1276:
that
Knowledge is, in theory, designed to be accessible to everyone. There is nothing inherently wrong with presenting qualitative descriptions that, while incomplete, give the reader a general picture of the concept without getting lost in the details (provided that the description is qualified as being "qualitative", "over-simplified", or in some other way incomplete). In particular, the introduction should attempt to be as accessible as possible to the widest audience. We can get into the technicalities later in the article (technicalities that may be
85:
64:
184:
174:
153:
31:
653:. This Knowledge entry speaks about "a distribution has positive skew (right-skewed) if the right (higher value) tail is longer and negative skew (left-skewed) if the left (lower value) tail is longer". To my opinion skewness has nothing to do with the size of either tail, but more with the 'weight' associated with the tail. A reformulation of the definition on Knowledge would help a lot. --
22:
1230:
the protective sheath around the grass (coleoptiles) or the grass itself that's doing the growing upward away from gravity. The point is the plant responds to gravity in a nonlinear way (from what I can gather from the plot alone and no units or legend). A less jargonny caption would be less distracting from the core point (the shape of the distribution).
1067:
terms of standard deviation; he probably doesn't have a grasp on that topic either and there's no use sending him to a similarly rigorous and opaque article. If I didn't have a slight acquaintace with the topic, this article would be completely mysterious. The only layman explanation I see in
Knowledge for why skewed distributions are important is in
1188:
mean = median ≠ mode. For the one-side-bounded uniform PMF, (e.g. a uniform PMF over the set 1,2,3,4,...,inf) the number of modes is infinite, but the values of the modes aren't all infinite,. However for this bounded-on-one-side uniform distribution, the mean and median go to positive infinity, in the limit (I think). So, again, mean = median ≠ mode.
1545:, was to remove a redundant step in the calculation, to improve readability; however, this introduced a source of accidental errors by other editors (see below) due to them not understanding the flow of the working. To fix this original source of editing-errors, I will introduce an extra step into the working.
649:
tail (tail at small end of the distribution) is more pronounced that the right tail (tail at the large end of the distribution), the function is said to have negative skewness. If the reverse is true, it has positive skewness. If the two are equal, it has zero skewness." The difficulty lies within the word
2311:
The first example figure shows a distribution with positive skewness, but the caption described it as "non-negative (positive) skewness." This phrasing implies these terms are synonymous, and will likely mislead people unfamiliar with the terminology. Since the distribution clearly has non-zero skew,
1066:
Nice to know a rank outsider can help a bit. Actually the article has a more fundamental problem. Apparently it's written by insiders for insiders, thus is full of mathematical rigor and no understanding for outsiders. That's why Daniel above didn't catch on to the fact that skew is denominated in
728:
The formula for G1 is incorrect. The coefficient on the g1 term should be inverted. See Zar, Biostatistical
Analysis, 4th ed., p. 71, 6.9, where G1 is Zar's sqrt(b1). Using the k-statistic results of Stuart and Ord, p.422, 12.29, which present k-statistics in terms of the sample moments m2 and m3,
2054:
Am I correct in stating that, in both cases, the sequence of numbers given is right-leaning? Of course, this might not necessarily imply that the information given is incorrect (though I think that a calculation of the skewness would have to be given in order to justify the assertions made). There
1916:
The "3" in this coefficient is simply wrong. The article here gives the impression there's a distinction between "mode skewness" and
Pearson's first skewness coefficient. There simply isn't. The "3" in the *second* Pearson skewness coefficient arises because Pearson noticed (in his 1895 paper ) that
1346:
I thus suggest changing the headline of this section and re-write it or delete it altogether. I'd personally just delete it. However, if the consensus is that the real world importance of skewed distributions is important and should be discussed, maybe the introduction would be a good place to write
1134:
Someone should provide an example where the mean is not to the right of the median of a right-skewed distribution (a picture would be best). Since the article has a paragraph devoted to discussing the misconception regarding mean and skewness, there should be an example. Relatedly, the term "mass of
620:
has a negative skewness, but according to the current definition, it should have a longer left tail, which clearly is not the case. I checked
Mathworld for their definition, and this one seems to contradict the definition of Knowledge. Even if I am mistaken, this definition should be clarified and a
531:
In addition to the article being unclear to a layperson, many of us are concerned with interpreting our data than the beauty of the underlying method, though agreeably, a basic understanding of the methodology and assumptions enables one to use the appropriate tool effectively. In this case, a quick
2129:
Perhaps the article could offer a calculation of the "Pearson's method coefficient of skewness" for these sequences, as this provides ONE objective definition of the skewness (though, to be fair, there are several definitions of skewness and applying them all in such a simple example would probably
2113:
Having had a chance to read the "Relationship of mean and median" section (which comes after the "Introduction" section), the following observations seem important: 1) The sequence (40, 49, 50, 51) has mean 47.5 and median 49.5, so mean < median, so (by the logic/definition of the "Relationship
1443:
The reason should be obvious from the definition as the centralized moment. Please correct me if I'm wrong! (The previous edit(s) may have originated from some confusion between non-centralized and centralized moments as the numerator can also be written as E - 3\mu E + 2\mu^3 and is in fact how it
1229:
Yea, but the shape of the distribution did help me visualize skewness. It might be better to replace the bio-science jargon in the caption with something like: "the tendency of wheat grass to respond to the force of gravity and grow upward" Someone reading up on skewness doesn't likely care if it's
585:
An objective introduction should not include apparently biased references to the inferior quality of unspecified textbooks, especially when the author has not specified the overall textbook population to which the textbooks are being compared. Surely, not ALL textbook discussions of the median are
488:
Secondly, the standardised third moment is a ratio. It is usually impossible that the expectation of a ratio can be written in a simple form that generalises to all distributions. In fact the estimator for the central third moment in the numerator is unbiased, and the variance in the denominator is
1894:
