Skewness Formula

The measure of skewness tells us the direction and the extent of skewness. In symmetrical distribution the mean, median, and mode are identical. the more the mean moves away from the mode, the larger the asymmetric or skewness.

Before learning let’s learn more about Mean, Median, and Mode first.

Mean

Mean is the average of the numbers in the data distribution, It is calculated by adding up all 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

Example: Find the mean of a dataset of exam scores: 70, 80, 85, 90, and 95.

Solution:

Mean = (70 + 80 + 85 + 90 + 95) / 5 = 84

So the mean of this dataset is 84.

Median

When arranging all the data in order (ascending and descending) the comes in the middle of the data is called the median.

Median is the middle value of a dataset when the values are arranged in order from smallest to largest. 

Examples for Odd Numbers in the Dataset

Example 1: Find the median of a dataset of exam scores: 70, 85, 80, 95, 90

Solution:

Firstly arrange all no. in order from smallest to largest: 70, 80, 85, 90, 95.

The mid value is 85. so, the median is 85.

Example 2: Find the median of a dataset: 5, 10, 15, 20, 25. 

Solution:

Firstly arrange all no. in order from smallest to largest: 5, 10, 15, 20, 25. 

The mid value is 15. so, the median is 15.

If there are an even number of values in the dataset, the median is calculated by taking the average of the two middle values.

Examples for Even Numbers in the Dataset

Example 1: Find the median of a dataset of exam scores: 70, 80, 85, 90.

Solution:

The median is calculated as (80 + 85) / 2 = 82.5

So the median of this dataset is 82.5.

Example 2: Find the median of a dataset: 2, 4, 6, 8, 10, 12.

Solution:

Firstly, we need to find the middle two numbers. So, 6, and 8 are mid values of the dataset 

Median = (6 + 8) / 2 = 7

So the median of this dataset is 7.

Mode

most frequently used number in data is called the mode of the data.

Example 1: We have a data set representing the number of pets owned by 10 people: 3, 1, 0, 2, 1, 1, 4, 2, 2, 1. Find the mode.

Solution:

So, the value that appears most frequently in the data set is 1. the value 1 appears four times. Therefore, the mode of this data set is 1.

Skewness Formula

The skewness formula is discussed in the image below,

Type of Skewness

Various types of skewness used in mathematics are,

• Positive Skewness
• Negative Skewness
• Zero Skewness

Positive Skewness

Positive Skewness means the tail on the right side of the distribution is longer. The mean and median will be greater than the mode. 

Condition for positive skewness = Mean > Median >Mode

The positive curve of skewness is shown in the image below,

Let’s take an example of the income distribution where a few people earn very high incomes and the majority earn lower incomes. so, this is often positively skewed. Analyzing skewed data can provide valuable insights into the underlying causes and potential solutions or interventions.

Negative Skewness

Negative Skewness means when the tail of the left side of the distribution is longer than the tail on the right side. The mean and median will be less than the mode.

Condition for negative skewness is Mode > Median > Mean

The curve shows negative skewness in the image below,

Let’s take an example of a match, during the match most of the players of a particular team scored runs above 50 and only a few of them scored below 10. In such a case, the data is generally represented with the help of a negatively skewed distribution. And this data is helpful to analyze the game’s performance.

Zero Skewness

It is also known as a “symmetric distribution”.It signifies that distribution of data is evenly distributed around the mean, with no long tails on either end of the distribution

Condition for zero skewness is Mean = Mode = Median

The curve for zero skews is shown in the image below,

Methods to Measure Skewness

Skewness can be measured using Karl Pearson’s Coefficient of Skewness.

Karl Pearson’s Co-efficient of Skewness 

The formula for measuring Skewness using Karl Pearson’s Co-efficient is discussed below in the image,

Conditions

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

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

Read More,

• Bayes’ Theorem
• Mean

Solved Examples on Skewness Formula

Example 1: Find the skewness for the given Data ( 2,4,6,6) 

Solution:

Mean of Data = (2 + 4 + 6 + 6) / 4

                       = 18 / 4

                       = 4.5

Number of terms (n) = 4 (even)

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

                          = [(4 /2)^th term + (4/2 +1)^th term] / 2

                          = [2^nd term + 3^rd term] / 2

                          = [4+6]/2

                          = 10/2

Median of Data  = 5

Mode of Data = Highest Frequency term = 6 (frequency 2)

S.D. = √[(2 – 5)² + (4-5)² + (6-5)² + (6-5)²/4]

       = √[(9 + 1 + 1 + 1)/4]

       = √(3)

       = 1.732

Skewness = 3(Mean – Median)/S.D.

