# Where does SKEW Excel equation come from?

I need to quote Excel's skew equation in a paper but I couldn't find any information about this equation:

$n$: size of the population
$s$: standard deviation
signed $x$: mean

Does anyone know if in statistics this equation is known with some name so I could add a quote on the paper indicating where the equation comes from?

here's the link of excel's skew article: https://support.office.com/en-us/article/SKEW-function-bdf49d86-b1ef-4804-a046-28eaea69c9fa

• See en.wikipedia.org/wiki/Skewness#Sample_skewness & the reference given there: Joanes & Gill (1998). "Comparing measures of sample skewness and kurtosis", JRSS D, 47, 1. Jun 23, 2015 at 9:44
• $n$ is better reported as size of the sample. What you call "signed $x$" is often spoken aloud as "$x$ bar", but it is better practice to write $\bar x$ in a paper or thesis. The wording "signed $x$" makes no obvious sense here. $s$ is indeed standard deviation but for full disclosure you need to explain in turn how it is calculated, i.e. whether with $n - 1$ or with $n$ in the divisor. Jun 23, 2015 at 10:36
• @Nick Cox Yes it's x bar, i couldn't remember the exact term. Btw the standard deviation is calculated with n−1 in the divisor.
– nsp
Jun 23, 2015 at 11:38
• @Scortchi-ReinstateMonica: do you want to post your comment(s) as an answer? Better to have a short answer than no answer at all. Anyone who has a better answer can post it. Jun 10, 2020 at 11:56

One definition of skewness is as the third standardized moment of a distribution: the ratio of the third central moment to the cube of the standard deviation (the square root of the second central moment): $$\frac{\operatorname{E}{(X-\operatorname{E}{X})^3}} {\left[\operatorname{E}{(X-\operatorname{E}{X})^2}\right]^\frac{3}{2}}$$

The corresponding property in a sample (or finite population) is given by

$$\frac{\sum (x - \bar{x})^3/n}{\left[\sum (x - \bar{x})^2/n\right]^\frac{3}{2}},$$

the Fisher—Pearson third standardized moment coefficient, or the Fisher–Pearson coefficient of skewness, or suchlike.

In samples of i.i.d. observations, this statistic is a consistent, though not unbiased, estimator of the skewness of the distribution from which the samples are drawn, provided of course that the first 3 moments exist. Now, the estimators of the second & third central moments that appear in the denominator & numerator are also consistent, & with biases that depend only on the sample size, $$\frac{n-1}{n}$$ & $$\frac{(n-1)(n-2)} {n^2}$$ respectively: substituting the de-biased estimators into the ratio results in

$$\frac{\frac{n^2}{(n-1)(n-2)}\sum (x - \bar{x})^3/n} {\left[\left(\frac{n}{n-1}\right)\sum (x - \bar{x})^2/n\right]^\frac{3}{2}} = \frac{n}{(n-1)(n-2)}\cdot\sum\left(\frac{x - \bar{x}} {s}\right)^3 ,$$

where $$s$$ is the sample standard deviation; & this is known as the adjusted Fisher–Pearson third standardized moment coefficient, &c. (I can't say I see the motive behind the adjustment, as it still doesn't give an unbiased estimator of skewness.)

See https://en.wikipedia.org/wiki/Skewness#Sample_skewness & the references given there: Joanes & Gill (1998), "Comparing measures of sample skewness and kurtosis", J. Royal Stat. Soc. D, 47 (1); & Doane & Seward (2011), "Measuring skewness: a forgotten statistic", J. Stat. Educ., 19 (2).

• History is more tangled than people want to believe, but a label Thiele-Pearson-Fisher would attribute credit a little more fairly and in time order. One of Pearson's more tangled moments [so to speak] was finding out that Thiele was there before him and giving a reference (good practice), but in such a way that implied that Thiele wasn't worth reading (poor show). The history is touched on -- and more crucially there are more references -- in stata-journal.com/article.html?article=st0204 (although I should add Picard in Annals of Mathematical Statistics 1951 to the references). Jun 13, 2020 at 8:31