# How to use the KURT function

*What is the KURT function?*

The KURT function lets you calculate the sample excess kurtosis in Excel. The kurtosis shows the tailedness of a probability distribution, in other words, how often outliers (extreme values) occur. This number is **not** related to the peak of the distribution but to it's tails.

#### Table of Contents

## 1. Introduction

**How to analyze the kurtosis value?**

A high kurtosis number tend to indicate more outliers (heavy tails) in a distribution, whereas a small kurtosis value indicate less outliers (thin tails).

**What is the difference between excess kurtosis and regular kurtosis?**

The difference between excess kurtosis and regular kurtosis is in the way they are calculated and interpreted.

Regular kurtosis, also known as kurtosis or the fourth standardized moment, is a measure of the "tailedness" of a probability distribution. It describes how heavy the tails of a distribution are compared to a normal distribution. A higher kurtosis than 3 indicates that the distribution has heavier tails, while a lower kurtosis than 3 indicates lighter tails.

Excess kurtosis, also known as kurtosis excess or kurtosis risk, is calculated by subtracting 3 from the regular kurtosis value. The value 3 is subtracted because the normal distribution has a kurtosis of 3.

The formula for excess kurtosis is: Excess Kurtosis = Kurtosis - 3

**What is the kurtosis and excess kurtosis of a normal distribution?**

- The kurtosis of a normal distribution is 3 and is called a "mesokurtic".

A distribution with a higher kurtosis than three is called "leptokurtic", meaning that it has fatter tails than a normal distribution. A distribution with a lower kurtosis than three is called "platykurtic", meaning that it has thinner tails than a normal distribution.

The kurtosis ranges from 1 to infinity. - The excess kurtosis of a normal distribution is 0.

Excel calculates the "excess" kurtosis meaning a value larger than 0 (zero) is called "leptokurtic" and a kurtosis value smaller than 0 (zero) is called "platykurtic".

The excess kurtosis ranges from -2 to infinity.

*What is the difference between kurtosis and skewness?*

Skewness measures the symmetry of a distribution whereas kurtosis measures outliers or the tails of the distribution relative to a normal distribution.

A symmetric distribution has zero skewness, meaning that the data values are evenly distributed on both sides of the mean. An asymmetric distribution has either positive or negative skewness, meaning that the data values are more concentrated on one side of the mean than the other.

**What are the formulas for the sample excess kurtosis and the population excess kurtosis?**

Sample Excess Kurtosis = (n(n+1) / ((n-1)(n-2)(n-3))) * Σ((x_i - μ)^{4} / s^{4}) - (3(n-1)^{2}) / ((n-2)(n-3))

n is the sample size

x_i is the i-th observation

μ is the sample mean

s is the sample standard deviation

Population Excess Kurtosis = Σ(x - x_{mean})^{4}/(Σ(x - x_{mean})^{2})^{2}-3

μ_4 is the fourth central moment about the mean (Σ[(X - μ)^{4}])

σ is the population standard deviation

μ is the population mean

## 2. KURT Function Syntax

KURT(*number1*, *number2*, ...)

## 3. KURT Function Arguments

Argument |
Description |

number1 |
Required. A single numerical value or a cell reference to multiple numerical values. |

number2 |
Optional. Up to 30 additional arguments. |

Text, logical values and empty cells are ignored.

The KURT function returns

- #DIV/0! error value if the number of values is less than 4.

## 4. KURT Function Example 1

The image above shows two distributions, the first one is a normal distribution and the second one has random values. The values are in cell range D19:D35. The chart shows the normal distribution and another distribution containing more random values.

Formula in cell D37:

This formula returns 3.47 meaning the kurtosis is higher than 0 (zero). This tells us that the distribution in D19:D35 has "heavy" tails compared to the normal distribution, in other words, this distribution has more outliers which seems to be the case if you examine blue curve in the chart above.

## 5. KURT Function Example 2

**You are a scientist and has collected the following measurements of a peculiar phenomenon. Calculate the kurtosis? The data is presented in the table below.
**

197 | 171 | 183 | 190 | 165 |

151 | 171 | 148 | 185 | 173 |

172 | 177 | 192 | 151 | 197 |

188 | 169 | 164 | 166 | 184 |

187 | 154 | 190 | 154 | 180 |

159 | 162 | 181 | 176 | 193 |

199 | 209 | 167 | 150 | 177 |

169 | 169 | 165 | 174 | 213 |

206 | 177 | 210 | 192 | 154 |

150 | 205 | 182 | 170 | 167 |

The image above shows the data in cell range B19:F28, the chart above shows a normal distribution and a histogram represented by a column chart. The yellow line displays the normal distribution and the blue columns represents the frequency table of the data given in B19:F28.

Formula in cell C30:

The formula returns -0.65 which indicates that the distribution has thin tails, in other words, it has less number of outliers than a normal distribution.

## 6. How is kurtosis calculated?

The math formula Excel uses to calculate the kurtosis is:

Xi is each value in the data set.

x ̅ is the average of all values.

n is the number of values.

s is the standard deviation. It is a measure of how spread out the values are.

Using the values 2, 4, 1, and 6 shown in the image above:

Average: 13/4 equals 3.25

n: 4

Standard deviation: 2.21735578260835

(How to use the STDEV.S function)

#### Step 1 - First part of the math formula

4*(4+1)/((4-1)*(4-2)*(4-3))

becomes

20/(3*2*1)

becomes

20/6 equals approx. 3.33333

#### Step 2 - Second part of the math formula

SUM(((B3:B6-AVERAGE(B3:B6))/H2)^4)

becomes

SUM((({2;4;1;6}-3.25)/2.21735578260835)^4)

becomes

SUM({-1.25;0.75;-2.25;2.75}/2.21735578260835)^4)

becomes

SUM(({-0.563734521002122;0.338240712601273;-1.01472213780382;1.24021594620467})^4)

becomes

SUM({0.10099468543522;0.0130889112324045;1.06020180982477;2.3658611031313})

and returns

3.54014650962367

#### Step 3 - Third part of the math formula

becomes

3*(4-1)^2/((4-2)*(4-3))

becomes

3*3^2/(2*1)

becomes

27/2 equals 13.5

#### Step 4 - Multiply and subtract

becomes

3.33333*3.54014650962367)-13.5

becomes

11.8004883654122-13.5

equals

-1.69951

This value matches the value calculated by the KURT function in cell D3, see the image below.

### Useful resources

KURT function - Microsoft

Excel's KURT( ) function computes the excess kurtosis, not the kurtosis

How to Interpret Excess Kurtosis and Skewness

### Functions in 'Statistical' category

The KURT function function is one of 73 functions in the 'Statistical' category.

### Excel function categories

### Excel categories

### 3 Responses to “How to use the KURT function”

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Great explanation on how to use the KURT function in Excel to calculate kurtosis and its relation to outliers. One question I have is, how can we use the kurtosis value to make meaningful interpretations about our data?

Since kurtosis measures outliers (i.e., rare extreme values), high kurtosis alerts you to the problem.

There are erroneous statements in the document concerning relation of the kurtosis to the shape/height of the peak; these should be ignored, or better yet, corrected by the author.

Ana Carol,

thank you.

Peter Westfall,

thank you for your comment, I appreciate it.

https://en.wikipedia.org/wiki/Kurtosis

I have corrected these statements.