By Applying Skewness Formula,

Skewness = 3(4.5 – 5)/1.732

                = 3(-0.5)/ 1.732

Skewness = – 0.866

So, the skewness of these data is negative.

Example 2: A boy collects some rupees in a week as follows (25,28,26,30,40,50,40) and finds the skewness of the given Data in question with the help of the skewness formula.

Solution:

Mean of Data = (25+28+26+30+40+50+40) / 7

                      = 239 / 7

                      = 34.14

Number of terms (n) =7 (odd) 

Arrange Data in ascending order = 25,26 ,28,30,40,40,50

The median of data is = 30

Mode of Data = Highest Frequency term = 40 (frequency 2)

S.D       = √(1/7 – 1) x ((25 – 34.1429)² + (28 – 34.1429)² + (26 – 34.1429)² + (30 – 34.1429)² + (40 – 34.1429)² +(534.1429)² + (40 – 34.1429)²)
           = √(1/6) x ((-9.1429)² + (-6.1429)² + (-8.1429)² + (-4.1429)² + (5.8571)² + (15.8571)² + (5.8571)²)
           = √(0.1667) x ((83.5926) + (37.7352) + (66.3068) + (17.1636) + (34.3056) + (251.4476) + (34.3056))
           = √(0.1667) x 524.8571
           = √87.4762
         . = 9.3529

   Skewness = 3(Mean – Median)/S.D.

By Applying Skewness Formula,

Skewness = 3(34.14 – 30)/9.3529

                = 1.32

Skewness = 1.32

So skewness for these data is positive

Example 3: Attendance of all classes of a school are as follows find their skewness? 

1st (35), 2nd(32), 3rd(38), 4th(39), 5th(43)

Class Name	Number of students
1 st	35
2 nd	32
3 rd	38
4 th	39
5 th	45

Solution:

Mean of Data =  (35 + 32 + 38 + 39 + 42)/5

                      = 186/5

                      = 37.2

Number of terms (n) = 5 (odd)

Arrange Data in ascending order = 32,35,38,39,42 

Median of Data  = 38

S.D. = √(1/5 – 1) x ((35 – 37.2)2 + (32 – 37.2)2 + (38 – 37.2)2 + (39 – 37.2)2 + (42 – 37.2)2)
       = √(1/4) x ((-2.2)2 + (-5.2)2 + (0.8)2 + (1.8)2 + (4.8)2)
       = √(0.25) x ((4.84) + (27.04) + (0.64) + (3.24) + (23.04))
       = √(0.25) x 58.8
       = √14.7
       = 3.8341

Skewness = ∑(y_i – y_mean) / (n – 1) x (sd)³

Skewness =((35 – 37.2)³ + (32 – 37.2)³ + (38 – 37.2)³ + (39 – 37.2)³ + (42 – 37.2)³) / (5 – 1)³ x 3.8341

Skewness = ((-2.2)³ + (-5.2)³ + (0.8)³ + (1.8)³ + (4.8)³ )/ (4)³ x 3.8341

Skewness =((-10.648) + (-140.608) + (0.512) + (5.832) + (110.592)) / 64 x 3.8341

Skewness =-34.32 / 245.3824

Skewness = -0.1522

So, the skewness of these data is negative.

FAQs on Skewness

Q1: Can skewness be zero?

Answer:

Yes, skewness can be zero. This occurs when the distribution is perfectly symmetrical. It is possible if there are an equal number of values on the left and right sides of the mean.

Q2: What does positive skewness mean?

Answer:

Positive skewness means the distribution is skewed to the right. There are more values on the right side of the mean than on the left side. 

Condition for Positive Skewness = Mean > median >mode

Q3: What does negative skewness mean?

Answer:

Negative skewness means the distribution is skewed to the left. There are more values on the left side of the mean than on the right side. 

Condition for negative skewness = Mode > median > mean

Q4: How is skewness used in finance and investment analysis?

Answer:

In finance and investment analysis, skewness is used to measure the degree of asymmetry in returns on investment. Skewed returns can have an impact on portfolio management and risk management strategies, and understanding the skewness of a particular investment can help investors to make better-informed decisions.

Q5: What is the formula for calculating skewness?

Answer:

The formula used to calculate Skewness is,

Skewness Formula= 3(Mean – Median)/S.D.

Q6: How is skewness used in hypothesis testing?

Answer:

Skewness is used in hypothesis testing to determine whether a sample of data is normally distributed or not. If the skewness value is close to zero, the data is likely to be normally distributed. If the skewness value is positive or negative, the data is likely to be skewed and may require non-parametric tests for hypothesis testing.