178:
4177:
4175:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4130:
4127:
4124:
4121:
4097:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4057:
4054:
4050:
4046:
4043:
4040:
4037:
4034:
4033:
4030:
4027:
4024:
4021:
4019:
4016:
4013:
4008:
4004:
4000:
3996:
3992:
3989:
3986:
3983:
3978:
3975:
3970:
3969:
3967:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3918:
3913:
3910:
3907:
3899:
3897:
3894:
3889:
3886:
3883:
3879:
3873:
3868:
3864:
3860:
3856:
3852:
3849:
3848:
3845:
3842:
3839:
3831:
3829:
3826:
3821:
3818:
3815:
3811:
3805:
3801:
3797:
3792:
3788:
3784:
3783:
3781:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3723:floating point
3704:
3701:
3697:
3692:
3687:
3684:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3531:
3528:
3525:
3522:
3519:
3516:
3499:
3498:
3485:
3482:
3477:
3474:
3464:
3451:
3446:
3443:
3440:
3437:
3435:
3432:
3431:
3428:
3425:
3422:
3419:
3412:
3409:
3404:
3401:
3396:
3393:
3390:
3384:
3383:
3381:
3369:
3365:
3364:
3349:
3343:
3340:
3335:
3332:
3328:
3324:
3319:
3316:
3311:
3308:
3293:
3282:
3279:
3276:
3273:
3270:
3267:
3257:
3253:
3252:
3237:
3231:
3228:
3223:
3220:
3216:
3212:
3207:
3204:
3199:
3196:
3181:
3170:
3167:
3164:
3161:
3158:
3155:
3145:
3141:
3140:
3124:
3121:
3115:
3110:
3104:
3099:
3094:
3091:
3086:
3081:
3078:
3074:
3068:
3065:
3059:
3053:
3050:
3045:
3042:
3027:
3013:
3010:
3004:
3000:
2995:
2991:
2987:
2982:
2978:
2974:
2969:
2966:
2960:
2948:
2941:
2940:
2927:
2922:
2916:
2913:
2908:
2905:
2901:
2896:
2892:
2889:
2884:
2881:
2875:
2861:
2845:
2842:
2837:
2832:
2829:
2823:
2811:
2807:
2806:
2791:
2788:
2782:
2779:
2774:
2771:
2756:
2743:
2740:
2728:
2724:
2723:
2708:
2702:
2699:
2694:
2691:
2687:
2681:
2678:
2671:
2668:
2663:
2660:
2645:
2634:
2631:
2628:
2625:
2622:
2617:
2614:
2602:
2598:
2597:
2584:
2581:
2576:
2573:
2563:
2558:
2554:
2553:
2542:
2539:
2534:
2531:
2519:
2514:
2510:
2509:
2498:
2495:
2491:
2487:
2484:
2481:
2471:
2456:
2453:
2450:
2447:
2444:
2441:
2436:
2433:
2430:
2405:
2402:
2399:
2396:
2393:
2390:
2380:
2327:
2321:
2318:
2315:
2312:
2309:
2306:
2301:
2298:
2295:
2289:
2283:
2280:
2277:
2274:
2271:
2268:
2263:
2251:
2248:
2245:
2242:
2239:
2236:
2179:
2175:
2171:
2151:
2148:
2145:
2142:
2139:
2135:
2132:
2111:
2107:
2103:
2075:
2072:
2069:
2054:
2053:
2042:
2039:
2034:
2031:
2028:
2023:
2019:
2016:
2013:
2010:
2007:
2004:
1998:
1995:
1991:
1985:
1973:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1899:
1896:
1893:
1882:
1881:is continuous.
