Example Page (Standard Deviation)
Here are a few examples of sample and population standard deviation to learn how to calculate them manually.
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
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