I'm going to try to improve the introduction. The figure with the two graphs in the section below should be moved up to the introduction, so that the text can directly describe the graphs in that section! Also, the numerical examples are very confusing. It currently sounds as though the numbers in
1187:
This seems incorrect for the 1,2,3,4,... example series, if by that you/we mean a uniform PMF over some set of natural numbers. A bounded uniform PMF has as many modes as it has possible values or bins. So the mode of the sequence 1,2,3,4,5 is the set 1,2,3,4,5. The median is 3. The mean is 3. So,
1302:
The
Knowledge article on Skewness cites reference #14 (in Czech) concerning Cyhelsky's Skewness Coefficient. However, the formulation yields a negative value for a right-sided skew, which is commonly described as "positive" skewness. I don't read Czech. Perhaps someone who does read Czech could
1275:
An editor recently made some modifications to the introduction which were later reverted by another editor (for different reasons than I am discussing here). The original changes attempted to be more mathematically specific and technically accurate. I would encourage editors, however, to remember
648:
the skewness is defined as: 2*sqrt(2)*(5*pi-16)/((3*pi-8)^(3.0/2.0)), which is approximately -0.485692828, clearly negative. The similar definition is supported by
Mathworld. When defining the skewness Mathworld says "Skewness is a measure of the degree of asymmetry of a distribution. If the left
2074:
On second thoughts, it does seem as if the quoted remarks are making assertions about the importance of the mean (and so the quoted remarks MAY be correct in some sense). However, this is a little confusing because the "Introduction" correlates the idea of negative skewness with right-leaning
2125:
It would be good to include some information about those situations where the different definitions of skewness agree with each other, and disagree with each other (including nonparametric skewness). Of course, this might make the article too long due to the number of definitions of skewness
1872:
I've removed the item on
Cyhelský's skewness as it does not have a proper academic published reference. In fact, a simple search on Google Scholar and Google Books, shows that no one mentions it, other than a couple of references back to this wikipedia article! I fear this is another case of
1342:
I just read the article and was slightly perturbed by the "Applications" section. The skewness is a mathematical measure of a probability distribution and hence has no application as such. The fact that some stochastic models make assumptions about zero skewness and may not be useful to model
1154:
The sentence "If there is zero skewness (i.e., the distribution is symmetric) then the mean = median. (If, in addition, the distribution is unimodal, then the mean = median = mode.)" is wrong in the same way that the subsequent paragraph beginning "many textbooks" is describing a common
1439:
Just to make sure I am not mistaken: here's the full history for someone to double check. Previously (before July) we had for the numerator of gamma_1: E - 3\mu\sigma^2 - \mu^3 which was changed to E - 3\mu\sigma^2 + 2\mu^3 which I have now reverted to E - 3\mu\sigma^2 - \mu^3
1161:
e.g. consider a r.v. taking the value -4 with probability 1/3, 1 with probability 1/2, and 5 with probability 1/6 (you could write the numbers on a die - -4 on two faces, 1 on three and 5 on the last face). The mean is 0, the third central moment is 0, but the median is 1.
2121:
However, this is clearly confusing in that, when a novice reads the article from beginning to end, they require concepts that are presented 'later', 'earlier on' in the article. Perhaps the "Introduction" section should mention the definition of *Non-parametric skewness*?
2050:
The page states: "We can transform this sequence into a negatively skewed distribution by adding a value far below the mean, e.g. (40, 49, 50, 51). Similarly, we can make the sequence positively skewed by adding a value far above the mean, e.g. (49, 50, 51, 60)."
1322:
Concerning my "Skew askew" post and quoting
Professor Rosanne - "Never mind." Data that skew to the right-side tail do have more values below the mean than above, and the Knowledge expression for the Cyhilsky coefficient does yield a positive value.
664:. I just simulated ten million samples from it (by simulating three component velocities from a normal distribution and calculating speed from them) and while the mean and the variance match the formulae given in the article, the skewness came out at
1071:
where it's only a side point. Where that explanation belongs, and a broader discussion of the significance of skew, with examples such as why most people are poorer than average but have more than the average number of legs, is in this article.
564:
One can always add a narrow "peak" to the density function, so that the skewness is not altered significantly but the mode is. Perhaps something with unimodality of the distribution? Or is it to be taken just as a rule of thumb?
1846:"there are negative values missing" (if the x-axis has lower/negative values on the left and higher/positive values on the right) and vice versa with positive skew. But there have to be better ways -- anyone came across any?
939:
999:
1207:
I understand that the graph demonstrates skewness, but most people don't know what the words "gravitropic" or "coleoptiles" mean, so the graph provides no additional understanding of the meaning of skewness.
2055:
is a possibility that the quoted comment is a mistake and that, in fact, both sequences have negative skew (as, indeed, they appear to using the inuitive notions introduced in the "Introduction" section).
2075:
distributions (and vice versa) and then goes onto give examples, which are both right-leaning, but with negative and positive skewness, respectively. This would seem to be pedagogically unsound.
532:
guide to the interpretation of the resultant statistic is important. How does a skewness of 1 compare to a skewness of 0.5 or -1 etc.? Perhaps that could be covered in the graph if not in the text.
135:
686:
proof to be perfectly adequate for all practical purposes, but it's still somehow satisfying to check the algebra even though when the two disagree it's nearly always my algebra that's at fault).
586:
inferior to the introduction presented by the smug author. The author should justify the comments about textbooks, including a justification of the value of the textbook comments themselves!
497:
Adding two graphs here to illustrate visually the difference between left and right skew would be enormously beneficial. I got them confused until someone drew it on the board in stats class.
846:
has a small-N formula for the standard error of skewness, and says the distribution of skewness/SES ~ N(0,1). Is there some better reference to point to for a statistical test of
Skewness?
240:
357:
2376:
2312:
describing it as non-negative is unspecific, redundant and verbose. I propose the caption be changed to simply state it has positive skewness, and drop the unnecessary "non-negative."
1816:
is not known. Therefore, this edit will be reverted back to the original outcome. Please reply here to explain why this edit was made, if you feel that the revert was not called for.
