Table of Contents
Skewness and kurtosis are measures of the shape of a distribution. Skewness is a measure of the asymmetry of a distribution while kurtosis is a measure of the tail thickness of the distribution. Skewness and kurtosis can be measured using various formulas such as the mean, median, mode, range, kurtosis and skewness coefficients, and sample standard deviation. These measures can be used to identify whether a distribution is normal, skewed, or leptokurtic.
In statistics, skewness and kurtosis are two ways to measure the shape of a distribution.
Skewness is a measure of the asymmetry of a distribution. This value can be positive or negative.
- Negative skew indicates that the tail is on the left side of the distribution, which extends towards more negative values.
- Positive skew indicates that the tail is on the right side of the distribution, which extends towards more positive values.
- A value of zero indicates that there is no skewness in the distribution at all, meaning the distribution is perfectly .
Kurtosis is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution.
- The kurtosis of a normal distribution is 3.
- If a given distribution has a kurtosis less than 3, it is said to be playkurtic, which means it tends to produce fewer and less extreme outliers than the normal distribution.
- If a given distribution has a kurtosis greater than 3, it is said to be leptokurtic, which means it tends to produce more outliers than the normal distribution.
Note: Some formulas (Fisher’s definition) subtract 3 from the kurtosis to make it easier to compare with the normal distribution. Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0.
When reporting the skewness and kurtosis of a given distribution in a formal write-up, we generally use the following format:
The skewness of [variable name] was found to be -.89, indicating that the distribution was left-skewed.
The kurtosis of [variable name] was found to be 4.26, indicating that the distribution was more heavy-tailed compared to the normal distribution.
Keep in mind the following when reporting the results:
- Round the values for skewness and kurtosis to two decimal places.
- Drop the leading 0 when reporting the values (e.g. use .79, not 0.79)
The following example shows how to use this format in practice.
Example: Reporting Skewness & Kurtosis
Suppose we’re analyzing the distribution of exam scores among students at a certain university.
Using statistical software, we calculate the values for the skewness and kurtosis of the distribution to be:
- Skewness: -1.391777
- Kurtosis: 4.170865
We would report these values as follows:
The skewness of the exam scores was found to be -1.39, indicating that the distribution was left-skewed.
The kurtosis of the exam scores was found to be 4.17, indicating that the distribution was more heavy-tailed compared to the normal distribution.
The following tutorials explain how to calculate skewness and kurtosis in different statistical software:
The following tutorials explain how to report other statistical results: