Skewness Formula: Methods to Measure Skewness, Types, and Solved Examples

Skewness represents deviating from a symmetrical bell curve. It shows an asymmetrical curve with a tail extending to the right (positive skew) or to the left (negative skew).

• It also indicates that a dataset's mean, median, and mode have different values.
• The skewness formula can be calculated using the mean of the distribution, the number of variables, and the standard deviation of the distribution.
• It is a statistical metric that may be used to indicate a probability distribution's asymmetry.
• Skewness is used to identify outliers in a dataset. It also indicates where the data set is trending.
• This knowledge is essential in finance since it allows investors to minimize the risks related to their investments.

Key Terms: Symmetry, Asymmetry, Variables, Probability, Skewness, Standard derivation, Mean, Distribution, Mode, Median, Skewness formula, Co-efficient of Skewness

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Mean, Median, and Mode

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To know about the skewness formula, we have to first learn the Mean, Median, and Mode.

Mean

The average of the numbers in the data distribution is called the mean. It is determined by adding all of the values in the dataset and dividing the sum by the number of values in the dataset.

Mean = Sum of all values in Dataset / Total number of values

Median

The median is the middle value of a dataset when the values are ordered from least to greatest.

Mode

The most frequently used number in data is known as the mode of the data.

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Skewness Formula

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The skewness formula is named so because the plotted graph appears skewed. Skewness is a statistical measure that shows an asymmetric probability distribution.

The formula of Skewness is given by

\(Skewness = \frac {3(Mean - Median)}{Standard\:Deviation\:(SD)}\)

The standard deviation used in the above formula is given by

\(Standard \: Deviation = \sqrt {\frac{\sum |x - \bar x|^2}{n}}\)

Where

• x is the random variable
• \(\bar x\) is the mean of the data
• n is the total number of data

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Methods to Measure Skewness

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Skewness can be calculated using Karl Pearson's Coefficient of Skewness.

Karl Pearson’s Co-efficient of Skewness 

The formula to calculate skewness using Karl Pearson's coefficient is given below.

• Using Mode

\(sk_1 = \frac {x\: - \:Mode}{Standard \: Deviation}\)

• Using Median

\(sk_2 = \frac {3(x\: - \:Median)}{Standard \: Deviation}\)

Conditions:

• The coefficient of skewness is zero for symmetrical distribution, if Mean = Mode = Median.
• The coefficient of skewness will be positive if Mean > Mode.
• Then the coefficient of skewness will be negative if Mean < Mode.

The sign of the Karl person`s coefficient of skewness is positive for positively skewed and negative for negatively skewed.

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Types of Skewness

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There are 3 types of Skewness:

• Positive Skewness
• Negative Skewness
• Zero Skewness

Positive Skewness

Positive skewness indicates that the tail on the right side of the distribution is longer. The mean and median will be larger than the mode.

The condition for positive skewness is

Mean > Median > Mode

Negative Skewness

Negative skewness occurs when the tail on the left side of the distribution is longer than the tail on the right. The mean and median will be smaller than the mode.

The condition for negative skewness is

Mode > Median > Mean

Zero Skewness

Zero skewness is also referred to as a "symmetric distribution". It indicates that data is distributed uniformly around the mean, with no lengthy tails on either end of the distribution.

The condition for zero skewness is

Mean = Mode = Median

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Solved Examples

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Ques. Find the skewness for the given Data (3, 4, 5, 5).

Ans. The given data is (3, 4, 5, 5)

Number of terms (n) = 4 (Here n is even)

Mean of Data = (3 + 4 + 5 + 5) / 4

⇒ Mean of Data = 17 / 4 = 4.25

Median of Data = {[n / 2]^th term + [n / 2 + 1]^th term} / 2

⇒ Median of Data = [(4 /2)^th term + (4/2 +1)^th term] / 2

⇒ Median of Data = [2^nd term + 3^rd term] / 2

⇒ Median of Data = [4 + 5] / 2 = 9/2 = 4.5

Standard deviation, S.D. = √[(3 – 4.5)² + (4 – 4.5)² + (5 – 4.5)² + (5 – 4.5)² / 4]

⇒ S.D = √[(2.25 + 0.25 + 0.25 + 0.25) / 4]

⇒ S.D = √(0.75) = 0.87

The formula of Skewness is given by

Skewness = 3(Mean – Median)/S.D.

On substituting the values, we get

Skewness = 3(4.25 – 4.5) / 0.87

⇒ Skewness = 3(-0.25)/ 0.87

⇒ Skewness = – 0.86

Therefore, the skewness of these data is negative.

Ques. Find the skewness for the given Data (2, 3, 4, 9).

Ans. The given data is (2, 3, 4, 9)

Number of terms (n) = 4 (Here n is even)

Mean of Data = (2 + 3 + 4 + 9) / 4

⇒ Mean of Data = 18 / 4 = 4.5

Median of Data = {[n / 2]^th term + [n / 2 + 1]^th term} / 2

⇒ Median of Data = [(4/2)^th term + (4/2 +1)^th term] / 2

⇒ Median of Data = [2^nd term + 3^rd term] / 2

⇒ Median of Data = [3 + 4] / 2 = 7/2 = 3.5

Standard deviation, S.D. = √[(2 – 3.5)² + (3 – 3.5)² + (4 – 3.5)² + (9 – 3.5)² / 4]

⇒ S.D = √[(2.25 + 0.25 + 0.25 + 30.25) / 4]

⇒ S.D = √(8.25) = 2.87

The formula of Skewness is given by

Skewness = 3(Mean – Median)/S.D.

On substituting the values, we get

Skewness = 3(4.5 – 3.5) / 2.87

⇒ Skewness = 3(1)/ 2.87

⇒ Skewness = 1.04

Therefore, the skewness of these data is positive.

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Things to Remember

• Skewness refers to the degree of asymmetry in a probability distribution.
• The Skewness formula is given by \(Skewness = \frac {3(Mean - Median)}{Standard\:Deviation\:(SD)}.\)
• Standard deviation is given by the formula \(Standard \: Deviation = \sqrt {\frac{\sum |x - \bar x|^2}{n}}.\)
• There are three types of skewness i.e. positive, negative, and zero.
• The coefficient of skewness is zero for symmetrical distribution, if Mean = Mode = Median.

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