How to Find Coefficient of Skewness (With Examples)

The Coefficient of Skewness can be calculated using the formula,

Coefficient of Skewness = (X̄ – Mode)/σ.

• Here, X̄ is the mean of the data values.
• σ is the standard deviation.

Sometimes, a given set of data values can have more than one possible value for the mode. This is explained in the second example below.

We cannot use the above formula in such cases because there is no unique value of the mode that we can choose. In such cases, we use the median to calculate the coefficient of skewness.

We can calculate the Coefficient of Skewness using the median by the formula,

Coefficient of Skewness = 3*(X̄ – Median)/σ.

We now explain how to use both of the above formulae by looking at some examples.

Example 1: Using the Mode

Consider the following set of data values:

1, 2, 2, 2, 6, 8, 7.

Step 1: Find the Mode of the data values.

Since the value 2 occurs the largest number of times we have that,

Mode = 2.

Step 2: Calculate the mean of the data values.

Mean X̄ = ΣX_i/N = (1 + 2 + 2 + 2 + 6 + 8 + 7)/7 = 28/7 = 4.

Step 3: We then find the standard deviation of the data values by taking the square root of the variance.

Variance = Σ(X_i–X̄)²/N

= [(1-4)² + (2-4)² + (2-4)² + (2-4)² + (6-4)² + (8-4)² + (7-4)²]/7

= [9 + 4 + 4 + 4 + 4 + 16 + 9]/7 = 50/7 = 7.1429.

We now take the square root to obtain the standard deviation.

Standard deviation σ = √Variance = √7.1429 = 2.6726.

Step 4: We now calculate the coefficient of skewness using the formula,

Coefficient of Skewness = (X̄ – Mode)/σ = (4-2)/2.6726 = 0.7483.

Example 2: Using the Median

Consider the following set of data values:

2, 2, 3, 4, 4, 5, 15.

Notice that here both data values 2 and 4 occur with the same frequency, that is, both occur twice. Thus we have two values for the mode and therefore we cannot use the formula for the coefficient of skewness involving the mode.

We, therefore, calculate the coefficient using the median.

Step 1: We first find the median of the data values. Note that the data is already arranged in ascending order.

Median = [(N+1)/2]th term = [(7+1)/2]th term = 4th term = 4.

Step 2: We then find the arithmetic mean for the data values.

Mean X̄ = ΣX_i/N = (2 + 2 + 3 + 4 + 4 + 5 + 15)/7 = 35/7 = 5.

Step 3: Calculate the standard deviation σ.

Variance = Σ(X_i–X̄)²/N

= [(2-5)² + (2-5)² + (3-5)² + (4-5)² + (4-5)² + (5-5)² + (15-5)²]/7

= [9 + 9 + 4 + 1 + 1 + 0 + 100]/7 = 124/7 = 17.7143.

Standard deviation σ = √Variance = √17.7143 = 4.2088.

Step 4: Use the formula for the coefficient of skewness involving the median.

Coefficient of Skewness = 3*(X̄ – Median)/σ = 3*(5-4)/4.2088

Coefficient of Skewness = 3/4.2088 = 0.7128.

Interpretation of the Coefficient of Skewness:

The coefficient of skewness allows us to understand the shape of the distribution of the data set.

• If the coefficient of skewness is zero, the data has a symmetric distribution with a peak (mode) at the center. The data is distributed symmetrically on both sides of the mode.
• If the coefficient of skewness is positive, we say that the data is positively skewed. A positively skewed distribution has a peak to the left and a tail to the right side.
• If the coefficient of skewness is positive, we say that the data is negatively skewed. A negatively skewed distribution has a peak to the right and a tail to the left side of the distribution.