Section author: Danielle J. Navarro and David R. Foxcroft

Skew and kurtosis

There are two more descriptive statistics that you will sometimes see reported in the psychological literature: skew and kurtosis. In practice, neither one is used anywhere near as frequently as the measures of central tendency and variability that we’ve been talking about. Skew is pretty important, so you do see it mentioned a fair bit, but I’ve actually never seen kurtosis reported in a scientific article to date.

Illustration of skewness

Fig. 15 Illustration of skewness. On the left we have a negatively skewed data set (skewness = -.93), in the middle we have a data set with no skew (well, hardly any: skewness = -.006), and on the right we have a positively skewed data set (skewness = 0.93).

Since it’s the more interesting of the two, let’s start by talking about the skewness. Skewness is basically a measure of asymmetry and the easiest way to explain it is by drawing some pictures. As Fig. 15 illustrates, if the data tend to have a lot of extreme small values (i.e., the lower tail is “longer” than the upper tail) and not so many extremely large values (left panel) then we say that the data are negatively skewed. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are positively skewed. That’s the qualitative idea behind skewness. If there are relatively more values that are far greater than the mean, the distribution is positively skewed or right skewed, with a tail stretching to the right. Negative or left skew is the opposite. A symmetric distribution has a skewness of 0. The skewness value for a positively skewed distribution is positive, and a negative value for a negatively skewed distribution.

One formula for the skewness of a data set is as follows

\[\mbox{skewness}(X) = \frac{1}{N \hat{\sigma}^3} \sum_{i=1}^N (X_i - \bar{X})^3\]

where N is the number of observations, X̄ is the sample mean, and \(\hat{\sigma}\) is the standard deviation (the “divide by N - 1” version, that is).

Perhaps more helpfully, you can use jamovi to calculate skewness: set the check box Skewness in the Statistics options under Exploration → Descriptives. For the afl.margins variable, the skewness figure is 0.780. If you divide the skewness estimate by the Std. error for skewness you have an indication of how skewed the data is. Especially in small samples (N < 50), one rule of thumb suggests that a value of 2 or less can mean that the data is not very skewed, and a value of over 2 that there is sufficient skew in the data to possibly limit its use in some statistical analyses. Though there is no clear agreement on this interpretation. That said, this does indicate that the afl.margins variable is somewhat skewed (0.780 / 0.183 = 4.262).

The final measure that is sometimes referred to, though very rarely in practice, is the kurtosis of a data set. Put simply, kurtosis is a measure of how thin or fat the tails of a distribution are, as illustrated in Fig. 16. By convention, we say that the “normal curve” (black lines) has zero kurtosis, and the degree of kurtosis is assessed relative to this curve.

Illustration of kurtosis

Fig. 16 An illustration of kurtosis. On the left, we have a “platykurtic” distribution (kurtosis = -.95) meaning that the distribution has “thin” or flat tails. In the middle we have a “mesokurtic” distribution (kurtosis is almost exactly 0) which means that the tails are neither thin or fat. Finally, on the right, we have a “leptokurtic” distribution (kurtosis = 2.12) indicating that the distribution has “fat” tails. Note that kurtosis is measured with respect to a normal distribution (black line).

The data in the left panel of Fig. 16 have a pretty flat distribution, with thin tails, so the kurtosis is negative and we call the data platykurtic. The data in the right panel have a distribution with fat tails, so the kurtosis is positive and we say that the data is leptokurtic. Only the data in the middle panel have neither thin or fat tails, so we say that it is mesokurtic and has kurtosis zero. This is summarised in the table below:

informal term technical name kurtosis value
“tails to thin” platykurtic negative
“tails neither thin or fat” mesokurtic zero
“tails too fat” leptokurtic positive

The equation for kurtosis is pretty similar in spirit to the formulas we’ve seen already for the variance and the skewness. Except that where the variance involved squared deviations and the skewness involved cubed deviations, the kurtosis involves raising the deviations to the fourth power:[1]

\[\mbox{kurtosis}(X) = \frac{1}{N \hat\sigma^4} \sum_{i=1}^N \left( X_i - \bar{X} \right)^4 - 3\]

I know, it’s not terribly interesting to me either.

More to the point, jamovi has a check box for Kurtosis just below the check box for Skewness, and this gives a value for kurtosis of 0.101 with a standard error of 0.364. This means that the afl.margins variable has only a small kurtosis, which is ok.

[1]The “-3” part is something that statisticians tack on to ensure that the normal curve has kurtosis zero. It looks a bit stupid, just sticking a “-3” at the end of the formula, but there are good mathematical reasons for doing this.