422:
391:
322:
291:
2294:
2231:
1543:
1416:
form and this has carried through into some later works. It is therefore impirtant to know what definition of sample skewness is being used, particularly for small samples.
2258:
1706:
1613:
1500:
1664:
1580:
1436:
Recently someone edited the formula for G1 and "corrected" the last term in the numerator. That "correction" was undone soon after, and the dance was repeated once more.
2191:
1778:
2366:
1135:
a distribution" is used in this article when there is no explanation what that is. The reason I say that is because it is vital to understanding of mean and skewness.
1920:
The MathWorld reference for this appears to simply be wrong. The "3" in relation to skewness involving the mode is not part of Pearson's work as far as I can see.
2381:
1845:
Positive and negative skew is something that is hard to remember -- does anyone know some helps to remind here? I usually think about it as "negative skew" =: -->
1814:
865:
Can somebody give a sense of skale to skewness? e.g. what does a skewness of 1 mean? What does a skewness of -1 mean? (rather than just positive or negative)
2391:
125:
35:
1917:
often the median lay about 1/3 of the way from the mean to the mode (and so median-skewness should be multiplied by 3 to make it like the mode-skewness).
1249:
Fixed, it appears someone else noticed the formula had issues since there was a citation needed note there. You can derive it from the original formula.
2406:
2126:
presented within the article (there are possibly 9 different definitions of skewness presented within the article, depending upon how you count them).
230:
1780:, but also introduced an error. I am not sure of the reason behind this edit, as the standard deviation is known, so the variance is easily calculated;
522:
Almost impossible for a lay person without knowledge of statistics to understand this article. There needs to be a more general introduction given. --
2155:
According to the standardized moment Knowledge entry, the 3rd moment is designated as \gamma_3 and not as \gamma_1 as done everywhere in this article.
759:. They also allege an unbiased estimator for finite populations using a different regularization factor. Could somebody please reconcile the two? --
2361:
2396:
1051:
I've removed the snakes, which appeared to be straightforward vandalism. Thanks for spotting them. How they survived for over a year I've no idea.
2371:
458:
672:. I should probably go through the algebra to derive the skewness from the raw moments before I change the M-B dist article, but i'll remove the
890:
206:
1303:"check" to see whether the Knowledge formulation should be revised to result in a negative coefficient when the data show a left-sided skew.
2401:
1451:
1354:
950:
101:
2333:
Why keep the image "A general relationship of mean and median under differently skewed unimodal distribution" if the relationship is wrong?
1895:
the examples are taken from the graph at the very top of the article. This doesn't make sense, obviously, since that graph lacks an x-axis.
489:
unbiased (but its 3/2 power is biased). By the linearisation method (or delta method) we can say that the ratio is approximately unbiased.
2386:
1826:
1380:
834:
A rough estimate of the "standard error of Skewness" (SES) for a normal distribution is sqrt(6/N), as seen on a few places out in the web:
593:
485:
Firstly, if the sample is from a finite population, the observations are dependent, while the proof of unbiasedness requires independence.
2334:
1256:
1169:
1092:
1876:
1853:
1215:
1136:
866:
542:
Comment from main article moved to appropriate page: Section to develop: Why should we care about skew? what difference does it make!
2315:
1158:
Specifically, if you define skewness in terms of third moment, mean doesn't necessarily equal median when the third moment is zero.
197:
158:
92:
69:
2027:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
2356:
1822:
15 February 2012 = My edit to revert back to using the variance, and to prevent editors making the recurrent editing-mistake.
661:
645:
630:
617:
613:
470:
1091:
It would be nice to have a figure here that shows a bunch of probability density diagrams and their Skewness as examples.--
806:
article, can someone put a similar table in this article. and talk about the range of skewness, and the rang of kurtosis.
1373:
According to the Knowledge page for variance, sample variance is the 1/(n-1) form, but the formula for g1 is using 1/n.
779:
What is a tetrete (in the description of the first figure)? I have never heard this term and cannot find a definition.
441:
44:
729:
you can do the algebra, getting g1 in terms of the ratio of k-statistics, and see that Zar is correct. -- J.D. Opdyke
2109:
Applying the Ideas of the "Relationship of mean and median" section to the "Introduction" section To Resolve Confusion
1455:
1358:
482:
The statement about unbiasedness of the estimate of skewness given needs further qualification for two reasons.
331:
1830:
1006:
683:
597:
1384:
433:
2338:
2094:
For the reasons described above, I have included a clarification needed tag in the "Introduction" section.
1939:(1895). "Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material".
1173:
1096:
1260:
1219:
1120:
1077:
1041:
676:
437:
2319:
1880:
1857:
1140:
882:
870:
512:
396:
365:
296:
265:
1343:
processes that exhibit non-zero skewness does not mean that skewness is "useful" or is being "applied".
533:
2263:
2200:
2138:
2099:
2080:
2060:
1983:
1068:
566:
50:
1505:
183:
1979:
700:
The incorrect sign (now corrected) can be confirmed algebraically by starting from the result for the
490:
84:
63:
2260:
notation comes from - but of course, there should be a source for it. Otherwise, "normalizing" it to
1948:
1849:
1447:
1444:
is written on MathWorld). We may have to watch this formula for changes again to avoid an edit war.
1376:
1350:
1252:
1235:
1211:
1193:
1165:
589:
559:
Skewness affects Mean the most and Mode the least. For a positivevely skewed distribution, Mean : -->
1898:
21:
2297:
2194:
2084:
1902:
1324:
1304:
784:
654:
622:
462:
205:
on Knowledge. If you would like to participate, please visit the project page, where you can join
100:
on Knowledge. If you would like to participate, please visit the project page, where you can join
2163:
1964:
1421:
1328:
1308:
1288:
1116:
1073:
1037:
709:
466:
189:
2236:
173:
152:
1669:
1585:
1472:
1994:
1639:
1555:
1021:
788:
1185:"...then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4,..."
