Variance Calculator

Calculating Sample Variance: A Step-by-Step Example

Consider the following data set representing the number of books read by a group of seven students during the summer: 4, 7, 10, 13, 16, 19, 22.

Step 1: Calculate the Mean

The mean, \( \bar{x} \), is the sum of all values divided by the number of values. In this case, we have:

$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i = \frac{4 + 7 + 10 + 13 + 16 + 19 + 22}{7} = \frac{91}{7} = 13 $$

Therefore, the sample mean of the number of books read by this group of students is \( \bar{x} = 13 \).

Step 2: Calculate the Sample Variance

The sample variance, \( s^2 \), represents the average squared deviation of each data point from the sample mean and is calculated as follows:

$$ s^2 = \frac{1}{n - 1} \sum_{i=1}^{n}(x_i - \bar{x})^2 = \frac{1}{n - 1} \left[(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \cdots + (x_n - \bar{x})^2\right] $$

$$ = \frac{1}{7-1} \left[(4 - 13)^2 + (7 - 13)^2 + \cdots + (19 - 13)^2 + (22 - 13)^2\right] $$

$$ = \frac{1}{6} \left[81 + 36 + \cdots + 36 + 81\right] = \frac{252}{6} = 42 $$

Therefore, the sample variance of the number of books read by this group of students is \( s^2 = 42 \). This value indicates that, on average, the data points deviate by about 42 units from the sample mean (in squared units).