*By WIKTORIA KORYGA (Predictive Solutions)*

**Kurtosis and skewness are measures of asymmetry that describe such properties as the shape and asymmetry of the distribution under analysis. They provide us with information on how the values of the variables deviate when compared to the mean value.**

## MEASURES OF ASYMMETRY AND CONCENTRATION OF THE DISTRIBUTION OF A VARIABLE

Kurtosis and skewness are measures of asymmetry that describe such properties as the shape and asymmetry of the distribution under analysis. They provide us with information on how the values of the variables deviate when compared to the mean value. Thus, they allow us to answer the question of whether the mean is in the center of the distribution (and therefore close to the median), how individual observations are dispersed around this mean, and how extreme are outlying observations.

## WHAT IS SKEWNESS AND WHAT DOES IT TELL US?

Skewness is a statistic that makes it possible to compare the distribution of the analyzed variable with a hypothetical normal distribution. It indicates the discrepancy between the mean value and the center of a given distribution. In turn, as is well known, the mean is characterized by its lack of robustness in the presence of extreme values. Therefore, if during the analysis of the distribution of a given variable we notice the presence of abnormally small or large values, we can conclude that the average has been “dragged” by these extreme values to the right or left. For example, in a situation with unusually small values, the average is “dragged” to the left side, When viewed on a graph, you will observe an elongated left tail of the distribution, or the occurrence of a left-skewed distribution.

## HOW TO INTERPRET THE COEFFICIENT OF SKEWNESS (ASYMMETRY)?

The skewness coefficient As can take negative values, equal zero, and take positive values. Depending on the value of the coefficient, it can be interpreted as follows:

1. As < 0 – Left-skewness

- Mo > Me >
- extended left tail of the distribution

2. As = 0 – Symmetric distribution

- Mo = Me =

3. As > 0 – Right-skewness

- Mo < Me <
- extended right tail of the distribution

Mo – mode

Me – median

– mean

*Figure 1. Types of distributions by value of skewness coefficient*

## WHAT IS KURTOSIS AND WHAT DOES IT TELL US?

We also use kurtosis to compare the distribution of the analyzed variable with a hypothetical normal distribution, in which the dispersion of observations around the mean is relatively uniform and there are no extreme outliers. Depending on the value of kurtosis, the plotted distribution can have a “fatter” or “thinner” tail, which is influenced by the intensity of extreme values.

Based on its value, we can distinguish three types of distributions:

- leptokurtic (K>0) – the distribution has a fatter tail, i.e., the intensity of extreme values is higher than in a normal distribution.
- mesocurtic (K=0) – the distribution is close to normal.
- platykurtic (K<0) – the distribution has a thinner tail than the normal distribution, i.e., the intensity of extreme values is lower than in the normal distribution.

*Figure 2. Types of distributions by value of kurtosis*

*Table 1. Selected descriptive statistics for the analyzed variabl*

*Figure 3. Histogram of the expenditure variable*

*Figure 4. Histogram of the age variable*

*Figure 5. Histogram of the price variable*

At the beginning of working with data, it is particularly useful to present the distributions of the analyzed variables in the form of histograms which allows one to easily and quickly grasp the most important properties, such as the discussed asymmetry or the way observations are concentrated.