Formula to Calculate Skewness

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

Mathematically, the skewness formula represents,

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where

  • Xi = ith Random VariableX= Mean of the DistributionN = Number of Variables in the DistributionƠ = Standard Distribution

Calculation of Skewness (Step by Step)

Example

  • Firstly, form a data distribution of random variables, and Xi denotes these variables. Next, figure out the number of variables available in the data distribution, denoted by N. 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. Next, determine the standard deviation of the distribution by using the deviations of each variable from the mean, i.e., Xi – X , and the number of variables in the distribution. Finally, the standard deviation is calculated, as shown below. Finally, the calculation of skewness is done based on the deviations of each variable from the mean, several variables, and the standard deviation of the distribution, as shown below.

Let us take the example of a summer camp in which 20 students assign certain jobs that they performed to earn money to raise funds for a school picnic. However, different students earned different amounts 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, one can calculate the mean of the distribution as,

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

Mean = $142.50

One can calculate the squares of the deviations of each variable 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, one can calculate the standard deviation by using the below formula,

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

ơ = 61.80

One can calculate the cubes of the deviations of each variable 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, the calculation of the 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 this article shows, one may use skewness to describe or estimate the symmetry of data distribution. It is crucial in risk management, portfolio management, trading, and option pricingOption PricingOption pricing refers to the process of determining the theoretical value of an options contract. read more. The measure is known as “Skewness” because the plotted graph gives a skewed display. A positive skew indicates that the extreme variables are larger than the 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, bringing down the mean value and resulting in a median larger than the meanMeanMean refers to the mathematical average calculated for two or more values. There are primarily two ways: arithmetic mean, where all the numbers are added and divided by their weight, and in geometric mean, we multiply the numbers together, take the Nth root and subtract it with one.read more. So, skewness ascertains the lack of symmetry or the extent of asymmetry.

This article has been a guide to Skewness Formula. Here, we discuss calculating skewness using its formula with practical examples and a downloadable Excel template. You can learn more about Excel modeling from the following articles: –

  • Explanation of Normal Distribution FormulaExplanation Of Normal Distribution FormulaNormal distribution is a distribution that is symmetric i.e. positive values and the negative values of the distribution can be divided into equal halves and therefore, mean, median and mode will be equal. It has two tails one is known as the right tail and the other one is known as the left tail.read moreFormulaFormulaStandard deviation (SD) is a popular statistical tool represented by the Greek letter ‘σ’ to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data’s reliability.read more of Standard Deviation Formula of CovarianceVariance vs Standard DeviationVariance Vs Standard DeviationVariance is a numeric value that defines every observation’s variability from the arithmetic mean, while Standard Deviation is a measure to determine how spread out the observations are from the arithmetic mean. read more