Question

determine the standard deviation ($\sigma$) by filling in the table as part of your calculation. consider the following data 6, 6, 10, 8, 10, 8

$x$	$\bar{x}$	$x - \bar{x}$	$(x - \bar{x})^2$

a. 1.63
b. 0.47
c. 0.94
d. 1.15

Explanation:

Step1: Calculate the mean

The data is 6, 6, 10, 8, 10, 8. The sum of the data is $6 + 6+10 + 8+10 + 8=48$. There are $n = 6$ data - points. The mean $\bar{x}=\frac{48}{6}=8$.

Step2: Calculate $x-\bar{x}$ and $(x - \bar{x})^2$ for each data - point

$x$	$\bar{x}$	$x-\bar{x}$	$(x - \bar{x})^2$
6	8	$6 - 8=-2$	$(-2)^2 = 4$
10	8	$10 - 8 = 2$	$2^2=4$
8	8	$8 - 8 = 0$	$0^2 = 0$
10	8	$10 - 8 = 2$	$2^2=4$
8	8	$8 - 8 = 0$	$0^2 = 0$

Step3: Calculate the variance

The variance $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}$. Here, $\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=4 + 4+4 + 0+4 + 0 = 16$. So, $\sigma^{2}=\frac{16}{6}=\frac{8}{3}\approx2.67$.

Step4: Calculate the standard deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}=\sqrt{\frac{8}{3}}\approx1.63$.

Answer:

a. 1.63