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 Field: Probability and statistics
One of the 500 most frequently viewed mathematics articles.


HOW ABOUT A POSITIVE SKEW VERSES NEGATIVE SKEW PICTURE —The preceding unsigned comment was added by Justcop4 (talkcontribs) .

Early comments[edit]

The statement about unbiasedness of the estimate of skewness given needs further qualification for two reasons.

Firstly, if the sample is from a finite population, the observations are dependent, while the proof of unbiasedness requires independence.

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 unbiased (but its 3/2 power is biased). [It is well known that the square root of the sample variance--the sample standard deviation--is biased; there is a correction for bias for specific distributions, but no general correction.] By the linearisation method (or delta method) we can say that the ratio is approximately unbiased. User:Terry Moore 11 Jun 2005

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.

Needs better intro[edit]

[ I hope I fixed this; this discussion section should be removed --Lingwitt 21:15, 23 January 2007 (UTC) ]

Generally discussions are not 'removed', but merely, eventually, 'archived'. — DIV ( 06:51, 19 July 2007 (UTC))

Almost impossible for a lay person without knowledge of statistics to understand this article. There needs to be a more general introduction given. --MateoP 21:52, 30 March 2006 (UTC)

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 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. 20:30, 24 June 2006 (UTC) GaryG

Comment from main article moved to appropriate page: Section to develop: Why should we care about skew? what difference does it make! Pgadfor 03:21, 14 May 2006 (UTC)

Missing assumption?[edit]

I think the following paragraph cannot hold under general conditions:

Skewness affects Mean the most and Mode the least. For a positivevely skewed distribution, Mean > Median > Mode and for a negatively skewed distribution, Mean < Median < Mode

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? 11:56, 23 June 2006 (UTC)

It's incorrect, so I've removed it. --Zundark 13:03, 23 June 2006 (UTC)

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 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! —Preceding unsigned comment added by (talk) 14:55, 13 August 2009 (UTC)

Positive versus negative skewness mixed up[edit]

When looking at the skewness for the Maxwell-Boltzmann-distribution it appeared to me that the definitions of positive and negative skewness got mixed up. The Maxwell-Boltzmann-distribution 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 Wikipedia. Even if I am mistaken, this definition should be clarified and a picture would definitily help. -- Pspijker 22:31 September 1st, 2006

What makes you think the Maxwell-Boltzmann-distribution has a negative skewness? --Henrygb 23:35, 16 September 2006 (UTC)
According to the Wikipedia entry for the Maxwell-Boltzmann-distribution 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 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 pronounced. This Wikipedia 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 Wikipedia would help a lot. --Pspijker 08:00, 26 September 2006 (CEST)
I'm pretty sure the problem is with the formula for skewness in Maxwell-Boltzmann distribution. 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 +0.484. I think the source of the error was most likely the MathWorld article on the M-B distribution. 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 {{expert subject}} tag from this article now (I'm originally a physicist so i consider this sort of experimental mathematics 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). Qwfp (talk) 15:27, 25 May 2010 (UTC)
The incorrect sign (now corrected) can be confirmed algebraically by starting from the result for the chi distribution. Melcombe (talk) 16:13, 25 May 2010 (UTC)

incorrect formula for G1[edit]

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, you can do the algebra, getting g1 in terms of the ratio of k-statistics, and see that Zar is correct. -- J.D. Opdyke

I am no expert on this but MathWorld gives a different formula for k3 and k2, both involving the sample mean. They also allege an unbiased estimator for finite populations using a different regularization factor. Could somebody please reconcile the two? --Yecril (talk) 08:05, 24 March 2012 (UTC)


What is a tetrete (in the description of the first figure)? I have never heard this term and cannot find a definition.

Looks as though it may have been a typographical error. DFH 14:04, 4 April 2007 (UTC)

range & table[edit]

Just like "Kurtosis of well-known distributions" in kurtosis article, can someone put a similar table in this article. and talk about the range of skewness, and the rang of kurtosis. Jackzhp 20:43, 13 July 2007 (UTC)

the variance of skewness[edit]

Can someone put a table to talk about the variance of skewness for different distribution? Jackzhp 23:57, 14 July 2007 (UTC)

The Distribution of Skewness and testing[edit]

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: and . 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? Drf5n (talk) 18:27, 17 June 2008 (UTC)


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) (talk) 21:09, 2 November 2008 (UTC)Daniel

Consider a Bernoulli distribution with probability of success
That has a skewness of 1. One with probability of success
has a skewness of 2, as does any exponential distribution.
To change the sign just take 1-p as the probability of success. --Rumping (talk) 21:01, 21 April 2009 (UTC)

Snake tail[edit]

Eh? Isn't a statistical snake something entirely different? Jim.henderson (talk) 19:11, 4 February 2009 (UTC)

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. Qwfp (talk) 22:53, 4 February 2009 (UTC)

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 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 Wikipedia for why skewed distributions are important is in Lake Woebegone effect 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. Jim.henderson (talk) 18:48, 8 February 2009 (UTC)

Example figures with skewness values[edit]

It would be nice to have a figure here that shows a bunch of probability density diagrams and their Skewness as examples.-- (talk) 12:40, 16 June 2009 (UTC) Gabor

Multidimensional case?[edit]

I gather that the multidimensional case of skewness is a third-order tensor. Is that right? Should this article discuss the multidimensional case? —Ben FrantzDale (talk) 11:45, 19 August 2009 (UTC)

Example of a right-skewed distribution where the mean is left of the median[edit]

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 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. (talk) 23:47, 25 September 2009 (UTC)

Introduction has a mistake related to the common mistake it points out[edit]

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 misunderstanding.