2176:
2134:
2133:
The order in which the ideas are presented in this article can clearly lead to confusion.
2095:
2076:
2056:
1956:
1725:
764:
701:
820:
Can someone put a table to talk about the variance of skewness for different distribution?
616:
it appeared to me that the definitions of positive and negative skewness got mixed up. The
1231:
1189:
851:
2103:
843:
1952:
1203:
What the heck does "gravitropic response of wheat coleoptiles" have to do with skewness?
669:
2002:
1056:
691:
1884:
1783:
2350:
2159:
2024:
1417:
1284:
705:
2036:
is noticibly better, with examination of various distributions and some references.
1936:
1017:
821:
807:
634:
576:
543:
2042:
Last edited at 23:28, 19 April 2007 (UTC). Substituted at 20:14, 1 May 2016 (UTC)
2114:
of mean and median" section), this sequence has NEGATIVE *nonparametric skew*.
682:
tag from this article now (I'm originally a physicist so i consider this sort of
359:; their expected values can even have the opposite sign from the true skewness."
2070:
Introduction Of Ideas Pertaining to the Mean's Relevance In Determining Skewness
1987:
760:
756:
734:
523:
202:
2046:
Possible Error on Page Pertaining to Sequences which Seem To Have Negative Skew
847:
179:
97:
2117:
2) The sequence (49, 50, 51, 60) has mean 52.5 and median 50.5, so mean : -->
839:
2152:
Tried to see if any of the other threads discussed this but did not find...
1998:
1052:
687:
509:
Generally discussions are not 'removed', but merely, eventually, 'archived'.
1960:
1130:
Example of a right-skewed distribution where the mean is left of the median
434:
https://github.com/scipy/scipy/blob/v1.1.0/scipy/stats/stats.py#L1002-L1074
934:{\displaystyle p={\frac {1}{2}}-{\frac {1}{\sqrt {20}}}\approx 0.2763932.}
561:
Mode and for a negatively skewed distribution, Mean < Median < Mode
2033:
803:
1115:. Is that right? Should this article discuss the multidimensional case?
994:{\displaystyle p={\frac {1}{2}}-{\frac {1}{\sqrt {8}}}\approx 0.1464466}
1466:
Below is the editing history, and my deductions to why they were made:
1150:
Introduction has a mistake related to the common mistake it points out
556:
I think the following paragraph cannot hold under general conditions:
1967:
1112:
1111:
I gather that the multidimensional case of skewness is a third-order
835:
262:
In the article, the following was stated: "In general, the ratios
660:
I'm pretty sure the problem is with the formula for skewness in
2342:
2323:
2300:
2167:
2142:
2064:
2006:
1906:
1861:
1834:
1459:
1425:
1388:
1362:
1332:
1312:
1292:
1264:
1239:
1223:
1197:
1177:
1144:
1124:
1100:
1081:
1060:
1045:
1025:
874:
855:
824:
810:
791:
768:
713:
695:
637:
601:
579:
569:
546:
536:
526:
515:
445:
2158:
I believe it should be modified, unless I missed something...
2118:
median, so this sequence has POSITIVE *nonparametric skew*.
1036:
Eh? Isn't a statistical snake something entirely different?
15:
945:
That has a skewness of 1. One with probability of success
1708:
for readability, and possibly to reduce people making the
2197:, and also not in linked articles. It's actually called
2019:
1625:
27 July 2011 = @08:18 Second incident of the same
668:
0.484. I think the source of the error was most likely
455:
HOW ABOUT A POSITIVE SKEW VERSES NEGATIVE SKEW PICTURE
1722:
1 August 2011 = @23:28 Variance converted back to
1245:
There was a bad error in the uncentered moment formula
844:
http://www.xycoon.com/skewness_small_sample_test_1.htm
2266:
2239:
2203:
2179:
1786:
1728:
1672:
1642:
1588:
1558:
1508:
1475:
953:
893:
401:
399:
370:
368:
336:
334:
301:
299:
270:
268:
802:
Just like "Kurtosis of well-known distributions" in
201:, a collaborative effort to improve the coverage of
96:, a collaborative effort to improve the coverage of
2288:
2252:
2225:
2185:
1808:
1772:
1700:
1658:
1607:
1574:
1537:
1494:
993:
933:
416:
385:
351:
316:
285:
2069:
1819:1 August 2011 = @23:29 Introduced error fixed.
2377:Knowledge level-5 vital articles in Mathematics
2023:, and are posted here for posterity. Following
1941:Philosophical Transactions of the Royal Society
1183:Regarding the attempt to correct the statement
2233:(with a tilde) there. I don't know where the
2017:The comment(s) below were originally left at
670:the MathWorld article on the M-B distribution
8:
1873:Knowledge being punked. Discussion please?
