Chapter 10.5, Problem 2CYU

To determine

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

Answer to Problem 2CYU

Mean is 60.75.

Median is 60.5.

Mode is 65.

Standard deviation is 5.26.

Range is 16

Explanation of Solution

Given information:

Frequency: 38,36,37,42,31,44,37,45,29,42,30,42

Constant to be added: +23

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

The Range is the difference between the lowest and highest values.

Calculation:

According to question,

Add 23 to all the frequency values,

New frequency:

61,59,60,65,54,67,60,68,52,65,53,65

For Mean,

Substituting the values in mean formula,

  $=\frac{61+59+60+65+54+67+60+68+52+65+53+65}{12}$

So,

  $=\frac{729}{12}$

On solving,

  $=60.75$

Hence, Mean is 60.75

For median,

Sorting the frequency

52,53,54,59,60,60,61,65,65,65,67,68

Substituting values,

  $=\frac{{\left(\frac{12}{2}\right)}^{th}term+{\left(\frac{12}{2}+1\right)}^{th}term}{2}$

On solving,

  $=\frac{{\left(6\right)}^{th}term+{\left(7\right)}^{th}term}{2}$

Substituting the values,

  $=\frac{60+61}{2}$

On solving,

  $=60.5$

Hence, Median is 60.5.

For mode,

Mode is the item with the maximum frequency. Substituting the values,

  $\mathrm{mod}e=6$

For standard deviation,

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(61-60.75)}^2+{(59-60.75)}^2+{(60-60.75)}^2+{(65-60.75)}^2+{(54-60.75)}^2+{(67-60.75)}^2$$ +{(60-60.75)}^2+{(68-60.75)}^2+{(52-60.75)}^2+{(65-60.75)}^2+{(53-60.75)}^2+{(65-60.75)}^2$\end{array}}{12}}$

So,

  $SD=\sqrt{\frac{\begin{array}{}$ {(0.25)}^2+{(-1.75)}^2+{(-0.75)}^2+{(4.25)}^2+{(-6.75)}^2+{(6.25)}^2$$ +{(-0.75)}^2+{(7.25)}^2+{(-8.75)}^2+{(4.25)}^2+{(-7.75)}^2+{(4.25)}^2$\end{array}}{12}}$

On solving,

  $SD=\sqrt{27.68}$

On solving,

  $SD=5.26$

Hence, Standard deviation is 5.26

For Range,

Range is the difference between the lowest and highest values.

  $range=\max -\min$

Substituting the values,

  $range=52-68$

On solving,

  $range=16$