Examples on Variance and Standard Deviation

Example 1: Find the variance and standard deviation of all the possibilities of rolling a die.

Solution:

All possible outcomes of rolling a die are {1; 2; 3; 4; 5; 6}.

This data set has six values (n) = 6

Before finding the variance, we need to find the mean of the data set.

Mean, x̅ = (1+2+3+4+5+6)/6 = 3.5

We can put the value of data and mean in the formula to get;

σ² = Σ (x_i – x̅)²/n

⇒ σ² = [(1-3.5)² + (2-3.5)² + (3-3.5)² + (4-3.5)² + (5-3.5)² + (6-3.5)²]/6

⇒ σ² = (6.25+2.25+0.25+0.25+2.25+6.25)/6

Variance (σ²) = 2.917

Now,

Standard Deviation (σ) = √ (σ²)

⇒ Standard Deviation (σ)  = √(2.917) 

⇒ Standard Deviation (σ) = 1.708

Example 2: Find the variance and standard deviation of all the even numbers less than 10.

Solution:

Even Numbers less than 10 are {0, 2, 4, 6, 8}

This data set has five values (n) = 5

Before finding the variance, we need to find the mean of the data set.

Mean, x̅ = (0+2+4+6+8)/5 = 4

We can put the value of data and mean in the formula to get;

σ² = Σ (x_i – x̅)²/n

⇒ σ² = [(0-4)² + (2-4)² + (4-4)² + (6-4)² + (8-4)²]/5

⇒ σ² = (16 + 4 + 0 + 4 + 16)/5 = 40/5

Variance (σ²) = 8

Now, Standard Deviation (σ) = √ (σ²)

⇒ Standard Deviation (σ)  = √(8) 

⇒ Standard Deviation (σ) = 2.828

Variance and Standard Deviation are the important measures used in Mathematics and Statics to find the meaning from a large set of data. The different formulas for Variance and Standard Deviation are highly used in mathematics to determine the trends of various values in mathematics. Variance is the measure of how the data points vary according to the mean while standard deviation is the measure of the central tendency of the distribution of the data. 

The major difference between variance and standard deviation is in their units of measurement. Standard deviation is measured in a unit similar to the units of the mean of data, whereas the variance is measured in squared units.

Here in this article, we will learn about variance and standard deviation including their definitions, formulas, and their differences along with suitable examples in detail.