Descriptive Statistics › Measures of Distribution Shape

Beyond center and spread, distribution shape gives useful information about data behavior. Two common measures, skewness and kurtosis, summarize asymmetry and tail weight, respectively [9].

One common sample measure of skewness uses central moments computed with a consistent $1/n$ denominator:

$$m_r = \frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^r, \qquad g_1 = \frac{m_3}{m_2^{3/2}}.$$

For nonconstant data, $m_2>0$ and the statistic $g_1$ is defined and unitless. Some software instead reports a finite-sample bias-adjusted estimator, so comparisons should use the same convention.

A nondegenerate symmetric distribution with a finite third moment has skewness zero, but zero skewness does not by itself prove symmetry. Positive skewness, or right skew, often accompanies a longer or heavier right tail. In many familiar unimodal cases the mean then exceeds the median, but that ordering is not guaranteed for every distribution.

Negative skewness, or left skew, often accompanies a longer or heavier left tail. Test scores on an easy exam can show this pattern when many students score near the upper boundary and a smaller number receive much lower scores.

There is no universal cutoff that turns a skewness value into “moderate” or “high.” Interpret its magnitude relative to the estimator, sample size, application, and the accompanying distribution plot.

Moment kurtosis standardizes the fourth central moment:

$$g_2 = \frac{m_4}{m_2^2}.$$

For nonconstant data, $m_2>0$ and $g_2$ is defined. Because fourth powers amplify large standardized deviations, kurtosis is sensitive to extreme observations. It is not a direct partition of probability between the center and tails, nor does it determine whether a density has a sharp or flat peak [10]. A normal distribution has population kurtosis 3. Many software packages instead report excess kurtosis, $g_2-3$, and may also apply a finite-sample bias correction; under the excess convention the normal reference is zero.

High kurtosis means that extreme standardized deviations contribute strongly to the fourth moment. It can be a warning that a normal model understates important extremes, but it does not guarantee a larger exceedance probability at every threshold. Risk analysis should therefore examine relevant tail probabilities and quantiles directly rather than rely on kurtosis alone.

Low kurtosis means that extreme standardized deviations make a smaller fourth-moment contribution than under the chosen reference. A continuous uniform distribution, for example, has raw kurtosis 1.8. This value does not by itself characterize the height or flatness of its density peak.

Kurtosis is one diagnostic for departures from a normal model, not a method-selection rule. When extremes matter, inspect distribution plots, tail quantiles, and model residuals and then choose methods whose assumptions fit the task.