1876:
1862:
1851:
1846:
1841:
1838:
1835:
1827:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1800:
1797:
1794:
1791:
1783:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1756:
1753:
1750:
1747:
1739:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1712:
1710:
1684:
1681:
1678:
1675:
1672:
1669:
1645:
1642:
1625:says that car
1614:
1611:
1608:
1605:
1602:
1596:
1593:
1590:
1587:
1582:
1579:
1576:
1573:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1514:
1511:
1508:
1505:
1500:
1497:
1494:
1491:
1488:
1430:
1427:
1424:
1421:
1418:
1415:
1409:
1406:
1403:
1400:
1395:
1392:
1389:
1386:
1351:
1348:
1345:
1342:
1339:
1336:
1330:
1327:
1324:
1321:
1316:
1313:
1310:
1307:
1268:
1265:
1263:
1260:
1245:
1242:
1237:
1234:
1229:
1226:
1223:
1220:
1217:
1211:
1208:
1203:
1200:
1197:
1194:
1191:
1175:
1172:
1152:
1148:
1134:
1131:
1128:
1125:
1118:
1114:
1107:
1103:
1099:
1094:
1090:
1083:
1076:
1072:
1067:
1064:
1058:
1041:
1034:
1014:
1011:
1008:
1005:
1002:
999:
994:
990:
987:
984:
962:
959:
936:
933:
926:
916:
905:
894:
886:
844:
841:
834:
824:
817:
808:
799:
792:
784:
765:
762:
750:
747:
744:
741:
738:
735:
732:
727:
723:
716:
710:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
664:
660:
657:
654:
650:
643:
640:
637:
634:
631:
628:
623:
619:
598:
595:
586:
582:
579:
553:
526:
519:
509:
504:
492:
488:
485:
479:
476:
473:
470:
461:
457:
438:
430:
427:
375:
368:
337:
334:
323:
319:
314:
303:
298:
295:
289:
276:
273:
270:
267:
258:
254:
234:
210:2 and 1 m
181:
180:
174:
138:
127:
121:
118:
26:
9:
6:
4:
3:
2:
6236:
6225:
6222:
6220:
6217:
6215:
6212:
6210:
6207:
6205:
6202:
6201:
6199:
6181:
6179:951-9205-24-1
6175:
6168:
6167:
6161:
6157:
6156:
6150:
6147:
6143:
6139:
6135:
6131:
6127:
6120:
6115:
6105:on 2019-07-13
6101:
6094:
6090:
6080:on 2010-06-15
6076:
6072:
6065:
6061:
6058:
6056:0-321-22773-5
6052:
6048:
6043:
6042:
6029:
6025:
6021:
6017:
6012:
6007:
6003:
5996:
5988:
5984:
5977:
5969:
5963:
5959:
5955:
5951:
5950:
5942:
5934:
5930:
5925:
5920:
5915:
5910:
5906:
5902:
5898:
5891:
5889:
5887:
5885:
5883:
5874:
5868:
5864:
5863:
5855:
5849:, p. 20.
5848:
5847:Tenhunen 1990
5843:
5836:
5831:
5823:
5817:
5813:
5812:
5804:
5798:
5796:
5789:
5782:
5777:
5775:
5773:
5766:, p. 14.
5765:
5760:
5754:, p. 10.
