Chapter 7.1, Problem 37E

(a)

To determine

To find:

The mean, median, mode, range, and standard deviation of the heights.

Answer to Problem 37E

Mean $=122$ , median $=109.2$ , mode $=98$ , range $=116$ , and standard deviation $=36$

Explanation of Solution

Given:

The table

  $x,\overline{x},x-\overline{x},{\left(x-\overline{x}\right)}^2$

Concept used:

The mean of the squared deviation or variance

  $=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}$

The square root of the variance

  $\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}$

Calculation:

Consider the data

  $60.5,57,59.2,62,64,58.5,57,58,55.5,56$

The mean is

  $\begin{array}{l}\overline{x}=\frac{60.5+57+59.2+62+64+58.5+57+58+55.5+56}{10}=\frac{588}{10}\\ \overline{x}=58.8\end{array}$

Median

  $\begin{array}{l}55.5,56,57,57,58,58.5,59.5,60.5,62,64\\ =\frac{58+58.5}{2}\\ =58.25\end{array}$

  $\begin{array}{l}=\frac{100+119}{2}\\ =109.5\end{array}$

Mode

  $\begin{array}{l}55.5,56,57,57,58,58.5,59.5,60.5,62,64\\ \to 57\end{array}$

Range:-difference of largest and smallest value

  $\begin{array}{l}=64-55.5\\ =8.5\end{array}$

The mean of the squared deviation or variance

  $\begin{array}{l}=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}\\ =\frac{2.89+3.24+\mathrm{...........}+7.84}{10}\\ =6.66\end{array}$

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
$60.5$	$58$	$1.7$	$2.89$
$57$	$58$	$-1.8$	$3.24$
$59.5$	$58$	$0.7$	$0.49$
$62$	$58$	$3.2$	$10.24$
$64$	$58$	$5.2$	$27.04$
$58.5$	$58$	$-0.3$	$0.09$
$57$	$58$	$-1.8$	$3.24$
$58$	$58$	$-0.8$	$0.64$
$55.5$	$58$	$-3.3$	$10.89$
$56$	$58$	$-2.8$	$7.84$

The square root of the variance

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}\\ \sigma =\sqrt{\frac{2.89+3.24+\mathrm{...........}+7.84}{10}}\\ \sigma =\sqrt{6.66}\\ \sigma \approx 2.58\end{array}$

(b)

To determine

To varify:

A new student is $7$ feet tall joins in this class.

Explanation of Solution

Given:

The table

  $x,\overline{x},x-\overline{x},{\left(x-\overline{x}\right)}^2$

Concept used:

The mean of the squared deviation or variance

  $=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}$

The square root of the variance

  $\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}$

Calculation:

A height of $7$ feet is equal to $84$ inches an outlier

So, all measures will increase exacpt for mode stays the same.

Consider the data

  $60.5,57,59.2,62,64,58.5,57,58,55.5,56,84$

The mean is

  $\begin{array}{l}\overline{x}=\frac{60.5+57+59.2+62+64+58.5+57+58+55.5+56+84}{11}=\frac{672}{11}\\ \overline{x}=61.1\end{array}$

Median

  $\begin{array}{l}55.5,56,57,57,58,58.5,59.5,60.5,62,64,84\\ =58.5\end{array}$

  $\begin{array}{l}=\frac{100+119}{2}\\ =109.5\end{array}$

Mode

  $\begin{array}{l}55.5,56,57,57,58,58.5,59.5,60.5,62,64,84\\ \to 57\end{array}$

Range:-difference of largest and smallest value

  $\begin{array}{l}=84-55.5\\ =28.5\end{array}$

The mean of the squared deviation or variance

  $\begin{array}{l}=\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}\\ =\frac{0.36+16.81+\mathrm{...........}+524.41}{11}\\ =\frac{643.91}{11}\end{array}$

$x$	$\overline{x}$	$x-\overline{x}$	${\left(x-\overline{x}\right)}^2$
$60.5$	$61.1$	$-0.6$	$0.36$
$57$	$61.1$	$-4.1$	$16.81$
$59.5$	$61.1$	$-1.6$	$2.56$
$62$	$61.1$	$0.9$	$0.81$
$64$	$61.1$	$2.9$	$8.41$
$58.5$	$61.1$	$-2.6$	$6.76$
$57$	$61.1$	$-4.1$	$16.81$
$58$	$61.1$	$-3.1$	$9.61$
$55.5$	$61.1$	$-5.6$	$31.36$
$56$	$61.1$	$-5.1$	$26.01$
$84$	$61.1$	$22.9$	$524.41$

The square root of the variance

  $\begin{array}{l}\sigma =\sqrt{\frac{{\left(x_1-\overline{x}\right)}^2+{\left(x_2-\overline{x}\right)}^2+\mathrm{.............}+{\left(x_n-\overline{x}\right)}^2}{n}}\\ \sigma =\sqrt{\frac{0.36+16.81+\mathrm{...........}+524.41}{11}}\\ \sigma \approx 7.65\end{array}$

As verified, all measures increased except for mode.