To Determine Skewness Mean and Deviation With a New Approach on Continuous Data Bibtex

Coefficient of Skewness

Coefficient of skewness is one way to measure the skewness of a distribution. Skewness can be defined as a measure of the asymmetry of a probability distribution. If the curve of a normal distribution is distorted towards the left or right then it is known as a skewed distribution.

The most important measure of skewness is the coefficient of skewness that was given by Karl Pearson. It is also known as Pearson's coefficient of skewness. In this article, we will learn more about the coefficient of skewness, its formulas, how to calculate it, and see certain associated examples.

1.	What is the Coefficient of Skewness?
2.	Coefficient of Skewness Formula
3.	How to Calculate Coefficient of Skewness?
4.	FAQs on Coefficient of Skewness

What is the Coefficient of Skewness?

The coefficient of skewness can be defined as a measure that is used to determine the strength and direction of the skewness of a sample distribution by using descriptive statistics such as the mean, median, or mode. The coefficient of skewness is used to compare a sample distribution to a normal one. If the value is very large it implies that there is a greater difference between the sample distribution as compared to a normal distribution.

Coefficient of Skewness Interpretation

Depending upon the value of the coefficient of skewness, the following inferences can be drawn about a distribution.

If the mean exceeds the mode and median (Mode < Median < Mean) then the distribution is positively skewed. In other words, if the coefficient of skewness is positive then the distribution is skewed to the right.
If the mode exceeds the median and mean (Mean < Median < Mode) then the distribution is negatively skewed. Thus, the coefficient of skewness will be negative and the distribution will be skewed to the left.
If the value of the mean, median, and mode are equal then the distribution is a normal distribution and the coefficient of skewness will be 0.

Coefficient of Skewness Formula

There are two formulas, that were developed by Karl Pearson, available to calculate the coefficient of skewness. One uses the mode while the other uses the mean. The Karl Pearson coefficient of skewness formulas are given below:

Using Mode

sk₁
= \(\frac{\overline{x}-Mode}{s}\)

Using Median

sk₂
= \(\frac{3(\overline{x}-Median)}{s}\)

where, \(\overline{x}\) is the mean and s is the standard deviation.

The first coefficient of skewness formula uses the mode. However, if there are not enough data points in the dataset then the mode is not considered to be a robust measure of central tendency. Furthermore, a dataset can have more than one mode. Thus, in most cases, researchers prefer using the second formula (with the median) to calculate the coefficient of skewness.

How to Calculate Coefficient of Skewness?

Depending upon the data available either of the two formulas can be used to calculate the coefficient of skewness. Suppose the mean of a data set is 60.5, the mode is 75, the median is 70 and the standard deviation is 10. The steps to calculate the coefficient of skewness are as follows:

Using Mode

Step 1: Subtract the mode from the mean. 60.5 – 75 = -14.5
Step 2: Divide this value by the standard deviation to get the coefficient of skewness. Thus, sk₁
= -14.5 / 10 = -1.45.

Using Median

Step 1: Subtract the median from the mean. 60.5 – 70 = -9.5
Step 2: Multiply this value by 3. This gives -28.5.
Step 3: Divide the value from step 2 by the standard deviation to obtain the coefficient of skewness. Thus, sk₂
= -28.5 / 10 = -2.85

Related Articles:

Data Collection Methods
Summary statistics
How to Find Median

Important Notes on Coefficient of Skewness

The coefficient of skewness is used to measure the extent and direction of skewness of a sample or distribution.
The coefficient of skewness can be positive, negative, or zero.
There are two formulas, given by Karl Pearson, that can be used to calculate the coefficient of skewness.

Examples on Coefficient of Skewness

Example 1:
Calculate the second coefficient of skewness for the following data.
1, 2, 3, 4, 5, 6, 7, 8, 9, 9

Solution:
Mean = 54 / 10 = 5.4

Variance = [(1 – 5.4)²
+ (2 – 5.4)²
+ (3 – 5.4)²
+ (4 – 5.4)²
+ (5 – 5.4)²
+ (6 – 5.4)²
+ (7 – 5.4)²
+ (8 – 5.4)²
+ (9 – 5.4)²
+ (9 – 5.4)²] / 10 – 1 = 7.44

Standard Deviation = √7.44 = 2.73

Median = ((n/2)^th
term + (n/2 + 1)^th
term)/2

= [5 + 6] / 2

= 5.5

sk₂
= \(\frac{3(\overline{x}-Median)}{s}\)

= 5.4 – 5.5 / 2.73

= -0.11

Answer:
sk₂
= -0.11
Example 2:
Calculate the first coefficient of skewness for the following data.
1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 12, 12, 13

Solution:
Mode = 9

Mean = 7.96

Standard Deviation = 2.98

sk₁
= \(\frac{\overline{x}-Mode}{s}\)

= (7.96 – 9) / 2.98

= -0.34

Answer:
sk₁
= -0.31
Example 3:
If the coefficient of skewness of a distribution is 0.32, the standard deviation is 6.5 and the mean is 29.6 then find the mode of the distribution.

Solution:
Using the formula for the first coefficient of skewness, the mode can be determined as follows:

sk₁
= \(\frac{\overline{x}-Mode}{s}\)

0.32 = (29.6 – mode) / 6.5

2.08 = 29.6 – mode

Mode = 27.52

Answer:
27.52

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FAQs on Coefficient of Skewness

What is the Meaning of Coefficient of Skewness?

The coefficient of skewness can be defined as a measure of skewness that indicates the strength and the direction of asymmetry in a probability distribution.

What is the Coefficient of Skewness Formula Using the Mode?

There are two formulas that can be used to calculate the coefficient of skewness. The Karl Pearson coefficient of skewness formula using the mode can be given as sk₁
= \(\frac{\overline{x}-Mode}{s}\).

What is the Coefficient of Skewness Formula Using the Median?

The coefficient of skewness using the median is a more robust measure of skewness than the coefficient that is calculated using the mode. The coefficient of skewness formula is given as \(\frac{3(\overline{x}-Median)}{s}\).

What Does a Negative Coefficient of Skewness Indicate?

A negative coefficient of skewness indicates that the distribution is negatively skewed. In other words, the distribution is skewed to the left and the mean < median < mode.

What Does a Zero Coefficient of Skewness Indicate?

A zero coefficient of skewness indicates that the distribution is symmetric. An example of a distribution that has a 0 coefficient of skewness is a normal distribution.

How to Calculate the Coefficient of Skewness Using the Mode?

The steps to calculate the coefficient of skewness using the mode are as follows:

Subtract the mode from the mean.
Divide this value by the standard deviation to get the coefficient of skewness.

How to Calculate the Coefficient of Skewness Using the Median?

The steps to calculate the coefficient of skewness using the mode are as follows:

Subtract the median from the mean and multiply this value by 3.
Divide this value by the standard deviation to get the Karl Pearson coefficient of skewness.

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Source: https://en.asriportal.com/62557/determine-the-coefficient-of-skewness-using-the-software-method/