Kurtosis
What is kurtosis?
Kurtosis is a statistic used to describe the shape of a distribution of continuous data. It measures the tendency of the data to be distributed toward the tails, or ends, of the distribution versus toward the middle. If your distribution has heavier tails and is more peaked compared to the normal distribution, the kurtosis is positive. If your distribution has light tails and is flatter in the middle than a normal distribution, the kurtosis is negative.
When should you use kurtosis?
Kurtosis measures the tail-heaviness of a set of data. This measure can be particularly useful when you expect a normal distribution, but your data look symmetric but not normal when visualizing with a histogram or box plot.
Kurtosis can be calculated for continuous or count (categorical numeric) data.
A distribution with positive kurtosis is called leptokurtic, since “lepto” means slender, referring to the peak. A distribution with negative kurtosis is called platykurtic, since “platy” means broad, again, referring to the peak.
It is important to note that data points in the tails of a distribution are not necessarily outliers. Outliers are observations that do not belong to the distribution; they belong to another distribution. Tail points, by contrast, are unlikely values that do belong to the distribution.
Many hypothesis tests with an assumption of normality are robust to departures from normality as long as the distribution is symmetric. To determine if a kurtosis value is of concern, you can create a bootstrap confidence interval on the kurtosis statistic and see if it contains zero.
Kurtosis is the fourth central moment about the mean. Statistical software, like JMP, often reports the kurtosis excess, which is defined as kurtosis minus 3 (with a correction for small sample sizes). A normal distribution has a kurtosis of 3 and kurtosis excess of zero.
How do you calculate kurtosis?
The formula for kurtosis is complicated and not generally calculated by hand. Instead, use statistical software, as displayed below.
Examples of kurtosis
Examine the kurtosis for the data in the Univariate Statistics Data. You can visualize kurtosis using the box plot and histogram and see the value of kurtosis in the Summary Statistics report.
Compare the length of the box to the length of the whiskers of the box plot. Which is longer? Compare the amount of data in the tails to the amount of data in the middle of the histogram. Which has more data?