352:{\displaystyle {\displaystyle \gamma _{1}}}
2148:Designation of the 3rd standardized moment
840:http://mathworld.wolfram.com/Skewness.html
608:Positive versus negative skewness mixed up
147:
58:
2280:
2269:
2268:
2265:
2244:
2238:
2217:
2206:
2205:
2202:
2178:
1995:Feel free to go ahead and fix the article
1797:
1785:
1764:
1739:
1727:
1680:
1671:
1650:
1641:
1599:
1587:
1566:
1557:
1529:
1516:
1507:
1486:
1474:
973:
960:
952:
913:
900:
892:
644:According to the Knowledge entry for the
406:
400:
398:
375:
369:
367:
341:
335:
333:
306:
300:
298:
275:
269:
267:
830:The Distribution of Skewness and testing
2367:Knowledge vital articles in Mathematics
2089:
1928:
149:
60:
19:
2090:Addition of 'Clarification Needed' Tag
575:It's incorrect, so I've removed it. --
328:estimators of the population skewness
2382:B-Class vital articles in Mathematics
1409:. Early workers in skewness used the
612:When looking at the skewness for the
417:{\displaystyle {\displaystyle G_{1}}}
386:{\displaystyle {\displaystyle b_{1}}}
317:{\displaystyle {\displaystyle G_{1}}}
286:{\displaystyle {\displaystyle b_{1}}}
7:
2329:Relationship between mean and median
1867:
1395:Yes. See the different formulae for
1087:Example figures with skewness values
195:This article is within the scope of
90:This article is within the scope of
2392:High-importance Statistics articles
2289:{\displaystyle {\tilde {\mu }}_{3}}
2226:{\displaystyle {\tilde {\mu }}_{3}}
783:Looks as though it may have been a
49:It is of interest to the following
1715:1 August 2011 = @23:16 Second
1632:27 July 2011 = @15:22 Second
1538:{\displaystyle 3\mu ^{3}-\mu ^{3}}
836:http://www.jalt.org/test/bro_1.htm
621:picture would definitily help. --
14:
2407:Mid-priority mathematics articles
2025:several discussions in past years
1016:as the probability of success. --
1005:has a skewness of 2, as does any
215:Knowledge:WikiProject Mathematics
2362:Knowledge level-5 vital articles
1469:22 April 2011 = My revision,
657:08:00, 26 September 2006 (CEST)
218:Template:WikiProject Mathematics
182:
172:
151:
110:Knowledge:WikiProject Statistics
83:
62:
29:
20:
2397:WikiProject Statistics articles
1868:Cyhelský's skewness coefficient
235:This article has been rated as
130:This article has been rated as
113:Template:WikiProject Statistics
2372:B-Class level-5 vital articles
2301:07:30, 10 September 2019 (UTC)
2274:
2211:
2168:01:59, 10 September 2019 (UTC)
1911:
1907:23:48, 16 September 2014 (UTC)
1862:09:52, 12 September 2012 (UTC)
1803:
1790:
1761:
1754:
1745:
1732:
1695:
1689:
1145:23:47, 25 September 2009 (UTC)
737:gives a different formula for
662:Maxwell-Boltzmann distribution
646:Maxwell-Boltzmann-distribution
638:23:35, 16 September 2006 (UTC)
631:Maxwell-Boltzmann-distribution
618:Maxwell-Boltzmann-distribution
614:Maxwell-Boltzmann-distribution
432:! See e.g. SciPy source code:
1:
2324:08:24, 20 February 2022 (UTC)
1993:Well spotted and researched!
1835:09:18, 15 February 2012 (UTC)
1369:sample variance in g1 formula
1240:21:00, 23 December 2014 (UTC)
1198:20:50, 23 December 2014 (UTC)
1012:To change the sign just take
209:and see a list of open tasks.
104:and see a list of open tasks.
2402:B-Class mathematics articles
2143:19:49, 3 February 2018 (UTC)
2104:19:11, 3 February 2018 (UTC)
2085:17:05, 3 February 2018 (UTC)
2065:16:53, 3 February 2018 (UTC)
2007:20:09, 1 February 2015 (UTC)
1988:15:58, 1 February 2015 (UTC)
1636:has not been fixed, but the
1333:00:10, 28 October 2010 (UTC)
1313:23:52, 27 October 2010 (UTC)
1082:18:48, 8 February 2009 (UTC)
1061:22:53, 4 February 2009 (UTC)
1046:19:11, 4 February 2009 (UTC)
885:with probability of success
875:21:09, 2 November 2008 (UTC)
2387:B-Class Statistics articles
2253:{\displaystyle \gamma _{1}}
1125:11:45, 19 August 2009 (UTC)
733:I am no expert on this but
633:has a negative skewness? --
602:14:55, 13 August 2009 (UTC)
2423:
2343:14:19, 2 August 2022 (UTC)
1701:{\displaystyle 3\mu ^{2}E}
1618:11 July 2011 = First
1608:{\displaystyle +2\mu ^{3}}
1548:9 July 2011 = First
1495:{\displaystyle +2\mu ^{3}}
1460:23:31, 1 August 2011 (UTC)
1426:13:02, 16 March 2011 (UTC)
1389:22:50, 15 March 2011 (UTC)
1224:05:57, 31 March 2010 (UTC)
1026:21:01, 21 April 2009 (UTC)
769:08:05, 24 March 2012 (UTC)
625:22:31 September 1st, 2006
527:21:52, 30 March 2006 (UTC)
2032:
1890:Reorganizing Introduction
1885:00:18, 18 June 2014 (UTC)
1659:{\displaystyle 3\mu ^{3}}
1575:{\displaystyle -\mu ^{3}}
1363:22:11, 7 March 2011 (UTC)
1265:21:53, 9 April 2010 (UTC)
1178:00:26, 2 March 2010 (UTC)
1101:12:40, 16 June 2009 (UTC)
856:18:27, 17 June 2008 (UTC)
825:23:57, 14 July 2007 (UTC)
811:20:43, 13 July 2007 (UTC)
792:14:04, 4 April 2007 (UTC)
629:What makes you think the
580:13:03, 23 June 2006 (UTC)
570:11:56, 23 June 2006 (UTC)
537:20:30, 24 June 2006 (UTC)
516:06:51, 19 July 2007 (UTC)
234:
167:
129:
78:
57:
1912:Pearson's first skewness
1293:14:49, 26 May 2010 (UTC)
1007:exponential distribution
816:the variance of skewness
724:incorrect formula for G1
714:16:13, 25 May 2010 (UTC)
696:15:27, 25 May 2010 (UTC)
684:experimental mathematics
547:03:21, 14 May 2006 (UTC)
446:16:15, 6 June 2018 (UTC)
241:project's priority scale
2186:{\displaystyle \gamma }
1773:{\displaystyle E-E^{2}}
1271:Comment on introduction
198:WikiProject Mathematics
2357:B-Class vital articles
2290:
2254:
2227:
2187:
2020:Talk:Skewness/Comments
1961:10.1098/rsta.1895.0010
1810:
1774:
1702:
1660:
1609:
1576:
1539:
1496:
1347:a sentence about it?
1107:Multidimensional case?
