Probability and Statistics > Statistics Definitions > Kurtosis

Kurtosis tells you how “peaked” your graph is, or how high the graph is around the mean. It’s also the fourth moment in statistics. A positive value means that you have too little data in your tails. A negative value means that you have too much data in your tail. This heaviness or lightness in the tails means that your data looks more peaked (or less peaked).

## What does it Mean?

Kurtosis is measured against the standard normal distribution. The standard normal distribution has a kurtosis of 3, so if your values are close to that then your graph is nearly normal. These nearly normal distributions are called **mesokurtic**.

## Mesokurtic Examples

Mesokurtic distributions are more technically defined as having a kurtosis of zero, although the distribution doesn’t have to be *exactly* zero in order for it to be classified as mesokurtic. The most common mesokurtic distributions are:

- The normal distribution.
- Any distribution with a Gaussian shape and zero probability at other places on the real line.
- The binomial distribution is mesokurtic for some values (i.e. for p = 1/2±√(1/12).

## Other Types

In addition to mesokurtic, the two other types of kurtosis are:

**Platykurtic**distributions, which have negative kurtosis. An example of a very platykurtic distribution is a uniform distribution, which has as much data in each tail as it does in the peak.**Leptokurtic**distributions, which have positive kurtosis. The most leptokurtic distribution is students’s t distribution which has the bulk of the data in the peak.

## What is Excess kurtosis?

Excess kurtosis is just kurt – 3. For example, the excess for the normal distribution is 0 – 3 = -3.

- Negative excess means there is less of a peak (more data in the tails).
- Positive excess means there is more of a peak (less data in the tails).

These graphs (from Wikipedia) should help you see the difference between negative, zero, and positive excess.

Uniform: excess = −1.2. | Normal: excess = 0. | Logistic: excess = 1.2 |

## How to Calculate by hand or with technology.

Kurtosis is the fourth moment, so to calculate it by hand, use the following formula:

The *4*th moment = (x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + . . . + x_{n}^{4})/n.

For Minitab and SPSS, you can find the option in the “Descriptive Statistics” tab.

## Kurtosis in Excel 2013

Watch the video or read the steps below:

There are two options in Excel for finding kurtosis: the KURT Function and the Data Analysis Toolpak (How to load the Data Analysis Toolpak).

### Kurtosis Excel 2013: KURT function

Step 1: Type your data into columns in an Excel worksheet.

Step 2: Click a blank cell.

Step 3: Type “=KURT(A1:A99)” where A1:99 is the cell locations for your data.

### Kurtosis Excel 2013: Data Analysis

Step 1: Click the “Data” tab and then click “Data Analysis.”

Step 2: Click “Descriptive Statistics” and then click “OK.”

Step 3: Click the Input Range box and then type the location for your data. For example, if you typed your data into cells A1 to A10, type “A1:A10” into that box

Step 4: Click the radio button for Rows or Columns, depending on how your data is laid out.

Step 5: Click the “Labels in first row” box if your data has column headers.

Step 6: Click the “Descriptive Statistics” check box.

Step 7: Select a location for your output. For example, click the “New Worksheet” radio button.

Step 8: Click “OK.”

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