# Example Page (Standard Deviation)

Here are a few examples of sample and population standard deviation to learn how to calculate them manually.

Example 1: For sample standard deviation

Determine the sample standard deviation of 15, 16, 19, 21, 23, 26, 30, 32, 36, 42.

Solution

Step 1: First of all, find the sample mean (x̅) by taking the sum of sample data values and divide it by the total number of observations.

Sum of sample values = 15 + 16 + 19 + 21 + 23 + 26 + 30 + 32 + 36 + 42

= 260

Total number of observation = n = 10

Sample mean of data set = x̅ = 260/10 = 130/5

= x̅ = 26

Step 2: Now find the deviation (difference of each data value from the mean) and take its square to make all the calculations positive.

 Data values (xi) (xi - x̅ ) (xi - x̅ )2 15 15 - 26 = -11 (-11)2 = 121 16 16 - 26 = -10 (-10)2 = 100 19 19 - 26 = -7 (-7)2 = 49 21 21 - 26 = -5 (-5)2 = 25 23 23 - 26 = -3 (-3)2 = 9 26 26 - 26 = 0 (0)2 = 0 30 30 - 26 = 4 (4)2 = 16 32 32 - 26 = 6 (6)2 = 36 36 36 - 26 = 10 (10)2 = 100 42 42 - 26 = 16 (16)2 = 256

Step 3: Now find the summation of the squared deviations.

∑ (xi - x̅)2 = 121 + 100 + 49 + 25 + 9 + 0 + 16 + 36 + 100 + 256

= 712

Step 4: Now divide the summation of the squared deviations by (N – 1) and the result will be the variance of the data set.

∑ (xi - x̅)2/(N-1) = 712 / 10 – 1

= 712 / 9

=  79.11

Step 6: Take the square root of the quotient and the summation of the squared deviations by N – 1 to get the sample standard deviation.

√ [∑ (xi - x̅)2 / (N–1)] = √79.11

=  8.894

Example 2: For population standard deviation

Determine the population standard deviation of 5, 9, 11, 17, 19, 22, 26, 29, 34, 38.

Solution

Step 1: First of all, find the population mean (μ) by taking the sum of population data values and divide it by the total number of observations.

Sum of population values = 5 + 9 + 11 + 17 + 19 + 22 + 26 + 29 + 34 + 38

= 210

Total number of observation = n = 10

Mean of population data set = μ = 210/10 = 105/5

= μ = 21

Step 2: Now find the deviation (difference of each data value from the mean) and take its square to make all the calculations positive.

 Data values (xi) xi - μ (xi - μ)2 5 5 - 21 = -16 (-16)2 = 256 9 9 - 21 = -12 (-12)2 = 144 11 11 - 21 = -10 (-10)2 = 100 17 17 - 21 = -4 (-4)2 = 16 19 19 - 21 = -2 (-2)2 = 4 22 22 - 21 = 1 (1)2 = 1 26 26 - 21 = 5 (5)2 = 25 29 29 - 21 = 8 (8)2 = 64 34 34 - 21 = 13 (13)2 = 169 38 38 - 21 = 17 (17)2 = 289

Step 3:Now find the sum of squared deviations.

∑ (xi - μ)2 = 256 + 144 + 100 + 16 + 4 + 1 + 25 + 64 + 169 + 289

= 1068

Step 4: Now divide the summation of the squared deviations by N.

∑ (xi - μ)2 / N = 1068 / 10

= 534 / 5

= 106.8

Step 6: Take the square root of the quotient and the summation of the squared deviations by N to get the population standard deviation.

√ [∑ (xi - μ)2 / N] = √106.8

= 10.334