Talk:Skewness

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

Early comments

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

[ 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 (128.250.204.118 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.216.129.143.26 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?

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? 88.101.32.104 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 63.240.104.100 (talk) 14:55, 13 August 2009 (UTC)

Positive versus negative skewness mixed up

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)

incorrect formula for G1

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

Tetrete?

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

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

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

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: http://www.jalt.org/test/bro_1.htm and http://mathworld.wolfram.com/Skewness.html . http://www.xycoon.com/skewness_small_sample_test_1.htm 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)

Scale

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

Consider a Bernoulli distribution with probability of success

$p={\frac {1}{2}}-{\frac {1}{\sqrt {20}}}\approx 0.2763932.$

That has a skewness of 1. One with probability of success

$p={\frac {1}{2}}-{\frac {1}{\sqrt {8}}}\approx 0.1464466$

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

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

It would be nice to have a figure here that shows a bunch of probability density diagrams and their Skewness as examples.--131.111.176.9 (talk) 12:40, 16 June 2009 (UTC) Gabor

Multidimensional case?

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

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

Introduction has a mistake related to the common mistake it points out

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 58.171.86.193 (talk) 00:26, 2 March 2010 (UTC)

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

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.