Q 3.2. - The Practice of Statistics for AP

Question

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, \mathrm{and} 84.$

Make a table that shows, for each value, its deviation from the mean and its squared deviation from the mean. 

Step-by-Step Solution

Answer

The deviation from the calculated mean is$\mathbf{ }\mathbf{5}\mathbf{.}\mathbf{59}$ and the square deviation of the mean is $\mathbf{31}\mathbf{.}\mathbf{24}$.

Step 1: Given information

Given data : 

The heights five starters are : $67, 72, 76, 76, \mathrm{and} 84.$

Step 2: Calculation

On the basis of provided information, the table is as follows;

$x$	$\left(x-\overline{x}\right)$	${\left(x-\overline{x}\right)}^2$
$67$	$-8$	$64$
$72$	$-3$	$9$
$76$	$1$	$1$
$76$	$1$	$1$
$84$	$9$	$81$
Total= $375$	$0$	$156$

The calculated mean is :

$\begin{array}{rcl}\overline{x}& =& \frac{62+72+76+76+84}{5}\\ & =& \frac{375}{5}\\ & =& \mathbf{75}\end{array}$

Hence the standard deviation from the calculated mean is;

$\begin{array}{rcl}\sigma & =& \frac{\sqrt{{\left(x_i-\mu \right)}^2}}{N}\\ & =& \sqrt{\frac{{\left(67-75\right)}^2+{\left(72-75\right)}^2+{\left(76-75\right)}^2+{\left(76-75\right)}^2+{\left(84-75\right)}^2}{5}}\\ & =& \sqrt{31.20}\\ & =& \mathbf{5}\mathbf{.}\mathbf{59}\end{array}$

Hence, the deviation from the calculated mean is $5.59$ and square of  deviation is $\mathbf{31}\mathbf{.}\mathbf{20}$

Step 3: Concept

By calculating each data point's divergence from the mean, the standard deviation is determined as the square root of variance: $\sigma =\sqrt{\frac{\Sigma {(xi-\mu )}^2}{N}}$