Chapter 10.3, Problem 4CYU

To determine

To calculate and compare mean and standard deviation of the two groups.

Answer to Problem 4CYU

Boy’s mean is 6.25 and standard deviation is 4.95

Girl’s mean is 3.56 and standard deviation is 2.06

Explanation of Solution

Given information: Boy’s data is,

5,16,3,8,4,12,2,15,0,1,9,3

Girl’s data is,

2,4,0,3,5,4,6,1,3,8,3,4

Formula used: Mean formula is,

  

Where, f is frequency

N is total number of items

X is item.

Standard deviation is,

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

Where, SD= standard deviation,

x= value

Calculation:

For Boys

Substituting the values in mean formula,

  $\overline{x}=\frac{5+16+3+8+4+12+2+15+0+1+9+3}{12}$

So,

  $\overline{x}=\frac{75}{12}$

On solving,

  $\overline{x}=6.25$

Hence, Mean for boys is 6.25

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(5-6.25)}^2+{(16-6.25)}^2+{(3-6.25)}^2+{(8-6.25)}^2+{(4-6.25)}^2+{(12-6.25)}^2+{(2-6.25)}^2$$ +{(15-6.25)}^2+{(0-6.25)}^2+{(1-6.25)}^2+{(9-6.25)}^2+{(3-6.25)}^2$\end{array}}{12}}$

So,

  $SD=\sqrt{\frac{\begin{array}{}$ {(-1.25)}^2+{(9.25)}^2+{(-3.25)}^2+{(1.75)}^2+{(-2.25)}^2+{(5.75)}^2+{(-4.25)}^2+{(-8.75)}^2+{(-6.25)}^2$$ +{(-5.25)}^2+{(2.75)}^2+{(3.25)}^2$\end{array}}{12}}$

On solving,

  $SD=\sqrt{\frac{294.79}{12}}$

So,

  $SD=\sqrt{24.56}$

On removing the under-root,

  $SD=4.95$

Hence, Standard deviation for boys is 4.95

For Girls,

Substituting the values in mean formula,

  $\overline{x}=\frac{2+4+0+3+5+4+6+1+3+8+3+4}{12}$

So,

  $\overline{x}=\frac{43}{12}$

On solving,

  $\overline{x}=3.56$

Mean for girls is 3.56

Substituting the values in standard deviation formula,

  $SD=\sqrt{\frac{\begin{array}{}$ {(2-3.56)}^2+{(4-3.56)}^2+{(0-3.56)}^2+{(3-3.56)}^2+{(5-3.56)}^2+{(4-3.56)}^2+{(6-3.56)}^2$$ +{(1-3.56)}^2+{(3-3.56)}^2+{(8-3.56)}^2+{(3-3.56)}^2+{(4-3.56)}^2$\end{array}}{12}}$

So,

  $SD=\sqrt{\frac{\begin{array}{}$ {(-1.56)}^2+{(0.44)}^2+{(3.56)}^2+{(-0.56)}^2+{(1.44)}^2+{(0.44)}^2+{(2.44)}^2+{(-2.56)}^2+{(-0.56)}^2$$ +{(4.44)}^2+{(-0.56)}^2+{(0.44)}^2$\end{array}}{12}}$

On solving,

  $SD=\sqrt{\frac{50.92}{12}}$

So,

  $SD=\sqrt{4.24}$

Removing the under-root,

  $SD=2.06$

Hence, Standard deviation for girls is 2.06

Comparing the mean we can see on an average boys do more pull-ups than girls, as mean of boys is greater than that of girls. On comparing standard deviation we can see, the girls have a lower standard deviation that means the points are closer to mean and boys have higher standard deviation that means the points are spread out over large range of values.