5753:
5748:
5741:
5736:
5729:
5724:
5717:
5712:
5696:
5692:
5685:
5677:
5671:
5667:
5666:
5658:
5651:
5646:
5639:
5634:
5627:
5622:
5620:
5618:
5616:
5614:
5599:(in Japanese)
5598:
5594:
5588:
5584:
5570:
5567:
5559:
5549:
5545:
5539:
5538:
5532:
5527:
5518:
5517:
5512:
5509:
5507:
5504:
5502:
5499:
5497:
5494:
5493:
5487:
5471:
5467:
5444:
5440:
5414:
5411:
5388:
5385:
5382:
5374:
5370:
5355:
5353:
5343:
5340:
5337:
5334:
5333:
5329:
5326:
5323:
5320:
5319:
5315:
5312:
5309:
5306:
5305:
5301:
5298:
5295:
5292:
5291:
5287:
5284:
5281:
5278:
5277:
5273:
5270:
5267:
5264:
5263:
5259:
5256:
5251:
5248:
5243:
5240:
5239:
5236:
5231:
5224:
5203:
5199:
5193:
5189:
5181:
5171:
5167:
5163:
5158:
5154:
5123:
5119:
5112:
5108:
5104:
5099:
5095:
5088:
5085:
5078:
5074:
5067:
5063:
5059:
5054:
5050:
5035:
5028:
5024:
5021:and thus the
5004:
5000:
4994:
4990:
4984:
4981:
4978:
4975:
4968:
4964:
4958:
4954:
4948:
4945:
4925:
4922:
4915:
4911:
4905:
4901:
4895:
4892:
4889:
4882:
4878:
4872:
4868:
4862:
4859:
4850:
4828:
4825:
4819:
4816:
4813:
4800:
4796:
4792:
4788:
4778:
4775:
4772:
4769:
4766:
4763:
4762:
4758:
4755:
4752:
4749:
4746:
4743:
4742:
4738:
4735:
4732:
4729:
4726:
4723:
4722:
4718:
4715:
4712:
4709:
4706:
4703:
4702:
4698:
4695:
4692:
4689:
4686:
4683:
4682:
4678:
4675:
4672:
4669:
4666:
4663:
4662:
4658:
4655:
4652:
4649:
4646:
4643:
4642:
4639:
4630:
4628:
4624:
4620:
4600:
4596:
4590:
4586:
4580:
4577:
4568:
4550:
4543:
4539:
4532:
4528:
4524:
4519:
4515:
4508:
4501:
4497:
4492:
4475:
4471:
4463:
4459:
4454:
4450:
4445:
4441:
4424:
4420:
4412:
4408:
4403:
4399:
4392:
4388:
4382:
4378:
4372:
4369:
4349:
4346:
4342:
4335:
4331:
4324:
4320:
4316:
4311:
4307:
4300:
4291:
4287:
4281:
4271:
4251:
4248:
4245:
4240:
4236:
4229:
4226:
4223:
4213:
4210:
4207:
4204:
4199:
4195:
4188:
4185:
4182:
4173:
4168:
4162:
4159:
4156:
4150:
4142:
4128:
4125:
4122:
4119:
4110:
4090:
4087:
4084:
4081:
4078:
4075:
4072:
4069:
4066:
4063:
4060:
4052:
4048:
4044:
4038:
4035:
4028:
4025:
4022:
4014:
4011:
4006:
3998:
3994:
3990:
3981:
3976:
3973:
3965:
3960:
3954:
3951:
3948:
3942:
3934:
3931:
3911:
3908:
3905:
3895:
3892:
3887:
3884:
3881:
3877:
3871:
3866:
3862:
3858:
3854:
3850:
3843:
3840:
3837:
3827:
3824:
3819:
3816:
3813:
3809:
3803:
3799:
3795:
3790:
3786:
3779:
3774:
3768:
3765:
3762:
3756:
3748:
3744:
3739:
3736:
3732:
3728:
3724:
3719:
3718:suggests.
