Standard Deviation Examples

Example 1:

Let us consider a data sample : 10,13,7,9,6

Solution:

We can calculate the mean, variance and standard deviation of the given sample data using the given formula.

Formula :

Mean :

Mean = Sum of X values / N(Number of Values)

Variance :

Variance = s²

Standard Deviation :

Substituting the values in the formula,

Number of Values, n = 5

Mean, M = (10 + 13 + 7 + 9 + 6) / 5 = 45 / 5 = 9 is the mean of the data set.

Standard Deviation, s = √∑(X-9)² / (5 - 1) = √[(10 - 9)² + (13 - 9)² + (7 - 9)² + (9 - 9)² + (6 - 9)²] / 4 = √[1² + 4² + (-2)² + 0 + -3²] / 4 = √[1 + 16 + 4 + 9] / 4 = √[30 / 4] = √7.5 = 2.73861 Hence, the value of Sample Standard deviation is 2.73861.

Variance of the Sample = 2.73861² = 7.5 Hence, the value of Sample Variance is 7.5

Example 2:

Let us consider a population : 20,12,9,5,1

Solution:

We can calculate the mean, variance and standard deviation of the given population using the formula.

Population Standard Deviation :

Formula :

Substituting the values in the formula,

Number of Values, n = 5

Mean, M = (20 + 12 + 9 + 5 + 1) / 5 = 47 / 5 = 9.4 is the mean of the population.

Standard Deviation, s = √∑(X - 9.4)² / 5
= √[(20 - 9.4)² + (12 - 9.4)² + (9 - 9.4)² + (5 - 9.4)² + (1 - 9.4)²] / 5
= √[10.6² + (2.6)² + (-0.4)² + -4.4² + (-8.4)²] / 5
= √[112.36 + 6.76 +0.16 + (19.36) + 70.56] / 5
= √[209.2 / 5]
= √41.84
= 6.46838
Hence, the value of Population Standard deviation is 6.46838.

Variance of the Population = 6.46838² =41.84 Hence, the value of Sample Variance is 41.84