*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.

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

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.

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]

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. |