Question

A data set with an outlier is shown. 20, 35, 40, 45, 45, 50, 75Which of the following best describes the effect on the mean of the data of the outlier is removed? A. The mean will increaseB. The mean will decrease C. The mean will remain the same D. There is not enough information to make a conclusion

Answer 1

To solve the problem, we heed to calculate the mean of the data set in the two situations i,e with the data set and without the data set.

An outlier is an extreme value in a data set that is either much larger or much smaller than all the other values

Using the formula for mean we have

`\text{mean}=\frac{sum\text{ of data set}}{frequence\text{ (n)}}`

The data set is giving as

hence

The mean with the outiers will be

Mean = 44.29

In the data set, we need to calculate the outliers, using the formula

`\begin{gathered} Q1=(1)/(4)(n+1) \\ \text{Where n=number of data=}7 \\ Q_1=(1)/(4)(7+1)=(1)/(4)*8=2 \\ \text{Hence } \\ Q_1=\text{ the second data=35} \end{gathered}`

Then for the upper quartile Q3, we have

`\begin{gathered} Q_3=(3)/(4)(n+1)=(3)/(4)(7+1)=(3)/(4)*8=6 \\ \text{Hence} \\ Q_3=6th\text{ data=50} \end{gathered}`

Then the inter-quartile range IQR is

`\text{IQR}=Q_3-Q_1=50-35=15`

Then applying the rule for the outliers, we have

`\begin{gathered} \text{lower outliers=Q}_1-(1.5*\text{IQR)}=35-(1.5*15)=12.5 \\ \text{Then } \\ \text{Higher outliers= Q}_3+(1.5* IQR)=50+(1.5*15)=72.5 \end{gathered}`

Hence

The outliers will be an of the dataset that is greater than the higher outliers or lower outliers,

Hence

The outliers is

`75`

Hence removing 75 from the data set we will have

`\text{Mean}=(20+35+40+45+45+50)/(6)=(235)/(6)=39.17`

Mean= 39. 17

Hence

Removing the outlier in the data set reduces the mean

Answer= The mean will decrease (Option B)