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for the following 3 values 222,1122,45444

WolframAlpha gives 0.706

Excel, using =SKEW(222,1122,45444) gives 1.729

What explains the difference?

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  • $\begingroup$ Is this question about empirical or maybe nonparametric skewness or about estimating skewness? $\endgroup$ – gwr Nov 14 '15 at 13:21
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They are using different methods to compute the skew. Searching in the help pages for skewness() within the R package e1071 yields:

Joanes and Gill (1998) discuss three methods for estimating skewness:

Type 1:
g_1 = m_3 / m_2^(3/2). This is the typical definition used in many older textbooks.
Type 2:
G_1 = g_1 * sqrt(n(n-1)) / (n-2). Used in SAS and SPSS.
Type 3:
b_1 = m_3 / s^3 = g_1 ((n-1)/n)^(3/2). Used in MINITAB and BMDP.
All three skewness measures are unbiased under normality.

#Why are these numbers different?
> skewness(c(222,1122,45444), type = 2)
[1] 1.729690
> skewness(c(222,1122,45444), type = 1)
[1] 0.7061429

Here's a link to the paper referenced if someone has the credentials to get it for further discussion or education: http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00122/abstract

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    $\begingroup$ It's not mathematically possible for "all three skewness measures to be unbiased," because (obviously) their expectations all differ. Perhaps you mean asymptotically unbiased? $\endgroup$ – whuber Apr 3 '11 at 2:10
  • $\begingroup$ @whuber - I'm going to defer to Friedrich.Leisch@R-project.org who maintains the e1071 package for clarification on what he meant specifically there. If my post wasn't clear, that comes from the help page for skewness() $\endgroup$ – Chase Apr 3 '11 at 2:15
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    $\begingroup$ @whuber I thought the same so I upvoted your comment then started reading the Joanes & Gill paper (start of section 3, p185) and realised we're wrong : a normal distribution has zero skewness, so any estimator that is a multiple of $g_1$ is unbiased under normality. Unfortunately there's no way to undo a comment upvote. $\endgroup$ – onestop Apr 3 '11 at 6:23
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    $\begingroup$ The point is that $g_1 = m_3 / m_2^{3/2}$, where $m_2$ and $m_3$ are the second and third moments about the mean, is the population skewness. As a sample statistic, it then raises similar issues to unbiased estimates of standard deviation, leading to corrections based on $n$ which have some justification but still do not produce unbiased estimates. But I think it is not very helpful to say that these are unbiased estimates of skewness for symmetric distributions; so too is 0, which has a lower variance but is inconsistent and is useless for estimating skewness of asymmetric distributions. $\endgroup$ – Henry Apr 3 '11 at 12:44
  • $\begingroup$ @onestop @Henry I agree with you. $\endgroup$ – whuber Apr 3 '11 at 17:45

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