3702:
3699:
3695:
3690:
3685:
3682:
3672:
3668:
3664:
3660:
3641:
3638:
3635:
3629:
3626:
3623:
3617:
3614:
3611:
3605:
3582:
3579:
3576:
3570:
3567:
3561:
3558:
3555:
3549:
3526:
3523:
3520:
3514:
3506:
3483:
3480:
3475:
3472:
3465:
3444:
3441:
3438:
3433:
3426:
3423:
3420:
3410:
3407:
3402:
3399:
3394:
3391:
3388:
3379:
3370:
3367:
3366:
3347:
3341:
3338:
3333:
3330:
3326:
3317:
3314:
3309:
3306:
3294:
3277:
3274:
3271:
3258:
3255:
3254:
3235:
3229:
3226:
3221:
3218:
3214:
3205:
3202:
3197:
3194:
3182:
3165:
3162:
3159:
3146:
3143:
3142:
3122:
3119:
3113:
3108:
3102:
3097:
3092:
3089:
3084:
3079:
3076:
3072:
3066:
3063:
3057:
3051:
3048:
3043:
3040:
3028:
3011:
3008:
3002:
2993:
2989:
2985:
2980:
2976:
2967:
2964:
2958:
2949:
2943:
2942:
2925:
2920:
2914:
2911:
2906:
2903:
2899:
2894:
2890:
2887:
2882:
2879:
2873:
2862:
2843:
2840:
2835:
2830:
2827:
2821:
2812:
2809:
2808:
2789:
2786:
2780:
2777:
2772:
2769:
2757:
2741:
2738:
2729:
2726:
2725:
2706:
2700:
2697:
2692:
2689:
2685:
2679:
2676:
2669:
2666:
2661:
2658:
2646:
2629:
2626:
2623:
2615:
2612:
2603:
2600:
2599:
2582:
2579:
2574:
2571:
2564:
2559:
2556:
2555:
2540:
2537:
2532:
2529:
2520:
2515:
2512:
2511:
2493:
2489:
2485:
2479:
2472:
2451:
2448:
2445:
2439:
2434:
2431:
2428:
2400:
2397:
2394:
2388:
2381:
2378:
2377:
2371:
2367:
2363:
2359:
2352:
2348:
2344:
2338:
2325:
2316:
2313:
2310:
2304:
2299:
2296:
2293:
2287:
2278:
2275:
2272:
2266:
2257:Actual change
2249:
2243:
2240:
2237:
2221:
2217:
2213:
2209:
2204:
2200:
2195:
2177:
2173:
2169:
2149:
2146:
2140:
2133:
2130:
2109:
2105:
2101:
2094:of the ratio
2073:
2070:
2067:
2040:
2037:
2032:
2029:
2026:
2021:
2014:
2011:
2008:
2002:
1996:
1993:
1989:
1974:
1956:
1953:
1950:
1944:
1941:
1935:
1932:
1929:
1926:
1923:
1917:
1897:
1894:
1891:
1883:
1877:
1863:
1849:
1839:
1836:
1833:
1823:
1820:
1814:
1811:
1808:
1802:
1795:
1792:
1789:
1779:
1776:
1770:
1767:
1764:
1758:
1751:
1748:
1745:
1735:
1732:
1726:
1723:
1720:
1714:
1708:
1698:
1697:
1696:
1679:
1676:
1673:
1667:
1660:(new value)
1659:
1655:
1651:
1641:
1638:
1636:
1632:
1628:
1609:
1606:
1603:
1600:
1594:
1591:
1588:
1580:
1577:
1574:
1560:
1556:
1552:
1533:
1530:
1527:
1524:
1521:
1518:
1512:
1509:
1506:
1498:
1495:
1492:
1486:
1475:
1471:
1463:
1459:
1454:
1452:
1448:
1444:
1428:
1422:
1419:
1416:
1413:
1407:
1404:
1401:
1393:
1390:
1387:
1373:
1369:
1365:
1349:
1343:
1340:
1337:
1334:
1328:
1325:
1322:
1314:
1311:
1308:
1294:
1290:
1282:
1278:
1274:
1259:
1256:
1243:
1235:
1232:
1227:
1224:
1221:
1218:
1215:
1206:
1198:
1195:
1189:
1171:
1169:
1165:
1161:
1156:
1145:
1132:
1126:
1123:
1116:
1112:
1105:
1101:
1097:
1092:
1088:
1081:
1074:
1070:
1065:
1056:
1044:the new one,
1040:
1033:
1028:
1025:
1012:
1006:
1003:
1000:
997:
992:
988:
985:
982:
970:
968:
958:
956:
952:
947:
934:
931:
903:
892:
875:
873:
867:
864:
860:
855:
842:
839:
806:
790:
774:
771:
770:percent error
764:Percent error
761:
748:
745:
739:
736:
733:
725:
721:
714:
708:
704:
692:
684:
676:
658:
655:
652:
641:
635:
632:
629:
621:
617:
596:
593:
584:
580:
577:
551:
537:
524:
507:
490:
486:
483:
477:
471:
468:
459:
442:
437:
426:
424:
420:
415:
378:
371:
355:
348:
335:
332:
321:
317:
312:
301:
296:
287:
280:actual change
274:
268:
265:
256:
239:
237:
229:
206:
199:
192:
188:
187:
177:
170:
166:
161:
160:
159:
157:
153:
149:
148:actual change
145:
141:
134:
130:
117:
115:
111:
107:
102:
99:
98:percent error
95:
91:
86:
84:
80:
76:
73:so the terms
72:
68:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
6183:. Retrieved
6165:
6154:
6132:(1): 43â46,
6129:
6125:
6107:. Retrieved
6100:the original
6082:. Retrieved
6075:the original
6046:
6001:
5995:
5986:
5976:
5948:
5941:
5904:
5900:
5861:
5854:
5842:
5830:
5810:
5803:
5795:close enough
5794:
5788:
5783:, p. 5.
