Chapter 10, Problem 11PT

To determine

To calculate mean, median, mode, range and standard deviation

Answer to Problem 11PT

Range is 16, mean is 26.7, median is 26.5, mode is 25 and 30, standard deviation is 4.80

Explanation of Solution

Given information:

The data set is,

19,22,10,17,26,24,12,22,18,17

The constant to be added is 8

Formula used:

Mean is,

  $\overline{x}=\frac{{\displaystyle \sum x}}{n}$

Median is,

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

Mode is,

  $z=Most\mathrm{Re}peatedValue$

Range is,

  $range=\max -\min$

Standard deviation,

  $sd=\sqrt{\frac{{\displaystyle \sum {(x-\overline{x})}^2}}{n}}$

Calculation:

Re-arranging the data,

10,12,17,17,18,19,22,22,24,26

Adding the constant,

18,20,25,25,26,27,30,30,32,34

The maximum is 34

The minimum is 18

For range, substituting the value,

  $range=34-18$

On solving,

  $range=16$

Hence, range is 16.

For mean, substituting the value,

  $\overline{x}=\frac{18+20+25+25+26+27+30+30+32+34}{10}$

So,

  $\overline{x}=\frac{267}{10}$

On solving,

  $\overline{x}=26.7$

Hence, mean is 26.7

For median, substituting the values,

  $M=\frac{{(\frac{10}{2})}^{th}+{(\frac{10}{2}+1)}^{th}}{2}$

On solving,

  $M=\frac{5^{th}+6^{th}}{2}$

Substituting the values,

  $M=\frac{26+27}{2}$

On solving,

  $M=26.5$

Hence, median is 26.5

For mode, substituting the value,

  $z=25$ and

  $z=30$

Hence, mode is 25 and 30

For standard deviation, substituting the value,

  $sd=\sqrt{\frac{\begin{array}{}$ {(18-26.7)}^2+{(20-26.7)}^2+{(25-26.7)}^2+{(25-26.7)}^2+{(26-26.7)}^2+{(27-26.7)}^2+{(30-26.7)}^2+$$ {(30-26.7)}^2+{(32-26.7)}^2+{(34-26.7)}^2$\end{array}}{10}}$ so,

  $sd=\sqrt{\frac{\text{75}\text{.69+44}\text{.89+2}\text{.89+2}\text{.89+0}\text{.49+0}\text{.09+10}\text{.89+10}\text{.89+28}\text{.09+53}\text{.29}}{10}}$

On solving,

  $sd=\sqrt{\frac{230.1}{10}}$

On solving,

  $sd=4.80$

Hence, standard deviation is 4.80