995:
935:
883:Bernoulli distribution
418:
387:
353:
318:
287:
93:WikiProject Statistics
2291:
2255:
2228:
2188:
1811:
1775:
1703:
1661:
1610:
1577:
1540:
1497:
1069:Lake Woebegone effect
996:
936:
755:, both involving the
461:comment was added by
419:
388:
354:
319:
288:
36:level-5 vital article
2296:might be helpful. --
2264:
2237:
2201:
2177:
1784:
1726:
1670:
1640:
1586:
1556:
1506:
1473:
951:
891:
397:
366:
332:
297:
266:
221:mathematics articles
2298:User:Haraldmmueller
1953:1895RSPTA.186..343P
785:typographical error
552:Missing assumption?
116:Statistics articles
2286:
2250:
2223:
2183:
2013:Assessment comment
1806:
1770:
1698:
1656:
1605:
1572:
1535:
1492:
1280:of some readers).
1155:misunderstanding.
991:
931:
501:Needs better intro
438:17 kutalmis bercin
414:
412:
383:
381:
349:
347:
314:
312:
283:
281:
190:Mathematics portal
45:content assessment
2277:
2214:
2173:I did not find a
2040:
2039:
1852:comment added by
1809:{\displaystyle E}
1719:has been removed.
1450:comment added by
1379:comment added by
1353:comment added by
1255:comment added by
1214:comment added by
1168:comment added by
983:
982:
968:
923:
922:
908:
798:range & table
592:comment added by
474:
258:Unbiased skewness
255:
254:
251:
250:
247:
246:
146:
145:
142:
141:
2414:
2295:
2293:
2292:
2287:
2285:
2284:
2279:
2278:
2270:
2259:
2257:
2256:
2251:
2249:
2248:
2232:
2230:
2229:
2224:
2222:
2221:
2216:
2215:
2207:
2192:
2190:
2189:
2184:
2130:not be wise).
2030:
2029:
2022:
1972:
1971:
1933:
1864:
1815:
1813:
1812:
1807:
1802:
1801:
1779:
1777:
1776:
1771:
1769:
1768:
1744:
1743:
1707:
1705:
1704:
1699:
1685:
1684:
1665:
1663:
1662:
1657:
1655:
1654:
1614:
1612:
1611:
1606:
1604:
1603:
1581:
1579:
1578:
1573:
1571:
1570:
1544:
1542:
1541:
1536:
1534:
1533:
1521:
1520:
1501:
1499:
1498:
1493:
1491:
1490:
1462:
1432:/* Definition */
1391:
1365:
1267:
1226:
1180:
1000:
998:
997:
992:
984:
978:
974:
969:
961:
940:
938:
937:
932:
924:
918:
914:
909:
901:
754:
745:
702:chi distribution
681:
675:
604:
491:User:Terry Moore
456:
423:
421:
420:
415:
413:
411:
410:
392:
390:
389:
384:
382:
380:
379:
358:
356:
355:
350:
348:
346:
345:
323:
321:
320:
315:
313:
311:
310:
292:
290:
289:
284:
282:
280:
279:
223:
222:
219:
216:
213:
192:
187:
186:
176:
169:
168:
163:
155:
148:
136:importance scale
118:
117:
114:
111:
108:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
2422:
2421:
2417:
2416:
2415:
2413:
2412:
2411:
2347:
2346:
2331:
2309:
2267:
2262:
2261:
2240:
2235:
2234:
2204:
2199:
2198:
2175:
2174:
2150:
2111:
2092:
2072:
2048:
2018:
2015:
1977:
1976:
1975:
1935:
1934:
1930:
1914:
1892:
1870:
1847:
1843:
1793:
1782:
1781:
1760:
1735:
1724:
1723:
1676:
1668:
1667:
1666:was changed to
1646:
1638:
1637:
1595:
1584:
1583:
1562:
1554:
1553:
1525:
1512:
1504:
1503:
1482:
1471:
1470:
1452:134.174.140.104
1445:
1434:
1415:
1408:
1401:
1374:
1371:
1355:134.174.140.104
1348:
1340:
1320:
1300:
1273:
1250:
1247:
1209:
1205:
1163:
1152:
1132:
1117:—Ben FrantzDale
1109:
1089:
1034:
949:
948:
889:
888:
863:
832:
818:
800:
777:
753:
750:
747:
744:
741:
738:
726:
679:
673:
610:
587:
554:
513:128.250.204.118
503:
480:
457:—The preceding
453:
402:
395:
394:
371:
364:
363:
337:
330:
329:
302:
295:
294:
271:
264:
263:
260:
220:
217:
214:
211:
210:
188:
181:
161:
132:High-importance
115:
112:
109:
106:
105:
73:High‑importance
72:
43:on Knowledge's
40:
30:
12:
11:
5:
2420:
2418:
2410:
2409:
2404:
2399:
2394:
2389:
2384:
2379:
2374:
2369:
2364:
2359:
2349:
2348:
2330:
2327:
2308:
2305:
2304:
2303:
2283:
2276:
2273:
2247:
2243:
2220:
2213:
2210:
2182:
2149:
2146:
2110:
2107:
2091:
2088:
2071:
2068:
2047:
2044:
2038:
2037:
2014:
2011:
2010:
2009:
1974:
1973:
1927:
1926:
1922:
1913:
1910:
1891:
1888:
1869:
1866:
1842:
1839:
1838:
1837:
1827:123.243.217.67
1823:
1820:
1817:
1805:
1800:
1796:
1792:
1789:
1767:
1763:
1759:
1756:
1753:
1750:
1747:
1742:
1738:
1734:
1731:
1720:
1713:
1697:
1694:
1691:
1688:
1683:
1679:
1675:
1653:
1649:
1645:
1630:
1623:
1616:
1602:
1598:
1594:
1591:
1569:
1565:
1561:
1546:
1532:
1528:
1524:
1519:
1515:
1511:
1489:
1485:
1481:
1478:
1467:
1433:
1430:
1429:
1428:
1413:
1406:
1399:
1381:128.111.110.55
1370:
1367:
1339:
1338:"Applications"
1336:
1319:
1316:
1299:
1296:
1278:over the heads
1272:
1269:
1246:
1243:
1204:
1201:
1151:
1148:
1131:
1128:
1108:
1105:
1088:
1085:
1064:
1063:
1033:
1030:
1029:
1028:
1010:
1003:
1002:
1001:
990:
987:
981:
977:
972:
967:
964:
959:
956:
943:
942:
941:
930:
927:
921:
917:
912:
907:
904:
899:
896:
862:
859:
831:
828:
817:
814:
799:
796:
795:
794:
776:
773:
772:
771:
751:
748:
742:
739:
725:
722:
721:
720:
719:
718:
717:
716:
677:expert subject
641:
640:
609:
606:
594:63.240.104.100
583:
582:
553:
550:
534:216.129.143.26
520:
519:
502:
499:
496:
479:
478:Early comments
476:
452:
449:
409:
405:
378:
374:
344:
340:
309:
305:
278:
274:
259:
256:
253:
252:
249:
248:
245:
244:
233:
227:
226:
224:
207:the discussion
194:
193:
177:
165:
164:
156:
144:
143:
140:
139:
128:
122:
121:
119:
102:the discussion
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2419:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2354:
2352:
2345:
2344:
2340:
2336:
2335:93.147.160.21
2328:
2326:
2325:
2321:
2317:
2313:
2307:Non-negative?