5759:
5747:
5735:
5723:
5711:
5699:. Retrieved
5694:
5684:
5664:
5657:
5652:, p. 9.
5645:
5633:
5601:. Retrieved
5596:
5587:
5562:
5553:
5534:
5361:
5349:
5249:
5241:
5229:
5222:
5033:
5026:
4851:
4799:log change 1
4798:
4795:log change 0
4794:
4790:
4786:
4784:
4636:
4626:
4618:
4569:
4289:
4283:
4143:
4111:
3935:
3932:
3746:
3742:
3740:
3720:
3670:
3666:
3662:
3658:
3504:
3502:
2365:
2361:
2357:
2350:
2346:
2342:
2339:
2222:
2215:
2211:
2207:
2202:
2198:
2196:
2190:is close to
2055:
1975:Normalized:
1657:
1653:
1649:
1647:
1639:
1634:
1630:
1626:
1558:
1554:
1550:
1473:
1469:
1461:
1457:
1455:
1450:
1446:
1442:
1371:
1367:
1363:
1292:
1288:
1280:
1276:
1272:
1270:
1257:
1177:
1157:
1146:
1038:
1031:
1029:
1026:
971:
966:
964:
948:
900:Experimental
876:
868:
862:
859:experimental
858:
856:
803:Experimental
775:
769:
767:
538:
443:
435:
432:
416:
385:(â6) â (â10)
373:
366:
352:2 to 1
349:
240:
232:
230:
184:
182:
172:
168:
164:
155:
151:
147:
143:
136:
132:
125:
123:
113:
109:
105:
103:
97:
87:
82:
78:
74:
55:
51:
47:
39:
35:
34:, the terms
29:
6219:Subtraction
6204:Measurement
6185:20 November
5835:Vartia 1976
5764:Vartia 1976
5752:Vartia 1976
5650:Vartia 1976
5548:introducing
5354:towards 0.
4623:centinepers
1445:costs 125%
1267:Comparisons
920:Theoretical
908:Theoretical
889: Error
863:theoretical
828:Theoretical
811:Theoretical
787: Error
222:$ 50 to 150
71:percentages
6198:Categories
6109:2010-05-05
6084:2010-05-05
6039:References
6011:2011.14807
5914:1806.00699
5603:2023-09-24
5556:March 2011
5531:references
4810:log change
4633:Additivity
4619:log points
4290:log change
4278:See also:
3902:when
3834:when
3725:values in
1629:costs 80%
1553:costs 20%
1468:since car
1366:costs 25%
1155:function.
120:Definition
114:log points
110:log change
44:quantities
6028:227228720
5701:March 26,
5415:∈
5412:λ
5375:λ
5164:−
5105:−
5089:−
5086:≠
5060:−
5023:magnitude
4985:
4979:−
4949:
4896:
4863:
4820:
4581:
4525:−
4455:∫
4451:≈
4404:∫
4373:
4347:≪
4317:−
4227:−
4211:−
4186:−
4126:±
4039:
4026:≠
4012:−
3982:⋅
3885:−
3855:∫
3851:−
3817:−
3787:∫
3627:−
3476:
3424:≠
3403:
3392:−
3315:−
3203:−
3049:−
2888:−
2778:−
2667:−
2575:−
2538:−
2432:−
2297:−
2262:Δ
1875:is fixed.
