Chapter 10.3, Problem 12HP

To determine

To find:The error in the given situation and provide the correct answer.

Answer to Problem 12HP

The error in the calculation is mean value and the correct answer is $Mean=15$ and Mean Absolute Deviation = 11.

Explanation of Solution

Given information:Alicia is finding the Mean Absolute Deviation of the data set $\left\{5,7,8,10,12,48\right\}$ .

Formula used:

The Mean of a given set of values is the average of the given set of values.

  $Mean=\frac{\text{sum}\text{of}\text{values}}{\text{number}\text{of}\text{values}}=\frac{{\displaystyle \sum x}}{n}$

The Mean Absolute Deviation of a given set of values is the average of deviation of each value from the mean value.

  $M.A.D=\frac{{\displaystyle \sum \left(|x-\overline{x}|\right)}}{n}$

Calculation:

The calculation done by Alicia is shown below

  

Alicia calculated the value of mean by the wrong formulabecause of which the answer is incorrect.

Hence the error in the calculation is mean value.

The correct value of $\text{mean}\left(\overline{x}\right)\&\text{mean}\text{absolute}\text{deviation}\left(M.A.D\right)$ for the set $\left\{5,7,8,10,12,48\right\}$ are shown below

  $\begin{array}{l}\text{set=}\left\{5,7,8,10,12,48\right\},n=6\left(even\right)\\ \\ Mean=\overline{x}=\frac{{\displaystyle \sum x}}{n}=\frac{\left(5+7+8+10+12+48\right)}{6}=15\\ \\ \text{The}\text{New}\text{set}\text{of}\text{values}\text{of}\left(|x-\overline{x}|\right)\text{are}\\ =\left\{|5-15|,|7-15|,|8-15|,|10-15|,|12-15|,|48-15|\right\}\\ =\left\{10,8,7,5,3,33\right\}\\ \\ M.A.D=\frac{{\displaystyle \sum \left(|x-\overline{x}|\right)}}{n}=\frac{\left(10+8+7+5+3+33\right)}{6}=\frac{66}{6}=11\\ M.A.D=11\mathrm{............}\left(1\right)\end{array}$

The Mean Absolute Deviation for the set $\left\{5,7,8,10,12,48\right\}$ is 11.

Hence the error in the calculation is mean value and the correct answer is $Mean=15$ and Mean Absolute Deviation = 11.