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 symmetrical.
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.
Along with reporting these values for skewness and kurtosis, we generally include some chart to visualize the distribution of values such as a histogram or boxplot so the reader can get a visual understanding of the distribution as well.
The following tutorials explain how to calculate skewness and kurtosis in different statistical software:
The following tutorials explain how to report other statistical results: