Question

What is the SAMPLE STANDARD DEVIATION (use your calculator) of this data set?

4, 6, 11, 6, 8, 7

(Round to the 2nd decimal place: Example 7.32

Answer 1

The sample size is  $n=6.$ The provided sample data along with the data required to compute the sample mean $\bar{x}$ and sample variance $s^2$ are shown in the table below:

$$\begin{matrix} & x & x^2 \\ & 4 & 16 \\ & 6 & 36 \\ & 6 & 36 \\ & 7 & 49 \\ & 8 & 4 \\ & 11 & 121 \\ Sum= & 42 & 322 \end{matrix}$$

The sample mean $\bar{x}$ is computed as follows:

$$\bar{x}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nx_i=\dfrac{42}{6}=7$$

The sample variance $s^2$ is

$$s^2=\dfrac{1}{n-1}\bigg(\displaystyle\sum_{i=1}^nx_i^2-\dfrac{1}{n}\big(\displaystyle\sum_{i=1}^nx_i\big)^2\bigg)=$$

$$=\dfrac{1}{6-1}\big(322-\dfrac{1}{6}(42)^2\big)=5.6$$

Therefore, the sample sandartd deviation $s$ is

$$s=\sqrt{s^2}=\sqrt{5.6}\approx2.37$$