# Moments coefficient of kurtosis using R with examples

In this tutorial, you will learn about what is moment coefficient of kurtosis and how to calculate moment coefficient of kurtosis in R.

## Moment Coefficient of Kurtosis

The literal meaning of kurtosis is peakedness or flatness of the data. The kurtosis measures how peaked or how flat the histogram is relative to the bell-shaped histogram. The bell-shaped histogram is based on normal (Gaussian) distribution.

• The value of kurtosis for a normal (Gaussian) distribution is 3.
• If the histogram has too many observations in the tails compared to normal histogram, then the kurtosis is larger than 3.
• If the histogram has short tails and most of the observations are tightly clustered around the mean, then the kurtosis is less than 3.

The moment coefficient of kurtosis (also known as Pearson's moment coefficient of kurtosis) is denoted by $\beta_2$ and is defined as

$$\beta_2=\dfrac{m_4}{m_2^2}$$

The moment coefficient of kurtosis $\gamma_2$ is defined as

$$\gamma_2=\beta_2-3$$

where

• $n$ total number of observations
• $\overline{x}$ sample mean
• $m_2 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^2$ is second sample central moment
• $m_4 =\dfrac{1}{n}\sum_{i=1}^n (x_i-\overline{x})^4$ is fourth sample central moment

To know more about moments, check the tutorial on how to compute raw and central moments using R with examples.

Note that, the normal curve (Gaussian curve) is a bell-shaped the value of $\beta_2$ is 3.

### Leptokurtic distribution

The distribution is said to be leptokurtic, if it has a higher peak than the normal curve.

### Platykurtic Distribution

The distribution is said to be platykurtic, if it has a lower peak than the normal curve.

### Mesokurtic Distribution

The distribution is said to be mesokurtic, if it is neither peaked nor flat.

## Interpretation of coefficient of skewness

• If $\beta_2 > 3$ or $\gamma_2 > 0$, then the distribution is leptokurtic.
• If $\beta_2 = 3$ or $\gamma_2 = 0$, then the distribution is mesokurtic.
• If $\beta_2 < 3$ or $\gamma_2 < 0$ then the distribution is platykurtic.

## Moment coefficient of kurtosis using R

To calculate the moment coefficient of kurtosis, we need to install the package moments. The function kurtosis() gives the $\beta_2=\dfrac{m_4}{m_2^2}$ coefficient for kurtosis. This function compute the estimators of Pearson's measure of kurtosis based on moments.

The syntax of the kurtosis() function is

kurtosis(x,na,rm=FALSE)

where,

• x : a numeric vector, matrix or data frame
• na.rm : logical value (default FALSE). Should missing values be removed?

## Numerical Problem moment coefficient of kurtosis using R

### Example 1 : Moment Coefficient of Kurtosis using R

The following data are the heights, correct to the nearest centimeters, for a group of children:
126, 129, 129, 132, 132, 133, 133, 135, 136, 137,
137, 138, 141, 143, 144, 146, 147, 152, 154, 161.
Find Moment coefficient of kurtosis and interprete the result.

# create a data vector
x <- c(126, 129, 129, 132, 132, 133, 133, 135, 136, 137,
137, 138, 141, 143, 144, 146, 147, 152, 154, 161)
# load the package
library(moments)
## Compute kurtosis
kurtosis(x)
[1] 2.831218

The $\beta_2$ coefficient of kurtosis is 2.8312182. As $\beta_2 < 3$, the data about height (in cm) is $\text{platykurtic}$.

### Example 2: Moment Coefficient of Kurtosis using R

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:
75, 80, 72, 78, 82, 85, 73, 75, 97, 87,
84, 76, 73, 79, 99, 86, 83, 76, 78, 73.
Find moment coefficient of kurtosis and interprete the result.

BS <-c(75, 80, 72, 78, 82, 85, 73, 75, 97, 87,
84, 76, 73, 79, 99, 86, 83, 76, 78, 73)
# load the package
library(moments)
## Compute skewness
kurtosis(BS)
[1] 3.571845

The $\beta_2$ coefficient of kurtosis is 3.5718448. As $\beta_2 > 3$, the data about Blood Sugar level is $\text{leptokurtic}$.

## Endnote

In this tutorial you learned about what is moment coefficient of kurtosis and how to calculate moment coefficient of kurtosis using R.