Question

we are going to calculate the standard deviation for the following set of sample data. 9 12 1 10 14 1) calculate the mean. $\bar{x}=$ 2) fill in the table below: fill in the differences of each data value from the mean, then the squared differences.

$x$	$x - \bar{x}$	$(x - \bar{x})^2$
12
1
10
14
	$sum(x - \bar{x})^2=$

1. calculate the sample standard deviation (s). $s=sqrt{\frac{sum(x - \bar{x})^2}{n - 1}}=$ (please round your answer to two decimal places)

Explanation:

Step1: Calculate the mean

The formula for the mean $\bar{x}$ of a sample $x_1,x_2,\cdots,x_n$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}$. Here, $n = 5$, $x_1=9,x_2 = 12,x_3=1,x_4 = 10,x_5=14$. So, $\bar{x}=\frac{9 + 12+1+10+14}{5}=\frac{46}{5}=9.2$.

Step2: Fill in the table

For $x = 9$: $x-\bar{x}=9 - 9.2=-0.2$, $(x - \bar{x})^2=(-0.2)^2 = 0.04$.
For $x = 12$: $x-\bar{x}=12 - 9.2 = 2.8$, $(x - \bar{x})^2=(2.8)^2=7.84$.
For $x = 1$: $x-\bar{x}=1 - 9.2=-8.2$, $(x - \bar{x})^2=(-8.2)^2 = 67.24$.
For $x = 10$: $x-\bar{x}=10 - 9.2 = 0.8$, $(x - \bar{x})^2=(0.8)^2=0.64$.
For $x = 14$: $x-\bar{x}=14 - 9.2 = 4.8$, $(x - \bar{x})^2=(4.8)^2=23.04$.
And $\sum(x - \bar{x})^2=0.04 + 7.84+67.24+0.64+23.04=98.8$.

Step3: Calculate the sample - standard deviation

The formula for the sample standard deviation $s$ is $s=\sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}$. Here, $n = 5$, so $s=\sqrt{\frac{98.8}{5 - 1}}=\sqrt{\frac{98.8}{4}}=\sqrt{24.7}\approx4.97$.

Answer:

1. $\bar{x}=9.2$

2.

$x$	$x-\bar{x}$	$(x - \bar{x})^2$
12	2.8	7.84
1	-8.2	67.24
10	0.8	0.64
14	4.8	23.04
	$\sum(x - \bar{x})^2 = 98.8$

1. $s\approx4.97$