Statistics How To

Kurtosis: Definition and How to Find it by Hand or Using Technology

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).


Negative kurtosis (left) and positive (right).

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
shape_uniform shape_normal shape_logistic

How to Calculate by hand or with technology.

Kurtosis is the fourth moment, so to calculate it by hand, use the following formula:
The 4th moment = (x14 + x24 + x34 + . . . + xn4)/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:

Excel 2013

Negative kurt(left) and positive kurt (right)

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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If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

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Kurtosis: Definition and How to Find it by Hand or Using Technology was last modified: October 12th, 2017 by Andale