Question

In Exercises $3.65-3,67$, 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

$1,2,4,4$

Short answer

$$\begin{gathered} a)Range=3 \\ b)samples\tan darddeviation=1.5 \end{gathered}$$

Step-by-step solution

Step 1. Given information. 

We have given data : 1,2,4,4

Step 2. Range of the data.   

$$\begin{gathered} Range=Max.value-Min.value \\ Range=4-1 \\ Range=3 \end{gathered}$$

Step 3. Mean  of the data.   

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \sum _{}\{x-\overline{x}{)}^2}}{n-1}} \\ Mean,\overline{x}=\frac{{\displaystyle \sum _{}}x}{n} \\ Mean=\frac{1+2+4+4}{4} \\ Mean=\frac{11}{4} \\ Mean=2.75 \\ sampiesize,n=4 \end{gathered}$$

Step 4. Standard deviation of the data. 

Variable $x$	Deviation from mean $x-\overline{x}$	Squared deviation role="math" localid="1651598578463" ${(x-\overline{x})}^2$
$1$	$-1.75$	$3.06$
$2$	$-0.75$	$0.56$
$4$	$1.25$	$1.56$
$4$	$1.25$	$1.56$

Thus, the standard deviation is

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \sum _{}}{(x-\overline{x})}^2}{n-1}} \\ =\sqrt{\frac{{\displaystyle 6.75}}{4-1}} \\ =\sqrt{2.25} \\ =1.5 \end{gathered}$$