Chapter 1: Q 3.2. (page 64)

The heights (in inches) of the five starters on a basketball team are $67, 72, 76, 76, and 84 .$
Make a table that shows, for each value, its deviation from the mean and its squared deviation from the mean.

Short Answer

Expert verified

The deviation from the calculated mean is $5.59$ and the square deviation of the mean is $31.24$ .

Step by step solution

Given information

Given data :

The heights five starters are : $67, 72, 76, 76, and 84 .$

Calculation

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

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

The calculated mean is :

localid="1649845044603" $\begin{array}{rcl} x & = & \frac{62 + 72 + 76 + 76 + 84}{5} \\ = & \frac{375}{5} \\ = & 75 \end{array}$

Hence the standard deviation from the calculated mean is;

localid="1649845109343" $\begin{array}{rcl} σ & = & \frac{\sqrt{{(x_{i} - μ)}^{2}}}{N} \\ = & \sqrt{\frac{{(67 - 75)}^{2} + {(72 - 75)}^{2} + {(76 - 75)}^{2} + {(76 - 75)}^{2} + {(84 - 75)}^{2}}{5}} \\ = & \sqrt{31.20} \\ = & 5.59 \end{array}$

Hence, the deviation from the calculated mean is $5.59$ and square of deviation is $31.20$

Concept

By calculating each data point's divergence from the mean, the standard deviation is determined as the square root of variance: $σ = \sqrt{\frac{Σ {(x i - μ)}^{2}}{N}}$