2306:
2302:
2299:
2281:
2271:
2245:
2241:
2218:
2208:
2196:
2180:
2172:
2171:
2170:
2169:
2165:
2161:
2156:
2153:
2147:
2145:
2144:
2140:
2136:
2131:
2127:
2123:
2119:
2115:
2108:
2106:
2105:
2101:
2097:
2087:
2086:
2082:
2078:
2067:
2066:
2062:
2058:
2052:
2045:
2043:
2035:
2031:
2028:
2026:
2021:
2012:
2008:
2004:
2000:
1996:
1992:
1991:
1990:
1989:
1985:
1981:
1969:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1937:Pearson, Karl
1932:
1929:
1925:
1921:
1918:
1909:
1908:
1904:
1900:
1896:
1889:
1887:
1886:
1882:
1878:
1874:
1865:
1863:
1859:
1855:
1851:
1840:
1836:
1832:
1828:
1824:
1821:
1818:
1798:
1794:
1787:
1765:
1757:
1751:
1748:
1740:
1736:
1729:
1721:
1718:
1714:
1711:
1692:
1686:
1681:
1677:
1673:
1651:
1647:
1643:
1635:
1631:
1628:
1624:
1621:
1617:
1600:
1596:
1592:
1589:
1567:
1563:
1559:
1551:
1547:
1530:
1526:
1522:
1517:
1513:
1509:
1487:
1483:
1479:
1476:
1468:
1465:
1464:
1463:
1461:
1457:
1453:
1449:
1441:
1437:
1431:
1427:
1423:
1419:
1412:
1405:
1398:
1394:
1393:
1392:
1390:
1386:
1382:
1378:
1368:
1366:
1364:
1360:
1356:
1352:
1344:
1337:
1335:
1334:
1330:
1326:
1317:
1315:
1314:
1310:
1306:
1297:
1295:
1294:
1290:
1286:
1281:
1279:
1270:
1268:
1266:
1262:
1258:
1257:171.66.85.193
1254:
1244:
1242:
1241:
1237:
1233:
1227:
1225:
1221:
1217:
1213:
1202:
1200:
1199:
1195:
1191:
1186:
1181:
1179:
1175:
1171:
1170:58.171.86.193
1167:
1159:
1156:
1149:
1147:
1146:
1142:
1138:
1129:
1127:
1126:
1122:
1118:
1114:
1106:
1104:
1102:
1098:
1094:
1093:131.111.176.9
1086:
1084:
1083:
1079:
1075:
1074:Jim.henderson
1070:
1062:
1058:
1054:
1050:
1049:
1048:
1047:
1043:
1039:
1038:Jim.henderson
1031:
1027:
1023:
1019:
1015:
1011:
1008:
1004:
988:
985:
979:
975:
970:
965:
962:
957:
954:
947:
946:
944:
928:
925:
919:
915:
910:
905:
902:
897:
894:
887:
886:
884:
880:
879:
878:
876:
872:
868:
860:
858:
857:
853:
849:
845:
841:
837:
829:
827:
826:
823:
815:
813:
812:
809:
805:
797:
793:
790:
786:
782:
781:
780:
774:
770:
766:
762:
758:
736:
732:
731:
730:
723:
715:
711:
707:
703:
699:
698:
697:
693:
689:
685:
678:
671:
667:
663:
659:
658:
656:
652:
647:
643:
642:
639:
636:
632:
628:
627:
626:
624:
619:
615:
607:
605:
603:
599:
595:
591:
581:
578:
574:
573:
572:
571:
568:
567:88.101.32.104
562:
557:
551:
549:
548:
545:
540:
538:
535:
529:
528:
525:
517:
514:
510:
507:
506:
505:
500:
498:
494:
492:
486:
483:
477:
475:
472:
468:
464:
460:
450:
448:
447:
443:
439:
435:
431:
427:
407:
403:
376:
372:
362:However, the
360:
342:
338:
327:
307:
303:
276:
272:
257:
242:
238:
232:
229:
228:
225:
208:
204:
200:
199:
191:
185:
180:
178:
175:
171:
170:
166:
160:
157:
154:
150:
137:
133:
127:
124:
123:
120:
103:
99:
95:
94:
89:
86:
82:
81:
77:
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2332:
2314:
2310:
2157:
2154:
2151:
2132:
2128:
2124:
2120:
2116:
2112:
2093:
2073:
2053:
2049:
2041:
2016:
1978:
1944:
1940:
1931:
1923:
1919:
1915:
1897:
1893:
1877:129.127.28.3
1875:
1871:
1854:134.2.234.65
1848:— Preceding
1844:
1717:mistaken fix
1716:
1710:mistaken fix
1709:
1634:mistaken fix
1633:
1627:mistaken fix
1626:
1620:mistaken fix
1619:
1550:mistaken fix
1549:
1446:— Preceding
1442:
1438:
1435:
1410:
1403:
1396:
1372:
1345:
1341:
1321:
1318:Skewer askew
1301:
1282:
1277:
1274:
1248:
1228:
1216:75.32.245.85
1206:
1184:
1182:
1160:
1157:
1153:
1137:71.64.105.56
1133:
1110:
1090:
1065:
1035:
1013:
867:199.212.7.17
864:
833:
819:
801:
778:
727:
665:
650:
611:
584:
563:
560:Median : -->
558:
555:
541:
530:
521:
508:
504:
495:
493:11 Jun 2005
487:
484:
481:
454:
429:
428:rather than
425:
361:
325:
261:
237:Mid-priority
236:
196:
162:Mid‑priority
131:
91:
51:WikiProjects
34:
2316:82.32.76.32
2135:ASavantDude
2096:ASavantDude
2077:ASavantDude
2057:ASavantDude
1980:Glenbarnett
1947:: 343–414.
