Variance and Standard Deviation of Ungrouped Data
Formulas: The variance is \({{\sigma }^{2}}=\frac{1}{n}\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{x}}{{)}^{2}}\). Then ‘σ’ the standard deviation is given by the positive square root of the variance \(\sigma =\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{x}}{{)}^{2}}}\).
Calculation of Variance and Standard Deviation for an Ungrouped Data
1. Find the variance and standard deviation of following data 5, 12, 3, 18, 6, 8, 2, 10.
Solution:
Mean: The mean of the given data is x̄ = ∑ xᵢ/n = (5 + 12 + 3 + 18 + 6 + 8 + 2 + 10)/ 8 = 64/8 = 8
2. Find the Variance
| xᵢ | 5 | 12 | 3 | 18 | 6 | 8 | 2 |
10 |
|
xᵢ – x̄ |
-3 | 4 | -5 | 10 | -2 | 0 | -6 | 2 |
| (xᵢ – x̄)² | 9 | 16 | 25 | 100 | 4 | 0 | 36 |
4 |
Solution: Here ∑ (xᵢ – x̄)² = 194
∴ variance (σ)² = \(\frac{1}{n}\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{x}}{{)}^{2}}\).
= ⅛ x 194
= 24.25
Standard Deviation: Hence standard deviation is \(\sigma =\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}{({{x}_{i}}-\overline{x}}{{)}^{2}}}\).
σ = √24.25 = 4.95