# Can Kurtosis Be Negative?

In statistics, kurtosis is used to describe the shape of a probability distribution.

Specifically, it tells us the degree to which data values cluster in the tails or the peak of a distribution.

The kurtosis for a distribution can be negative, equal to zero, or positive.

## Zero Kurtosis

If a distribution has a kurtosis of 0, then it is equal to the normal distribution which has the following bell-shape:

## Positive Kurtosis

If a distribution has positive kurtosis, it is said to be leptokurtic, which means that it has a sharper peak and heavier tails compared to a normal distribution.

This simply means that fewer data values are located near the mean and more data values are located on the tails.

The most well-known distribution that has a positive kurtosis is the t distribution, which has a sharper peak and heaver tails compared to the normal distribution.

## Negative Kurtosis

If a distribution has negative kurtosis, it is said to be platykurtic, which means that it has a flatter peak and thinner tails compared to a normal distribution.

This simply means that more data values are located near the mean and less data values are located on the tails.

One extreme example of a distribution that has a negative kurtosis is the uniform distribution, which has no peak at all and is a completely flat distribution.

## When to Use Kurtosis in Practice

In practice, we often measure the kurtosis of a distribution in the exploratory phase of analysis when we’re just trying to get a better understanding of the data.

So, if we see that the kurtosis is positive then we know we’re working with a distribution that has fewer data values located near the center and more data values that are spread out along the tails.

Conversely, if we see that the kurtosis is negative then we know we’re working with a distribution that has more data values located near the center and less data values in the tails.

### Additional Resources

To find the skewness and kurtosis for a given distribution, you can enter the raw data values into this Skewness and Kurtosis Calculator, which will tell you both the skewness and kurtosis for the distribution.

One of the most popular statistical tests that is used to determine whether or not a particular distribution has skewness and kurtosis that matches a normal distribution is the Jarque Bera Test.

Khan Academy also has a nice video series that describes how to classify the shapes of distributions.

## 4 Replies to “Can Kurtosis Be Negative?”

1. Peter Westfall says:

Kurtosis does not tell you about peakedness or flatness. It measures heaviness of tails only.

Further, “heavier tails” does NOT mean “more data in the tails.” Rather, it means that the tails extend much farther than the normal distribution would predict. A single data point that is ten standard deviations from the mean is evidence of extremely heavy tails.

2. Peter Westfall says:

Kurtosis does not measure peakedness or flatness at all. You can have an infinitely peaked distribution with very low kurtosis (eg, beta(.5,1)), and you can have a distribution that appears perfectly flat-topped over nearly all the data, but with infinite kurtosis (eg, .9999U(0,1) +.0001Cauchy). Kurtosis measures tail extremity.

3. Peter Westfall says:

Negative kurtosis does not imply a flatter peak. For example, the beta(.5,1) distribution is infinitely peaked but has negative kurtosis.

Further, positive kurtosis does not imply a sharper peak. For example, the .9999U(0,1) +.0001Cauchy distribution appears perfectly flat-topped over nearly all the data, but has infinite kurtosis. Kurtosis measures tails only.

4. PETER WESTFALL says:

Kurtosis tells you nothing about peakedness or flatness. It only tells you about tail weight.