Chapter 10.5, Problem 11PPS

To determine

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

Answer to Problem 11PPS

Mean is 36.8.

Median is 38.

Mode is 12.

Standard deviation is 19.98

Range is 56

Explanation of Solution

Given information:

Frequency: 11, 7, 3, 13, 16, 8, 3, 11, 17, 3

Constant to be multiplied: 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,

Multiply 4 to all the frequency values,

New frequency:

44,28,12,52,64,32,12,44,68,12

Sorting the frequency

12,12,12,28,32,44,44,52,64,68

For Mean,

Substituting the values in mean formula,

  $=\frac{44+28+12+52+64+32+12+44+68+12}{10}$

On solving,

  $=36.8$

Hence, Mean is 36.8.

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,

  $=\frac{32+44}{2}$

On solving,

  $=38$

Hence, Median is 38.

For mode,

Mode is the item with the maximum frequency, which is 12.

For standard deviation,

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(44-36.8)}^2+{(28-36.8)}^2+{(12-36.8)}^2+{(52-36.8)}^2+{(64-36.8)}^2$$ +{(32-36.8)}^2+{(12-36.8)}^2+{(44-36.8)}^2+{(68-36.8)}^2+{(12-36.8)}^2$\end{array}}{10}}$

On solving,

  $SD=\sqrt{\frac{3993.6}{10}}$

So,

  $SD=\sqrt{399.36}$

On solving,

  $SD=19.98$

Hence, Standard deviation is 19.98

For Range,

Substituting in the range formula,

  $range=68-12$

On solving,

  $range=56$