Specifically, if you define skewness in terms of third moment, mean doesn't necessarily equal median when the third moment is zero.

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. —Preceding unsigned comment added by (talk) 00:26, 2 March 2010 (UTC)

Regarding the attempt to correct the statement "...then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4,..." 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, 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. Hobsonlane (talk) 20:50, 23 December 2014 (UTC)

What the heck does "gravitropic response of wheat coleoptiles" have to do with skewness?[edit]

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. —Preceding unsigned comment added by (talk) 05:57, 31 March 2010 (UTC)

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 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). Hobsonlane (talk) 21:00, 23 December 2014 (UTC)

There was a bad error in the uncentered moment formula[edit]

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. —Preceding unsigned comment added by (talk) 21:53, 9 April 2010 (UTC)

Comment on introduction[edit]

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 that Wikipedia 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 over the heads of some readers).

--Mcorazao (talk) 14:49, 26 May 2010 (UTC)

Skew askew?[edit]

The Wikipedia article on Skewness cites reference #14 (in Czech) concerning Cyhelsky's Skewness Coefficient. However, the formulation [(the number of observations below the mean minus the number of observations above the mean)/total number of observations] 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 "check" to see whether the Wikipedia formulation should be revised to result in a negative coefficient when the data show a left-sided skew. Thinners (talk) 23:52, 27 October 2010 (UTC)

Skewer askew[edit]

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 Wikipedia expression for the Cyhilsky coefficient does yield a positive value. Thinners (talk) 00:10, 28 October 2010 (UTC)


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 processes that exhibit non-zero skewness does not mean that skewness is "useful" or is being "applied".

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 a sentence about it? —Preceding unsigned comment added by (talk) 22:11, 7 March 2011 (UTC)

sample variance in g1 formula[edit]

According to the Wikipedia page for variance, sample variance is the 1/(n-1) form, but the formula for g1 is using 1/n. —Preceding unsigned comment added by (talk) 22:50, 15 March 2011 (UTC)

Yes. See the different formulae for g1 and G1. Early workers in skewness used the g1 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. Melcombe (talk) 13:02, 16 March 2011 (UTC)

/* Definition */[edit]

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.

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[X^3] - 3\mu\sigma^2 - \mu^3 which was changed to E[X^3] - 3\mu\sigma^2 + 2\mu^3 which I have now reverted to E[X^3] - 3\mu\sigma^2 - \mu^3

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[X^3] - 3\mu E[X^2] + 2\mu^3 and is in fact how it is written on MathWorld). We may have to watch this formula for changes again to avoid an edit war. — Preceding unsigned comment added by (talk) 23:31, 1 August 2011 (UTC)

Below is the editing history, and my deductions to why they were made:
22 April 2011 = My revision, changed to , 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.
9 July 2011‎ = First mistaken fix, changed to .
11 July 2011‎ = First mistaken fix removed.
27 July 2011‎ = @08:18 Second incident of the same mistaken fix.
27 July 2011‎ = @15:22 Second mistaken fix has not been fixed, but the was changed to for readability, and possibly to reduce people making the mistaken fix.
1 August 2011‎ = @23:16 Second mistaken fix has been removed.
1 August 2011‎ = @23:28 Variance converted back to , 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; 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.
1 August 2011 = @23:29 Introduced error fixed.
15 February 2012 = My edit to revert back to using the variance, and to prevent editors making the recurrent editing-mistake.
—Ricketts (talk) 09:18, 15 February 2012 (UTC)


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" => "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? — Preceding unsigned comment added by (talk) 09:52, 12 September 2012 (UTC)

Cyhelský's skewness coefficient[edit]

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 Wikipedia being punked. Discussion please? (talk) 00:18, 18 June 2014 (UTC)

Reorganizing Introduction[edit]

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 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.

Brad (talk) 23:48, 16 September 2014 (UTC)

Pearson's first skewness[edit]

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 [1] ) that 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).

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.


  1. ^ Pearson, Karl (1895). "Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material". Philosophical Transactions of the Royal Society. 186: 343–414. Bibcode:1895RSPTA.186..343P. doi:10.1098/rsta.1895.0010. JSTOR 90649. 

Glenbarnett (talk) 15:58, 1 February 2015 (UTC)

Well spotted and researched! Feel free to go ahead and fix the article. Qwfp (talk) 20:09, 1 February 2015 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Skewness/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Kurtosis is noticibly better, with examination of various distributions and some references.

Last edited at 23:28, 19 April 2007 (UTC). Substituted at 20:14, 1 May 2016 (UTC)