*19*

In statistics,Â **skewnessÂ **andÂ **kurtosisÂ **are two ways to measure the shape of a distribution.

**SkewnessÂ **is a measure of the asymmetry of a distribution. This value can be positive or negative.

- A negative skew indicates that the tail is on the left side of the distribution, which extends towards more negative values.
- A positive skew indicates that the tail is on the right side of the distribution, which extends towards more positive values.
- A value of zero indicates that there is no skewness in the distribution at all, meaning the distribution is perfectly symmetrical.

**KurtosisÂ **is a measure of whether or not a distribution is heavy-tailed or light-tailed relative to a normal distribution.

- The kurtosis of a normal distribution is 3.
- If a given distribution has a kurtosis less than 3, it is said to beÂ
*playkurtic*, which means it tends to produce fewer and less extreme outliers than the normal distribution. - If a given distribution has a kurtosis greater than 3, it is said to beÂ
*leptokurtic*, which means it tends to produce more outliers than the normal distribution.

**Note:Â **Some formulas (Fisherâ€™s definition) subtract 3 from the kurtosis to make it easier to compare with the normal distribution. Using this definition, a distribution would have kurtosis greater than a normal distribution if it had a kurtosis value greater than 0.

This tutorial explains how to calculate both the skewness and kurtosis of a given dataset in R.

**Example: Skewness & Kurtosis in R**

Suppose we have the following dataset:

data = c(88, 95, 92, 97, 96, 97, 94, 86, 91, 95, 97, 88, 85, 76, 68)

We can quickly visualize the distribution of values in this dataset by creating a histogram:

hist(data, col='steelblue')

From the histogram we can see that the distribution appears to be left-skewed. That is, more of the values are concentrated on the right side of the distribution.

To calculate the skewness and kurtosis of this dataset, we can use **skewness() **and **kurtosis() **functions from the **moments **library in R:

library(moments) #calculate skewness skewness(data) [1] -1.391777 #calculate kurtosis kurtosis(data) [1] 4.177865

The skewness turns out to beÂ **-1.391777** and the kurtosis turns out to be **4.177865**.

Since the skewness is negative, this indicates that the distribution is left-skewed. This confirms what we saw in the histogram.Â

Since the kurtosis is greater than 3, this indicates that the distribution has more values in the tails compared to a normal distribution.

TheÂ **moments** library also offers theÂ **jarque.test()** function, which performs a goodness-of-fit test that determines whether or not sample data have skewness and kurtosis that matches a normal distribution. The null and alternative hypotheses of this test are as follows:

**Null Hypothesis**: The dataset has a skewness and kurtosis that matches a normal distribution.

**Alternative Hypothesis**: The dataset has a skewness and kurtosis that *does not* match a normal distribution.

The following code shows how to perform this test:

**jarque.test(data)
Jarque-Bera Normality Test
data: data
JB = 5.7097, p-value = 0.05756
alternative hypothesis: greater
**

The p-value of the test turns out to beÂ **0.05756**. Since this value is not less than Î± = .05, we fail to rejec the null hypothesis. We do not have sufficient evidence to say that this dataset has a skewness and kurtosis that is different from the normal distribution.Â

*You can find the complete documentation for theÂ moments library here.*

**Bonus: Skewness & Kurtosis Calculator**

You can also calculate the skewness for a given dataset using theÂ Statology Skewness and Kurtosis Calculator, which automatically calculates both the skewness and kurtosis for a given dataset.Â