1830:iff
1786:iff
1742:iff
1633:what car
1613:%
1586:$
1572:$
1555:less than
1537:%
1531:−
1522:−
1504:$
1490:$
1487:−
1426:%
1399:$
1385:$
1368:more than
1347:%
1320:$
1306:$
1241:%
1222:…
1210:%
1202:%
1196:−
1193:%
1130:%
1124:×
1098:−
1063:Δ
1010:%
986:−
951:numerator
932:×
904:−
885:%
840:×
807:−
783:%
656:−
589:reference
556:reference
487:−
333:−
294:Δ
203:100
183:The term
62:and is a
52:reference
5933:46928080
5718:, p. 141
5490:See also
5404:for any
3731:equality
2134:′
1262:Examples
423:fraction
419:unitless
64:unitless
56:starting
48:standard
6146:2683905
5742:, p.140
5544:improve
1637:costs.
955:commute
953:do not
410:
398:
394:
382:
196:1
30:In any
6214:Ratios
6176:
6144:
6053:
6026:
5964:
5931:
5869:
5818:
5672:
5533:, but
5288:+11.1
5274:â10.0
2205:, say
993:100000
989:100000
983:110000
718:
429:Domain
412:= â0.4
146:their
67:number
6170:(PDF)
6142:JSTOR
6122:(PDF)
6103:(PDF)
6096:(PDF)
6078:(PDF)
6067:(PDF)
6024:S2CID
6006:arXiv
5929:S2CID
5909:arXiv
5579:Notes
5330:â100
5316:+900
5285:+10.5
5271:â10.5
4679:15.5
3542:is:
1871:when
1652:from
1219:0.333
218:$ 100
154:, or
142:some
135:with
81:, or
60:ratio
6187:2022
6174:ISBN
6051:ISBN
5962:ISBN
5867:ISBN
5816:ISBN
5703:2021
5670:ISBN
5313:+230
5302:â90
5299:â230
4779:â25
4759:125
4699:4.5
4088:>
4076:>
3909:<
3841:>
3729:for
3505:mean
2379:Name
2201:and
2071:>
1895:>
1837:<
1821:<
1749:>
1733:>
1557:car
1525:0.20
1417:1.25
1370:car
1338:0.25
1271:Car
1166:and
935:100.
843:100.
768:The
543:and
359:â50%
158:is
131:and
112:and
92:and
38:and
6134:doi
6016:doi
5954:doi
5919:doi
5344:+â
5228:or
5032:or
4776:â50
4767:â50
4750:100
4739:â1
4736:â10
4727:â10
4719:21
3669:to
3661:to
3323:min
3266:min
3211:max
3154:max
1604:0.8
1595:000
1581:000
1513:000
1499:000
1423:125
1408:000
1394:000
1329:000
1315:000
1151:or
1149:%CH
1127:100
1001:0.1
512:ref
495:ref
464:ref
439:ref
407:â10
391:â10
376:ref
369:ref
357:is
326:ref
306:ref
261:ref
235:ref
175:ref
139:ref
128:ref
54:or
50:or
6200::
6140:,
6130:39
6128:,
6124:,
6069:.
6022:.
6014:.
6004:.
5985:.
5960:.
5927:.
5917:.
5905:10
5903:.
5899:.
5881:^
5771:^
5693:.
5612:^
5595:.
5486:.
5341:+â
5338:10
5327:ââ
5321:10
5310:10
5293:10
5282:10
5265:10
4982:ln
4946:ln
4893:ln
4860:ln
4849:.
4817:ln
4773:50
4764:50
4756:50
4753:50
4747:50
4744:50
4733:10
4724:10
4716:10
4713:10
4710:20
4707:10
4704:10
4696:â5
4693:10
4687:â5
4684:10
4673:10
4670:15
4664:10
4627:L%
4578:ln
4370:ln
4141:,
4036:ln
3749::
3473:ln
3400:ln
2364:,
2349:,
2220:.
