Chapter 3.1, Problem 15E

To determine

To find:

The mean, median, and mode for the given data set.

Answer to Problem 15E

Solution:

The mean age is 36.7 years, the median age is 32 years, and the data is multimodal with modal ages 29 years, 31 years, and 32 years.

Explanation of Solution

Given information:

The given data set is,

Ages of 20 Americans entrepreneurs (in years)

$\begin{array}{l}28394353\\ 35323429\\ 33313231\\ 25223029\\ 41362347\end{array}$

Definition:

Mean is the average of the given data set.

Median is the mid value of ordered data set.

Mode is the most frequently occurred value in the ordered data set.

If all the data values occur only once, then the data set has no mode. If only one value occurs most often, then the data set is unimodal. If exactly two values occur equally often, then the data set is bimodal. If more than two values occur equally often, then the data set is multimodal.

Formula used:

The formula to calculate the mean of set of data is,

$\begin{array}{rcl}\text{Mean}\left(\overline{x}\right)& =& \frac{x_1+x_2+\mathrm{...}+x_n}{n}\\ & =& \frac{{\displaystyle \sum _{i=1}^n}{x}_i}{n}\end{array}$

Where, $x_i$ is the $i^{\text{th}}$ data value and $n$ is the number of data values in the data set.

Calculation:

Substitute $28,39,\mathrm{...},47$ for $x_1,x_2,x_3,\mathrm{...},x_{20}$ respectively in the formula.

Here number of data in the given data set is 20, substitute 20 for $n$ in the formula.

$\begin{array}{rcl}\text{Mean}\left(\overline{x}\right)& =& \frac{{\displaystyle x_1+x_2+...+x_n}}{{\displaystyle n}}\\ & =& \frac{{\displaystyle \begin{array}{l}28+39+43+53+35+32+34+29+33+31+\\ 32+31+25+22+30+29+41+36+23+47\end{array}}}{{\displaystyle 20}}\\ & =& \frac{{\displaystyle 673}}{{\displaystyle 20}}\\ & \approx & 33.7\end{array}$

Mean age is 36.7 years.

Formula used:

The formula to calculate the median of set of ordered data is,

$\text{Median}=\{\begin{array}{l}{\left(\frac{n+1}{2}\right)}^{\text{th}}\text{value},n=\text{odd}\\ \frac{1}{2}\left[{\left(\frac{n}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{n+1}{2}\right)}^{\text{th}}\text{value}\right],n=\text{even}\end{array}$

$n$ is the number of data values in the data set.

Calculation:

Here given set of data is,

$\begin{array}{l}28394353\\ 35323429\\ 33313231\\ 25223029\\ 41362347\end{array}$

Arrange the given data in increasing order.

$\begin{array}{l}22,23,25,28,29,29,30,31,31,32,\\ 32,33,34,35,36,39,41,43,47,53\end{array}$

Number of values in data set is 20, it is even.

Substitute 20 for $n$ in the formula for even number of data set.

$\begin{array}{rll}\text{Median}& =& \frac{1}{2}\left[{\left(\frac{n}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{value}\right]\\ & =& \frac{1}{2}\left[{\left(\frac{20}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{20}{2}+1\right)}^{\text{th}}\text{value}\right]\\ & =& \frac{1}{2}\left[{10}^{\text{th}}\text{value}+{11}^{\text{th}}\text{value}\right]\\ & =& \frac{1}{2}\left[32+32\right]\end{array}$

Simplify further,

$\begin{array}{rll}\text{Median}& =& \frac{64}{2}\\ & =& 32\end{array}$

The median age is 32 years.

The most frequently occurred data is the mode.

Here, each of the data 29, 31, 32 occur 2 times.

Modes of the given data set are 29 years, 31 years, and 32 years.

3 data occur most often, so the data set is multimodal.