We have recently explored calculating the five number summary and drawing a box and whisker plot.

Once we have our minimum, lower quartile, median, upper quartile and maximum, we can use this information to determine whether a data point can be considered an outlier.

To do this, we calculate what we like to call the fences.

Once we have calculated the lower fence and the upper fence, any data that falls within the fence lines is not an outlier.

Any data that fall outside the fence lines will be considered an outlier.

Calculating the fences

Lower fence = Lower quartile $-1.5\times$−1.5× Interquartile Range

Upper fence = Upper quartile $+1.5\times$+1.5× Interquartile Range

Worked Example:

Consider the box plot given below.

By calculating the lower and upper fence, detemine whether the data points $4$4 and $32$32 can be considered outliers.

Think: Firstly we need to calculate the lower and upper fence.

Do:

Lower fence = $11-1.5\times8$11−1.5×8

Lower fence = $-1$−1

Upper fence = $19+1.5\times8$19+1.5×8

Upper fence = $31$31

Think: Does $4$4 lie inside or outside the fence?

Do: Since $4$4 is above the lower fence of $-1$−1, it is inside the fence and hence not an outlier.

Think: Does $32$32 lie inside or outside the fence?

Do: Since $32$32 is above the upper fence of $31$31, it is outside the fence and hence an outlier.

Worked Examples

Question 1

Consider the dot plot below.

Determine the median, lower quartile score and the upper quartile score.

Median $=$=$\editable{}$

Lower quartile $=$=$\editable{}$

Upper quartile $=$=$\editable{}$

Hence, calculate the interquartile range.

Calculate $1.5\times IQR$1.5×IQR, where IQR is the interquartile range.

An outlier is a score that is more than $1.5\times IQR$1.5×IQR above or below the Upper Quartile or Lower Quartile respectively. State the outlier.

Question 2

A set of data has the following box plot.

2

4

6

8

10

12

14

16

18

Calculate the interquartile range.

Calculate the value of the lower fence.

Calculate the value of the upper fence.

Question 3

Consider the following set of data:

$9$9$5$5$3$3$2$2$6$6$1$1

Complete the five-number summary for this data set.