2214:,
2194:.
1910:,
1631:of
1610:80
1589:50
1575:40
1534:20
1507:50
1493:10
1453:.
1447:of
1402:40
1388:50
1344:25
1323:40
1309:10
1228:33
1170:.
1153:Î%
1007:10
965:A
441::
396:=
336:1.
238::
228:.
205:cm
171:â
167:=
150:,
77:,
6189:.
6136::
6112:.
6087:.
6030:.
6018::
6008::
5989:.
5970:.
5956::
5935:.
5921::
5911::
5875:.
5824:.
5705:.
5678:.
5628:.
5606:.
5569:)
5563:(
5558:)
5554:(
5540:.
5472:1
5468:F
5445:0
5441:F
5419:R
5392:)
5389:y
5386:,
5383:x
5380:(
5371:F
5335:0
5324:0
5307:1
5296:1
5279:9
5268:9
5253:1
5250:V
5245:0
5242:V
5233:0
5230:V
5226:1
5223:V
5204:1
5200:V
5194:0
5190:V
5182:2
5178:)
5172:0
5168:V
5159:1
5155:V
5151:(
5124:1
5120:V
5113:1
5109:V
5100:0
5096:V
5079:0
5075:V
5068:0
5064:V
5055:1
5051:V
5037:1
5034:V
5030:0
5027:V
5005:1
5001:V
4995:0
4991:V
4976:=
4969:0
4965:V
4959:1
4955:V
4926:0
4923:=
4916:1
4912:V
4906:0
4902:V
4890:+
4883:0
4879:V
4873:1
4869:V
4837:)
4829:+
4826:1
4823:(
4814:=
4770:0
4730:0
4690:5
4676:5
4667:5
4601:0
4597:V
4591:1
4587:V
4551:=
4544:0
4540:V
4533:0
4529:V
4520:1
4516:V
4509:=
4502:0
4498:V
4493:V
4488:d
4476:1
4472:V
4464:0
4460:V
4446:V
4442:V
4437:d
4425:1
4421:V
4413:0
4409:V
4400:=
4393:0
4389:V
4383:1
4379:V
4350:1
4343:|
4336:0
4332:V
4325:0
4321:V
4312:1
4308:V
4301:|
4252:1
4249:=
4246:c
4241:L
4237:/
4233:)
4230:L
4224:K
4221:(
4214:1
4208:=
4205:c
4200:K
4196:/
4192:)
4189:L
4183:K
4180:(
4174:{
4169:=
4166:)
4163:L
4160:,
4157:K
4154:(
4151:H
4129:1
4123:=
4120:c
4091:0
4085:L
4082:,
4079:0
4073:K
4070:,
4067:0
4064:=
4061:c
4056:)
4053:L
4049:/
4045:K
4042:(
4029:0
4023:c
4018:)
4015:1
4007:c
4003:)
3999:L
3995:/
3991:K
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3985:(
3977:c
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3961:=
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3955:L
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3943:H
3912:L
3906:K
3896:t
3893:d
3888:1
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3878:t
3872:1
3867:L
3863:/
3859:K
3844:L
3838:K
3828:t
3825:d
3820:1
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3804:L
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3796:K
3791:1
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3775:=
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3760:(
3757:H
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3700:x
3696:1
3691:=
3686:x
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3645:)
3642:x
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3630:R
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3348:)
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3236:)
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2790:x
2787:y
2781:1
2773:x
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2141:1
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2106:/
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2022:|
2018:)
2015:y
2012:,
2009:1
2006:(
2003:R
1997:y
1994:d
1990:d
1972:.
1960:)
1957:y
1954:,
1951:x
1948:(
1945:R
1942:=
1939:)
1936:y
1933:a
1930:,
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1918:R
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1528:=
1519:=
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1216:=
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1133:.
1117:1
1113:V
1106:1
1102:V
1093:2
1089:V
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998:=
925:|
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