Question

In Exercises $3.65-3.70$, we have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range

b) sample standard deviation

4,2,0,2,2

Short answer

$$\begin{gathered} a)Range=4 \\ b)samples\tan darddeviation=1.41 \end{gathered}$$

Step-by-step solution

Step 1. Given information. 

We have given data :$4,2,0,2,2$

Step 2. Range of the data.  

$$\begin{gathered} Range=Max.value-Min.value \\ Range=4-0 \\ Range=4 \end{gathered}$$

Step 3. Mean  of the data.    

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \sum _{}\{x-\overline{x}{)}^2}}{n-1}} \\ Mean,\overline{x}=\frac{{\displaystyle \sum _{}x}}{n} \\ Mean=\frac{4+2+0+2+2}{5} \\ Mean=\frac{10}{55} \\ Mean=2 \\ sampiesize,n=5 \end{gathered}$$

Step 4. Standard deviation of the data.  

Variable $x$	Deviation from mean $\left(x-\overline{x}\right)$	Squared deviation ${\left(x-\overline{x}\right)}^2$
$$\begin{gathered} 4 \\ 2 \\ 0 \\ 2 \\ 2 \end{gathered}$$	$$\begin{gathered} 2 \\ 0 \\ -2 \\ 0 \\ 0 \end{gathered}$$	$$\begin{gathered} 4 \\ 0 \\ 4 \\ 0 \\ 0 \end{gathered}$$

Thus, the standard deviation is

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \sum _{}}{\left(x-\overline{x}\right)}^2}{n-1}} \\ =\sqrt{\frac{8}{5-1}} \\ =\sqrt{2} \\ =1.4142 \end{gathered}$$