Variance and standard deviation for ungrouped data
Let $x_i, i=1,2, \cdots , n$ be $n$ observations.
Formula
Sample variance of $X$ is denoted by $s_{x}^2$ and is given by
$s_x^2 =\dfrac{1}{n-1}\sum_{i=1}^{n}(x_i -\overline{x})^2=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\dfrac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)$
where,
$\overline{x}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$is the sample mean
The sample standard deviation of $X$ is defined as the positive square root of the sample variance. The sample standard deviation of $X$ is given by
$s_x =\sqrt{s_x^2}$
Example 1
The age (in years) of 6 randomly selected students from a class are
22,25,24,23,24,20.
Find the variance and standard deviation.
Solution
| $x_i$ | $x_i^2$ | |
|---|---|---|
| 22 | 484 | |
| 25 | 625 | |
| 24 | 576 | |
| 23 | 529 | |
| 24 | 576 | |
| 20 | 400 | |
| Total | 138 | 3190 |
Sample mean
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{138}{6}\\ &=23\text{ years} \end{aligned} $$
The average of age of students is $23$ years.
Sample variance
Sample variance of $X$ is
$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{5}\bigg(3190-\frac{(138)^2}{6}\bigg)\\ &=\dfrac{1}{5}\big(3190-\frac{19044}{6}\big)\\ &=\dfrac{1}{5}\big(3190-3174\big)\\ &= \frac{16}{5}\\ &=3.2 \end{aligned} $$
Sample standard deviation
The standard deviation is the positive square root of the variance.
The sample standard deviation is
$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{3.2}\\ &=1.7889 \text{ years} \end{aligned} $$
Thus the standard deviation of age of students is $1.7889$ years.
Example 2
The following data gives the hourly wage rates (in dollars) of 10 employees of a company.
20,21,24,25,18,22,24,22,20,22.
Find the variance and standard deviation of hourly wage rates.
Solution
| $x_i$ | $x_i^2$ | |
|---|---|---|
| 20 | 400 | |
| 21 | 441 | |
| 24 | 576 | |
| 25 | 625 | |
| 18 | 324 | |
| 22 | 484 | |
| 24 | 576 | |
| 22 | 484 | |
| 20 | 400 | |
| 22 | 484 | |
| Total | 218 | 4794 |
Sample mean
The sample mean of $X$ is
$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{218}{10}\\ &=21.8\text{ dollars} \end{aligned} $$
The average of hourly wage rates is $21.8$ dollars.
Sample variance
Sample variance of $X$ is
$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{9}\bigg(4794-\frac{(218)^2}{10}\bigg)\\ &=\dfrac{1}{9}\big(4794-\frac{47524}{10}\big)\\ &=\dfrac{1}{9}\big(4794-4752.4\big)\\ &= \frac{41.6}{9}\\ &=4.6222 \end{aligned} $$
Sample standard deviation
The standard deviation is the positive square root of the variance.
The sample standard deviation is
$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{4.6222}\\ &=2.1499 \text{ dollars} \end{aligned} $$
Thus the standard deviation of hourly wage rates is $2.1499$ dollars.
