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.
- A negative skew indicates that the tail is on the left side of the distribution, which extends towards more negative values.
- A 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.
This tutorial explains how to calculate both the skewness and kurtosis of a given dataset in R.
Example: Skewness & Kurtosis in R
Suppose we have the following dataset:
data = c(88, 95, 92, 97, 96, 97, 94, 86, 91, 95, 97, 88, 85, 76, 68)
We can quickly visualize the distribution of values in this dataset by creating a histogram:
From the histogram we can see that the distribution appears to be left-skewed. That is, more of the values are concentrated on the right side of the distribution.
To calculate the skewness and kurtosis of this dataset, we can use skewness() and kurtosis() functions from the moments library in R:
library(moments) #calculate skewness skewness(data)  -1.391777 #calculate kurtosis kurtosis(data)  4.177865
The skewness turns out to be -1.391777 and the kurtosis turns out to be 4.177865.
Since the skewness is negative, this indicates that the distribution is left-skewed. This confirms what we saw in the histogram.
Since the kurtosis is greater than 3, this indicates that the distribution has more values in the tails compared to a normal distribution.
The moments library also offers the jarque.test() function, which performs a goodness-of-fit test that determines whether or not sample data have skewness and kurtosis that matches a normal distribution. The null and alternative hypotheses of this test are as follows:
Null Hypothesis: The dataset has a skewness and kurtosis that matches a normal distribution.
Alternative Hypothesis: The dataset has a skewness and kurtosis that does not match a normal distribution.
The following code shows how to perform this test:
jarque.test(data) Jarque-Bera Normality Test data: data JB = 5.7097, p-value = 0.05756 alternative hypothesis: greater
The p-value of the test turns out to be 0.05756. Since this value is not less than α = .05, we fail to rejec the null hypothesis. We do not have sufficient evidence to say that this dataset has a skewness and kurtosis that is different from the normal distribution.
You can find the complete documentation for the moments library here.
Bonus: Skewness & Kurtosis Calculator
You can also calculate the skewness for a given dataset using the Statology Skewness and Kurtosis Calculator, which automatically calculates both the skewness and kurtosis for a given dataset.