Chapter 3.1, Problem 1E

To determine

To find:

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

Answer to Problem 1E

Solution:

The mean of the given data is 5, the median is 4.5, and the data is unimodal with mode 1.

Explanation of Solution

Given information:

The given data set is,

$10,1,5,9,4,1$

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 $i^{th}$ data value and $n$ is the number of data values in the data set.

Calculation:

Substitute $10,1,5,9,4,1$ for $x_1,x_2,x_3,x_4,x_5,x_6$ respectively in the formula.

Here number of data in the given data set is 6, substitute 6 for $n$ in the formula.

$\begin{array}{rll}\text{Mean}\left(\overline{x}\right)& =& \frac{x_1+x_2+\mathrm{...}+x_n}{n}\\ & =& \frac{10+1+5+9+4+1}{6}\\ & =& \frac{30}{6}\\ & =& 5\end{array}$

Mean of the given data set is 5.

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,

$10,1,5,9,4,1$.

Arrange the given data in increasing order.

$1,1,4,5,9,10$.

Number of values in data set is 6, it is even.

Substitute 6 for $n$ in the formula for even number of data set.

$\begin{array}{rll}\text{Median}& =& \frac{1}{2}\left[{\left(\frac{n}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{value}\right]\\ & =& \frac{1}{2}\left[{\left(\frac{6}{2}\right)}^{\text{th}}\text{value}+{\left(\frac{6}{2}+1\right)}^{\text{th}}\text{value}\right]\\ & =& \frac{1}{2}\left[3^{\text{rd}}\text{value}+4^{\text{th}}\text{value}\right]\\ & =& \frac{1}{2}\left[4+5\right]\end{array}$

Simplify further,

$\begin{array}{rll}\text{Median}& =& \frac{9}{2}\\ & =& 4.5\end{array}$

The median of the given data set is 4.5.

The most frequently occurred data is the mode.

Here, 1 occurs 2 times.

Mode of the given data set is 1.

Only one data occurs most often, so the data set is unimodal.