The principal measure of distribution shape used in statistics are skewness and kurtosis. The measures are functions of the 3rd and 4th powers of the difference between sample data values and the distribution mean (the 3rd and 4th central moments). With sample data, outliers (extreme values) may result in relatively high values for these measures, so they must be approached with some caution. For some continuous probability distributions, such as the Beta and Gamma distributions, shape measures are effectively distribution parameters; for others, such as the Normal, Uniform and Exponential distributions, there are no shape parameters, so their shape is fixed and their skewness and kurtosis do not change (are 0 or a constant). It is widespread practice in software packages to report all four central moment measures: mean, variance, skewness and kurtosis, although the precise basis for these computations may vary. As with all such measures, it is essential to check the software documentation to ensure the appropriate interpretation of the measure is made. Many software packages provide skewness and kurtosis measures as standard summary statistics, and these are often used to identify aspects of non-Normality in the data. The R environment does not provide all such measures directly, but they are relatively straightforward to add as functions computed from the mean (see further, Crawley, 2007, pp285 and 289 [CRA1]).

If a frequency distribution is unimodal and symmetric about the mean it has a skewness of 0. Values greater than 0 suggest skewness of a unimodal distribution to the left, with a long right tail (see diagram, below), whilst values less than 0 indicate skewness to the right with a longer tail to the left. Skewness is generally calculated as a function of the 3rd moment about the mean (denoted by α3 with a ^ symbol above it for the sample skewness). Alternative formulas for the moment-based skewness are provided below. The general formula (e.g. for a probability density function) is:

Given a set of data values, {xi} with known population mean and variance the summation form for momental skewness is:

If the mean and variance are estimated the above formula would be of the form:

but this is a biased estimate. An adjusted estimator for the population skewness (that used in SPSS and Excel, for example) is:

These formulas for sample skewness are derived from the source data, not from data that has been grouped into frequencies. In the latter case formulas would need to include the frequencies for each value, which is not the norm in most software packages.

Left skewed frequency distribution - Gamma(2,1)

The graph above shows a strongly left-skewed distribution which has a positive skewness value of 1.414 (i.e. √2). If one generates a sample of random values with a Gamma distribution (either by using the cumulative distribution function or using a software function provided for this purpose, e.g. the MATLab function gamrnd), the computed skewness of the sample will not be 1.414, but will be close to this value - the approximation will depend on the quality of the random sampling procedure and the size of the sample. Some software packages provide the estimated standard error for their skewness computation. In the case of SPSS this is:

For n>50, this expression is very close to √(6/n), which remains greater than 0.1 for n<500. The ratio of the skewness to its standard error provides a (weak) test of data Normality. An absolute value of >2 indicates that the data is non-Normal.

In addition to skewness computed from the 3rd moment of the distribution or sample data, other measures of skewness have been defined which are simpler to compute, but also have their drawbacks. These include measure based on the mean, mode and median, for example (mean-mode)/standard deviation (known as Pearson's mode skewness).

Bowley's and Pearson's Skewness

Bowley's skewness measure, which is based on the location of the upper and lower quartiles (Q3 and Q1) relative to the median (Med) has the merit of lying in the fixed range [-1,+1]. It is computed as:

or as

Pearson's skewness measure is similar, being the mean minus the mode divided by the standard deviation - the problem of determining the mode (if one exists) makes this a relatively poor measure in many instances:

A number of software packages, such as Mathematica, provide functions to compute these variants, but this is unusual. It is also important to note that all skewness computations are susceptible to unexpected values when distributions are more complex, for example some multi-modal discrete distributions - for a fuller discussion of this issue see von Hippel (2005, [VH1]).

Kurtosis is a measure of the peakedness of a frequency distribution. More pointy distributions (known as leptokurtic) tend to have high kurtosis values, especially if the tails of the distribution are larger than those of the Normal distribution, whereas more rounded flatter distributions with thinner tails have lower kurtosis (described as platykurtic). Kurtosis is a function of the 4th moment about the mean and is usually only meaningful for samples when the sample size is reasonably large (e.g. >50). It is customary to subtract 3 from the raw kurtosis value (which is the kurtosis of the Normal distribution) to give a figure relative to the Normal distribution known as the excess kurtosis. For the figure shown in the previous subsection (the Gamma distribution) the kurtosis is 3 (the same as for a Normal distribution, so the excess kurtosis is 0. Software packages vary in their implementation of this function (i.e. with or with 3 deducted, and with or without sample size adjustment). Kurtosis is generally denoted by α4 with a ^ symbol above it for the sample kurtosis). Alternative formulas for the moment-based kurtosis are provided below. The general formula (e.g. for a probability density function) is:

Given a set of data values, {xi} with known population mean and variance the summation form for momental kurtosis is:

If the mean and variance are estimated the above formula would be of the form:

but, as with skewness, this is a biased estimate. An adjusted estimator for the population kurtosis (that used in SPSS and Excel, for example) is:

As can be seen, for n>50, the expression for a is very close to n and for b, very close to 3. Some software packages provide the estimated standard error (SE) for their kurtosis computation. In the case of SPSS this is approximately twice the SE of the skewness (see above). The ratio of the kurtosis to its standard error provides a (weak) test of data Normality. An absolute value of >2 indicates that the data is definitely non-Normal.

References

[CRA1] Crawley M (2007) The R Book. J Wiley & Sons, New York

[VH1] von Hippel (2005) Mean, Median, and Skew: Correcting a Textbook Rule. J. of Statistics Education, 13, 2

Weisstein, E W "Skewness" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Skewness.html

Weisstein, E W "Kurtosis" From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Kurtosis.html