1582:changed to
1502:changed to
1375:—Preceding
1349:—Preceding
1298:Skew askew?
1251:—Preceding
1210:—Preceding
1164:—Preceding
881:Consider a
757:sample mean
588:—Preceding
212:Mathematics
203:mathematics
159:Mathematics
2351:Categories
1924:References
1825:—Ricketts
1232:Hobsonlane
1190:Hobsonlane
1032:Snake tail
929:0.2763932.
651:pronounced
107:Statistics
98:statistics
70:Statistics
2193:anywhere
989:0.1464466
735:MathWorld
424:are both
324:are both
39:is rated
2160:Scharleb
2034:Kurtosis
1850:unsigned
1841:Reminder
1622:removed.
1448:unsigned
1418:Melcombe
1377:unsigned
1351:unsigned
1325:Thinners
1305:Thinners
1285:Mcorazao
1253:unsigned
1212:unsigned
1166:unsigned
804:kurtosis
775:Tetrete?
706:Melcombe
655:Pspijker
623:Pspijker
590:unsigned
471:contribs
463:Justcop4
459:unsigned
426:unbiased
1949:Bibcode
1018:Rumping
877:Daniel
822:Jackzhp
808:Jackzhp
635:Henrygb
577:Zundark
544:Pgadfor
539:GaryG
511:— DIV (
239:on the
134:on the
41:B-class
1113:tensor
1103:Gabor
761:Yecril
524:MateoP
430:biased
326:biased
47:scale.
2195:there
1968:90649
1965:JSTOR
861:Scale
848:Drf5n
28:This
2339:talk
2320:talk
2164:talk
2139:talk
2100:talk
2081:talk
2061:talk
2003:talk
1999:Qwfp
1984:talk
1903:talk
1899:Brad
1881:talk
1858:talk
1831:talk
1456:talk
1422:talk
1402:and
1385:talk
1359:talk
1329:talk
1309:talk
1289:talk
1261:talk
1236:talk
1220:talk
1194:talk
1174:talk
1141:talk
1121:talk
1097:talk
1078:talk
1057:talk
1053:Qwfp
1042:talk
1022:talk
871:talk
852:talk
838:and
765:talk
746:and
710:talk
692:talk
688:Qwfp
598:talk
467:talk
442:talk
393:and
293:and
126:High
1957:doi
1945:186
1014:1-p
842:.
789:DFH
473:) .
451:Pic
231:Mid
2353::
2341:)
2322:)
2275:~
2272:μ
2242:γ
2212:~
2209:μ
2181:γ
2166:)
2141:)
2102:)
2083:)
2063:)
2005:)
1997:.
1986:)
1963:.
1955:.
1943:.
1905:)
1883:)
1860:)
1833:)
1749:−
1678:μ
1648:μ
1597:μ
1564:μ
1560:−
1552:,
1527:μ
1523:−
1514:μ
1484:μ
1458:)
1424:)
1387:)
1361:)
1331:)
1311:)
1291:)
1283:--
1263:)
1238:)
1222:)
1196:)
1176:)
1143:)
1123:)
1099:)
1080:)
1059:)
1044:)
1024:)
986:≈
971:−
926:≈
920:20
911:−
873:)
854:)
787:.
767:)
712:)
704:.
694:)
680:}}
674:{{
600:)
469:•
444:)
436:--
339:γ
2337:(
2318:(
2282:3
2246:1
2219:3
2162:(
2137:(
2098:(
2079:(
2059:(
2001:(
1982:(
1970:.
1959::
1951::
1901:(
1879:(
1856:(
1829:(
1804:]
1799:2
1795:X
1791:[
1788:E
1766:2
1762:]
1758:X
1755:[
1752:E
1746:]
1741:2
1737:X
1733:[
1730:E
1712:.
1696:]
1693:X
1690:[
1687:E
1682:2
1674:3
1652:3
1644:3
1629:.
1615:.
1601:3
1593:2
1590:+
1568:3
1531:3
1518:3
1510:3
1488:3
1480:2
1477:+
1454:(
1420:(
1414:1
1411:g
1407:1
1404:G
1400:1
1397:g
1383:(
1357:(
1327:(
1307:(
1287:(
1259:(
1234:(
1218:(
1192:(
1172:(
1139:(
1119:(
1095:(
1076:(
1055:(
1040:(
1020:(
1009:.
980:8
976:1
966:2
963:1
958:=
955:p
916:1
906:2
903:1
898:=
895:p
869:(
850:(
763:(
752:2
749:k
743:3
740:k
708:(
690:(
666:+
596:(
518:)
465:(
440:(
408:1
404:G
377:1
373:b
343:1
308:1
304:G
277:1
273:b
243:.
138:.
53::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
