Chapter 10.5, Problem 4CYU

To determine

To find mean median mode, range and standard deviation.

Answer to Problem 4CYU

Mean=20.5

Median=20.75

Mode=21

Range=6.5

Standard deviation=1.83

Explanation of Solution

Given information:

42,39,45,44,37,42,38,37,41,49,42,36

Constant=0.5

Formula used:

Mean= $\frac{{\displaystyle \sum _{i=1}^n{X}_i}}{n}$

Where, n=total number of terms,

  ${\displaystyle \sum _{i=1}^n{X}_i}$ = sum of all terms

Mode= most repeating numbers.

Median= $\frac{{(n+1)}^{th}}{2}observation$

Range= maximum- minimum

Standard deviation, ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$

Where $x_i^{}$ = each value from data

  $\mu$ =mean of data

  $N$ = size of data

Calculation:

After multiplying with constant,

New data is 21,19.5,22.5,22,18.5,21,19,18.5,20.5,24.5,21,18

Substituting the value,

Mean= $\frac{21+19.5+22.5+22+18.5+21+19+18.5+20.5+24.5+21+18}{12}=\frac{246}{12}=20.5$

Sort the number in ascending order,

18,18.5,18.5,19,19.5,20.5,21,21,21,22,22.5,24.5

Median= $\frac{{(\frac{N}{2})}^{th}ob+{(\frac{N}{2}+1)}^{th}ob}{2}=\frac{{(\frac{12}{2})}^{th}+{(\frac{12}{2}+1)}^{th}}{2}=\frac{6^{th}+7^{th}}{2}=$

  $\frac{20.5+21}{2}=20.75$

Mode= 21(because it is most repetitive number)

Range= $\max -\min =24.5-18=6.5$

For standard deviation,

values	Difference= $x_{}^{}-\mu$	( $x_{}^{}-\mu {)}^2$
21	0.5	0.25
19.5	-1	1
22.5	2	4
22	1.5	2.25
18.5	-2	4
21	0.5	0.25
19	-1.5	2.25
18.5	2	4
20.5	0	0
24.5	4	16
21	0.5	0.25
18	-2.5	6.25

  ${\displaystyle \sum _{}^{}{(x_{}^{}-\mu )}^2}$ =40.5

  ${\sigma }_{}=\sqrt{\frac{{\displaystyle \sum _{}^{}{(x_i^{}-\mu )}^2}}{N}}$ = ${\sigma }_{}=\sqrt{\frac{40.5}{12}=}1.83$