Chapter 10.5, Problem 3CYU

To determine

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

Answer to Problem 3CYU

Mean is 17.25.

Median is 28.5.

Mode is 9.

Standard deviation is 9.41

Range is 24

Explanation of Solution

Given information:

Frequency: 6,10,3,7,4,9,3,8,5,11,2,1

Constant to be multiplied: 3

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,

Multiplying 3 to all the frequency values,

New frequency:

18,30,9,21,12,27,9,24,15,33,6,3

Sorting the frequency

3,6,9,9,15,18,21,24,27,30,33

For Mean,

Substituting the values in mean formula,

  $=\frac{18+30+9+21+12+27+9+24+15+33+6+3}{12}$

So,

  $=\frac{207}{12}$

Solving,

  $=17.25$

Hence, Mean is 17.25

For median,

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,

  $=\frac{18+21}{2}$

On solving,

  $=19.5$

Hence, Median is 19.5.

For mode,

Mode is the item with the maximum frequency, which is 9.

For standard deviation,

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(18-17.25)}^2+{(30-17.25)}^2+{(9-17.25)}^2+{(21-17.25)}^2+{(12-17.25)}^2+{(27-17.25)}^2$$ +{(9-17.25)}^2+{(24-17.25)}^2+{(15-17.25)}^2+{(33-17.25)}^2+{(6-17.25)}^2+{(3-17.25)}^2$\end{array}}{12}}$

So,

  $SD=\sqrt{\frac{\begin{array}{}$ {(0.75)}^2+{(12.75)}^2+{(-8.25)}^2+{(3.75)}^2+{(-5.25)}^2+{(9.75)}^2$$ +{(-8.25)}^2+{(6.75)}^2+{(-2.25)}^2+{(15.75)}^2+{(-11.25)}^2+{(-14.25)}^2$\end{array}}{12}}$

Solving,

  $SD=\sqrt{88.68}$

On solving,

  $SD=9.41$

Hence, Standard deviation is 9.41

For Range,

Range is the difference between the lowest and highest values.

  $range=\max -\min$

Substituting the values,

  $range=33-9$

On solving,

  $range=24$