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.

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.

How to use the SKEW function

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 / s⁴) - (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.

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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.

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