Chapter 10.5, Problem 9PPS

To determine

To calculate mean, median, mode, range and standard deviation after adding the given constant.

Answer to Problem 9PPS

Mean is 22.5.

Median is 21.5.

Mode is 17.

Standard deviation is 7.36

Range is 24

Explanation of Solution

Given information:

Frequency: 27,21,34,42,20,19,18,26,25,33

Constant to be added: -4

Formula used: Mean formula is,

  $\overline{X}=\frac{{\displaystyle \sum fX}}{N}$

Where, f is frequency

N is total number of items

X is item.

Median formula is,

  $median=\frac{{\left(\frac{n}{2}\right)}^{th}term+{\left(\frac{n}{2}+1\right)}^{th}term}{2}$

, where N cumulative frequency

Mode is the item with the maximum frequency.

Standard deviation is,

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

Where, SD= standard deviation,

x= value

  $range=\max -\min$

Calculation:

According to question,

Add -4 to all the frequency values,

New frequency:

23,17,30,38,16,15,14,22,21,29

Sorting the frequency

14,15,16,17,21,22,23,29,30,38

For Mean,

Substituting the values in mean formula,

  $=\frac{23+17+30+38+16+15+14+22+21+29}{10}$

So,

  $=\frac{225}{10}$

On solving,

  $=22.5$

Hence, Mean is 22.5

For median,

Substituting values,

  $=\frac{{\left(\frac{10}{2}\right)}^{th}term+{\left(\frac{10}{2}+1\right)}^{th}term}{2}$

On solving,

  $=\frac{{\left(5\right)}^{th}term+{\left(6\right)}^{th}term}{2}$

Substituting the values,

  $=\frac{21+22}{2}$

On solving,

  $=21.5$

Hence, Median is 21.5.

For mode,

Grouping frequencies into classes of interval 5.

15-19: 4

20-24:3

25-29:1

30-34:1

35-39:1

“15-19” appear most often, so mode is 17

For standard deviation,

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(23-22.5)}^2+{(17-22.5)}^2+{(30-22.5)}^2+{(38-22.5)}^2+{(16-22.5)}^2$$ +{(15-22.5)}^2+{(14-22.5)}^2+{(22-22.5)}^2+{(21-22.5)}^2+{(29-22.5)}^2$\end{array}}{10}}$

So,

  $SD=\sqrt{\frac{\begin{array}{}$ {(0.5)}^2+{(-5.5)}^2+{(7.5)}^2+{(15.5)}^2+{(-6.5)}^2$$ +{(-7.5)}^2+{(-8.5)}^2+{(-0.5)}^2+{(1.5)}^2+{(-6.5)}^2$\end{array}}{10}}$

On solving,

  $SD=\sqrt{60.277}$

On solving,

  $SD=7.76$

Hence, Standard deviation is 7.76

For Range,

Substituting in the range formula,

  $range=38-14$

On solving,

  $range=24$

Hence, range is 24