Solved Examples of Standard Deviation

Example 1: Calculate the population standard deviation for the following data set: {6, 8, 10, 12, 14}

Solution:

Mean = (6 + 8 + 10 + 12 + 14) / 5 = 50 / 5 = 10

Now, calculate the squared deviations from the mean,

• (6 – 10)² = 16
• (8 – 10)² = 4
• (10 – 10)² = 0
• (12 – 10)² = 4
• (14 – 10)² = 16

Thus, Σ(xi – μ)² = 16 + 4 + 0 + 4 + 16 = 40

Divide the sum of squared differences by N – 1, and take square root to find standard deviation,

σ = √(40/5) = √8 ≈ 2.83

So, the population standard deviation of the data set is approximately 2.83.

Example 2: Calculate the sample standard deviation for the following data set: {12, 15, 18, 21, 24}

Solution:

Mean = (12 + 15 + 18 + 21 + 24) / 5 = 90 / 5 = 18

Now, calculate the squared deviations from the mean,

• (12 – 18)² = 36
• (15 – 18)² = 9
• (18 – 18)² = 0
• (21 – 18)² = 9
• (24 – 18)² = 36

Thus, Σ(xi – μ)² = 36 + 9 + 0 + 9 + 36 = 90

Divide the sum of squared differences by N – 1, and take square root to find standard deviation,

s = √(90/4) = √22.5 ≈ 4.74

Therefore, the sample standard deviation of the data set is approximately 4.74.

Standard Deviation is a measure of how data is spread out around the mean. It is a statistical tool used to determine the amount of variation or dispersion of a set of values from the mean. A low standard deviation indicates that the data points are clustered closely around the mean, while a high standard deviation means that the data points are spread across a wide range of values.

In this article, we will discuss how to calculate Standard Deviation using a formula.