Skewness Formula

Skewness Formula is a Statistical formula which is a calculate of the Probability distribution of the given set of variables and the same can be positive, negative or undefined.

Formula to Calculate Skewness

The term “skewness” refers to the statistical metric that is used to measure the asymmetry of a probability distribution of random variables about its own mean, and its value can be positive, negative, or undefined. The calculation of the skewness equation is done on the basis of the mean of the distribution, the number of variables, and the standard deviation of the distribution.

Mathematically, the skewness formula is represented as,

where

• X_i = i^th Random Variable
• X= Mean of the Distribution
• N = Number of Variables in the Distribution
• Ơ = Standard Distribution

Calculation of Skewness (Step by Step)

• Step 1: Firstly, form a data distribution of random variables, and these variables are denoted by X_i.
• Step 2: Next, figure out the number of variables available in the data distribution, and it is denoted by N.
• Step 3: Next, calculate the mean of the data distribution by dividing the sum of all the random variables of the data distribution by the number of variables in the distribution. The mean of the distribution is denoted by X.

• Step 4: Next, determine the standard deviation of the distribution by using the deviations of each variable from the mean, i.e., X_i – X and the number of variables in the distribution. The standard deviation is calculated, as shown below.

• Step 5: Finally, the calculation of skewness is done on the basis of the deviations of each variable from the mean, a number of variables, and the standard deviation of the distribution, as shown below.

Example

Let us take the example of a summer camp in which 20 students assigned certain jobs that they performed to earn money to raise funds for a school picnic. However, different students earned a different amount of money. Based on the information given below, determine the skewness in the income distribution among the students during the summer camp.

Solution:

The following is the data for the calculation of skewness.

Number of variables, n = 2 + 3 + 5 + 6 + 4= 20

Let us calculate the midpoint of each of the intervals

• ($0 + $50) / 2 = $25
• ($50 + $100) / 2 = $75
• ($100 + $150) / 2 = $125
• ($150 + $200) / 2 = $175
• ($200 + $250) / 2 = $225

Now, the mean of the distribution can be calculated as,

Mean= ($25 * 2 + $75 * 3 + $125 * 5 + $175 * 6 + $225 * 4) / 20

Mean = $142.50

The squares of the deviations of each variable can be calculated as below,

• ($25 – $142.5)² = 13806.25
• ($75 – $142.5)² = 4556.25
• ($125 – $142.5)² = 306.25
• ($175 – $142.5)² = 1056.25
• ($225 – $142.5)² = 6806.25

Now, the standard deviation can be calculated by using the below formula as,

ơ = [(13806.25 * 2 + 4556.25 * 3 + 306.25 * 5 + 1056.25 * 6 + 6806.25 * 4) / 20]^1/2

ơ = 61.80

The cubes of the deviations of each variable can be calculated as below,

• ($25 – $142.5)³ = -1622234.4
• ($75 – $142.5)³ = -307546.9
• ($125 – $142.5)³ = -5359.4
• ($175 – $142.5)³ = 34328.1
• ($225 – $142.5)³ = 561515.6

Therefore, Calculation of skewness of the distribution will be as follows,

= (-1622234.4 * 2 + -307546.9 * 3 + -5359.4 * 5 + 34328.1 * 6 + 561515.6 * 4) /[ (20 – 1) * (61.80)³]

Skewness will be –

Skewness = -0.39

Therefore, the skewness of the distribution is -0.39, which indicates that the data distribution is approximately symmetrical.

Relevance and Uses of Skewness Formula

As seen already in this article, skewness is used to describe or estimate the symmetry of data distribution. It is very important from the perspective of risk management, portfolio management, trading, and option pricing. The measure is called “Skewness” because the plotted graph gives a skewed display. A positive skew indicates that the extreme variables are larger than skews. The data distribution is such a way it escalates the mean value in a way that it will be larger than the median resulting in a skewed data set. On the other hand, a negative skew indicates that the extreme variables are smaller, which brings down the mean value, which results in a median larger than the mean. So, skewness ascertains the lack of symmetry or the extent of asymmetry.

Recommended Articles

This has been a guide to Skewness Formula. Here we discuss how to calculate skewness using its formula with practical example and a downloadable excel template. You can learn more about excel modeling from the following articles –

• Explanation of Normal Distribution Formula
• Formula of Standard Deviation 
• Formula of Covariance
• Variance vs Standard Deviation