# Sample Variance Formula

Before going to learn the sample variance formula, let us recall what is sample variance. In a practical situation, when the population size **N** is large it becomes difficult to obtain value **x _{i}** for every observation in the population and hence it becomes difficult to calculate the variance for the population. In such cases, we can estimate the variance by calculating it on a sample of size

**n**taken from the population of size

**N**. This estimated variance is called the sample variance (

**S**). Let us explore the sample variance formula for a given sample below.

^{2}## What Is the Sample Variance Formula?

The sample variance formula involves the sample size and the mean. Given a sample of data (observations) for the random variable **x**, its sample variance is given by

\(S^2 = \dfrac{1}{n-1} \sum^{n}_{i=1}(x_i - \bar{x})^2\)

where

**S**is the sample variance^{2 }**n**is the sample size**x**is the_{i}**i**value of the random variable^{th}**x**and**x̄**is the sample mean.

Let us see the applications of the sample variance formula in the following section.

## Solved Examples Using the Sample Variance Formula

**Example 1: **There are 45 students in a class. 5 students were randomly selected from this class and their heights (in cm) were recorded as follows:

131 | 148 | 139 | 142 | 152 |

### Calculate the sample mean and the sample variance of their heights (in cm).

**Solution:**

Sample size (n) = 5

Sample Mean = \(\dfrac{131 + 148 + 139 + 142 + 152}{5}\) = \(\dfrac{712}{5}\) = 142.4 cm

Using the sample variance formula,

Sample Variance =** **\(\dfrac{1}{n-1} \sum^{n}_{i=1}(x_i - \bar{x})^2\) = \(\dfrac{1}{5-1} \sum^{5}_{i=1}(x_i - 142.4)^2 \)

= \( \dfrac{(131 - 142.4)^2 + (148 - 142.4)^2 + (139 - 142.4)^2 + (142 - 142.4)^2 + (152 - 142.4)^2}{4} \) = 66.3 cm^{2}

**Answer:** Sample Mean = 142.4 cm, Sample Variance = 66.3 cm^{2}

**Example 2: **There are 8 small-sized trees in a garden. Given below are the heights (in cm) of these trees. (Take approximations wherever necessary)

812 | 836 | 982 | 769 | 920 | 1021 | 942 | 720 |

### a. Find the population standard deviation of their heights.

### b. Three trees were randomly chosen from this population and their heights are listed as follows. Find the sample standard deviation of their heights. (Use √2594 = 51)

1021 | 982 | 920 |

**Solution:**

**a. **Population size (**N**) = 8

Let **X**_{i }be the height of the i^{th} tree.

Population Mean (**X̄**) = \( \dfrac{812+836+982+769+920+1021+942+720}{8} \) = 875.26 cm

Population Variance = \( \dfrac{\sum^{N}_{i=1} (X_i - \bar{X})}{N} \) = \( \dfrac{\sum^{8}_{i=1} (X_i - 875.26 )}{8} \) =

\( \frac{(812 - 875.26 )^2 + (836 - 875.26 )^2 + (982 - 875.26 )^2 + (769 - 875.26 )^2 + (920 - 875.26 )^2 + (1021 - 875.26 )^2 + (942- 875.26 )^2 + (720 - 875.26 )^2}{8} \) = 9999.8 cm^{2} ≈ 10000 cm^{2}

Population Standard Deviation = \(\sqrt{10000}\) = 100 cm

**b. **Sample size (**n**) = 3

Let **x**_{i }be the height of the i^{th} tree.

Sample Mean (**x̄**) = \( \dfrac{1021+982+920}{3} \) = 974.3 cm

Using the sample variance formula,

Sample Variance = \( \dfrac{\sum^{n}_{i=1} (x_i - \bar{x})}{n-1} \) = \( \dfrac{\sum^{3}_{i=1} (x_i - 974.3)}{3-1} \) = \( \dfrac{(1021 - 974.3)^2 + (982 - 974.3)^2 + (920 - 974.3)^2 + }{2} \) = 2594.3cm^{2}

Sample Standard Deviation = \(\sqrt{2594.3}\) = 51 cm

**Answer:** (a) The population standard deviation of the height of the trees is approximately 100 cm. (b) The sample standard deviation of the height of the trees is approximately 51 cm.

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