Variance and Standard Deviation of Ungrouped Data

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