Standard Deviation

Standard deviation is defined as the positive square root of the variance, denoted $\sigma$ (sigma): $\sigma = \sqrt{\text{Var}(X)}$ It is preferred to variance for interpretation because it carries the same unit as the original data, whereas variance is expressed in squared units. The working formulas take three closely related forms depending on how the data is presented.

1. Raw data: if a variable X takes the values $x_1, x_2, \ldots, x_n$ with arithmetic mean $\bar{x} = \frac{1}{n}\sum x_i$, then $\text{Var}(X) = \sigma^2 = \frac{1}{n}\sum_{i=1}^{n}x_i^2 - \bar{x}^2, \qquad \text{S.D.} = \sigma = \sqrt{\text{Var}(X)}$
2. Ungrouped frequency distribution: if $x_1, x_2, \ldots, x_n$ are the values of X with corresponding frequencies $f_1, f_2, \ldots, f_n$, and $N=\sum f_i$ is the total frequency, then $\text{Var}(X) = \sigma^2 = \frac{1}{N}\sum_{i=1}^{n}f_i(x_i-\bar{x})^2 = \frac{\sum f_ix_i^2}{N} - \bar{x}^2, \quad \text{where } \bar{x} = \frac{\sum f_ix_i}{N}$
3. Grouped frequency distribution: exactly the same formula as (ii) is used, except that $x_1,\ldots,x_n$ now stand for the mid-values of the class intervals and $f_1,\ldots,f_n$ are the corresponding class frequencies. Standard deviation is again $\sigma = \sqrt{\text{Var}(X)}$ in every case. Worked Example (Ex.1) — raw data: For 9, 12, 15, 18, 21, 24, 27, n = 7 and $\bar{x} = 126/7 = 18$. Building the deviations-from-mean table gives squared deviations 81, 36, 9, 0, 9, 36, 81, which total 252. So $\text{Var}(X) = 252/7 = 36$ and $\text{S.D.} = \sigma = \sqrt{36} = 6$. Worked Example (Ex.2) — raw data, alternate method: Marks out of 25 of 5 students are 10, 13, 17, 20, 23. Here n = 5, $\sum x_i = 83$ so $\bar{x}=83/5=16.6$, and $\sum x_i^2 = 100+169+289+400+529=1487$. Then $\text{Var}(X) = 1487/5 - (16.6)^2 = 297.4 - 275.56 = 21.84$, and $\text{S.D.} = \sqrt{21.84} \approx 4.67$. Worked Example (Ex.3) — ungrouped frequency distribution: A die is rolled 30 times, giving scores 1-6 with frequencies 2, 6, 2, 5, 10, 5 (N = 30). Building f.x and f.x² columns gives $\sum f_ix_i = 120$ and $\sum f_ix_i^2 = 554$. So $\bar{x} = 120/30 = 4$, $\text{Var}(X) = 554/30 - 4^2 = 18.47 - 16 = 2.47$, and $\text{S.D.} = \sqrt{2.47} \approx 1.57$. …