What is the Skewness Formula?
The following article provides an outline for Skewness Formula. In statistics, the term “skewness” refers to the ratio that helps measure the asymmetry of the distribution of real-valued random variables around its mean. For example, the value of the statistical metricJUnit Ignore can be positive, negative, or undefined based on the fact that the tail of the distribution is right-aligned, left-aligned, or no deviation, respectively. In case the value of skewness is less than -1 or greater than 1, then the distribution is called highly skewed, while a zero value of skewness means that the distribution is symmetrical about its mean.
To put it simply, a standard data distribution that is perfectly symmetrical resembles the shape of a bell, and as such, it is called the bell curve. In such a scenario, the skewness is zero, and the mean, median, and mode are equal. However, this doesn’t happen in practical cases, and the mean, median, and mode tend to deviate from each other. So now, the skewness indirectly tells you how the median and mode are distributed around the mean.
The formula for skewness can be expressed by using the mean of the distribution, no. of random variables, and standard deviation of the distribution. Mathematically, it is represented as,
Skewness = ∑iN(
- Xi = ith random variable
- X̅ = mean of distribution
- N = No. of random variables in the distribution
- Ơ = Standard deviation of the distribution
The skewness formula can be computed based on the following steps:
Step 1: First, accumulate a certain number of random variables to create a data distribution table. Xi denotes these variables.
Step 2: Next, determine the no. of random variables accumulated in the data distribution table. It is denoted by N.
Step 3: Next, compute the distribution’s mean by summing up all the variables available in the distribution table and dividing it by the no. of random variables. It is denoted by, and mathematically it is represented as,
Step 4: Next, compute the distribution’s standard deviation based on each variable’s deviations from the distribution’s mean (i.e. ) and the no. of random variables available in the data distribution. Mathematically, it is represented as,
Step 5: Finally, the skewness formula can be derived by using deviations of each variable from the distribution’s mean (step 3), no. of random variables (step 2), and distribution’s standard deviation (step 4), as shown below.
Examples of Skewness Formula
Given below is an example of the Skewness Formula:
Let us take the example of a small office with a total of 35 employees. Each falls in a different salary bracket based on their no. of industry experience and educational background. The distribution table with the salary information is given below. Determine the skewness of the salary distribution based on the provided information.
|Annual Salary bracket (in $’ 000)||No. of employees|
|$10 – $20||4|
|$20 – $30||10|
|$30 – $40||12|
|$40 – $50||6|
|$50 – $60||3|
Given, No. of variables, N = 4 + 10 + 12 + 6 + 3 = 35
Now, let us compute the midpoints for each annual salary bracket (interval)
1st interval: ($10 + $20) / 2 = $15
2nd interval: ($20 + $30) / 2 = $25
3rd interval: ($30 + $40) / 2 = $35
4th interval: ($40 + $50) / 2 = $45
5th interval: ($50 + $60) / 2 = $55
Now, the mean of the distribution can be calculated as,
Mean, = ($15 * 4 + $25 * 10 + $35 * 12 + $45 * 6 + $55 * 3) / 35
Now, squares of the deviations of each interval midpoint from the distribution’s mean can be calculated as below,
1st interval: ($15 – $33.3)2 = 334.4
2nd interval: ($25 – $33.3)2 = 68.7
3rd interval: ($35 – $33.3)2 = 2.9
4th interval: ($45 – $33.3)2 = 137.2
5th interval: ($55 – $33.3)2 = 471.5
Now, the standard deviation (in $’ 000) can be calculated by using the above formula as
ơ = [(334.4 * 4 + 68.7 *10 + 2.9 * 12 + 137.2 * 6 + 471.5 * 3) / 35]1/2
Now, cubes of the deviations of each interval midpoint from the distribution’s mean can be calculated below,
($15 – $33.3)3 = -6114.1
($25 – $33.3)3 = -568.8
($35 – $33.3)3 = 5.0
($45 – $33.3)3 = 1607.5
($55 – $33.3)3 = 10238.5
Now, the skewness of the salary distribution can be calculated by using the above formula,
= (- 6114.1 * 4 – 568.8 * 10 + 5.0 * 12 + 1607.5 * 6 + 10238.5 * 3) /[ (35 – 1) * (11.08)3]
Therefore, the salary distribution table exhibits symmetry as indicated by a skewness value of fairly close to zero.
Relevance and Use of Skewness Formula
The skewness formula is a fundamental statistical concept used to assess the symmetry of a data distribution around its mean. The concept revolves around the relationship between mean, median, and mode. The concept finds extensive portfolio management, risk management, option pricing, and trading applications.
A positive skew indicates that the extreme outcomes are more significant, resulting in a mean value higher than the data distribution’s median value. On the other hand, a negative skew indicates that the extreme outcomes are smaller, resulting in a mean value lower than the median value of the data distribution. In positive skewness, the median and mode are to the left of the mean, while in negative skewness, they are on the right side.
This has been a guide to Skewness Formula. So, by now, you know what skewness is, how to interpret its value, and what implications it has on a distribution. You may also look at the following article to learn more –