Chapter 12, Problem 6CT

To determine

To find: the mean, median, mode and range of the hours that the gardener works.

Answer to Problem 6CT

The mean is $44.4$ ,median is $44$ , mode is $44$ and range here is $18$ .

Explanation of Solution

Given:

The hours a gardener worked over the past $14$ weeks are:

  $39,52,41,44,47,36,51,44,50,40,53,46,44,35$ .

Concept used:

Arithmetic mean:

Mean: $\stackrel{-}{x}=\frac{{\displaystyle \sum xi}}{N}$ .

Median $=\left\{\begin{array}{c}\frac{N}{2}\text{th term; }N\text{ is odd}\\ \frac{N}{2}\text{th term; }N\text{ is even}\end{array}\right\}$ .

Mode= the value in the data set that occurs most frequently.

Calculation:

The hours a gardener worked over the past $14$ weeks are:

  $39,52,41,44,47,36,51,44,50,40,53,46,44,35$ .

Mean is the average of the data:

  $\begin{array}{l}\stackrel{-}{x}=\frac{39+52+41+44+47+36+51+44+50+40+53+46+44+35}{14}.\\ \Rightarrow \frac{622}{14}\approx \mathrm{44.4.}\\ \text{the mean }\stackrel{-}{x}=\mathrm{44.4.}\end{array}$

Median is calculated by making the order in increasing order and then choose the middle value.

If the number of the data is even then there will have two middle value so, by taking the mean of two middle value get the median as:

The data in increasing order is:

  $35,36,39,40,41,44,44,44,46,37,50,51,52,53$ .

Total number of data is $14$ which is even so there will be two middle value which is $44,44$ by taking mean of two middle value:

  $\frac{44+44}{2}=44$ .

The median is $44$ .

Mode is the value that occurs maximum or most often.

Taking the increasing order of the data:

  $35,36,39,40,41,44,44,44,46,37,50,51,52,53$ .

Her the maximum value that occur is $44$ .

The mode is $44$ .

Range is the difference between largest and smallest value.

Taking the increasing order of the data:

  $35,36,39,40,41,44,44,44,46,37,50,51,52,53$ .

Here largest value is $53$ and smallest value is $35$ .

  $53-35=18$ .

The range here is $18$ .

Since, the data set has no outlies, then the mean best describes the data set.

Hence, mean is $44.4$ ,median is $44$ , mode is $44$ and range here is $18$ .