In statistics, we tend to assume that our data will fit some kind of trend and that most things will fit into a "normal" range. This is why we look at measures of central tendency, such as the mean, median and mode .

An outlier is an event that is very different from the norm and results in a score that is really above or below average. For example, if there are five people in a group and four people were between 120 \text{ cm} and 130 \text{ cm}, whereas Jim was 165 \text{ cm}, Jim would be an outlier as he is much taller than everyone else in the group.

Examples

Example 1

The dot plot shows the temperature (\degree \text{C}) in a town over a several week period. Identify the temperature that is an outlier.

The image shows a dot plot with the data for temperature in a town. Ask your teacher for more information.

Worked Solution

Create a strategy

Identify the temperature that is much greater or smaller than these scores.

Apply the idea

We can see that the dot for 21 is far away from the rest of the dots.\text{Outlier}=21 \degree \text{C}

Idea summary

An outlier is an event that is very different from the norm and results in a score that is really above or below average.

Effects of outliers

Outliers can skew or change the shape of our data. This can be a problem (especially for small data sets) because the mean, median and range might not properly represent the situation. We can counteract this by removing outliers. Removing outliers will have the following effects:

Removing a really low outlier	Removing a really high outlier
The range will decrease.	The range will decrease.
The median might increase.	The median might decrease.
The mean will increase.	The mean will decrease.
The mode will not change.	The mode will not change.

Examples

Example 2

Consider the following set of data: 53,\,46,\,25,\,50,\,30,\,30,\,40,\,30,\,47,\,109

Find the mean, median, mode, and range.

Worked Solution

Create a strategy

To find the mean, use the formula: \text{Mean}=\dfrac{\text{Sum of score}}{\text{Number of scores}}

To find the median find the middle score, to find the mode find the most frequent score.

To find the range, use the formula: \text{Range}=\text{Highest score}-\text{Lowest score}

Apply the idea

\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{53+46+25+50+30+30+40+30+47+109}{10}	Use the formula
	\displaystyle =	\displaystyle \dfrac{460}{10}	Evaluate the addition
	\displaystyle =	\displaystyle 46	Evaluate the division

To find the median, order the scores: 25,\,30,\,30,\,30,\,40,\,46,\,47,\,50,\,53,\,109

The middle scores are: 40,\,46

\displaystyle \text{Median}	\displaystyle =	\displaystyle \dfrac{40+46}{2}	Find the average of the middle values
	\displaystyle =	\displaystyle 43	Evaluate

To find the mode, choose the score which occurs most often.

\text{Mode}=30

To find the range:

\displaystyle \text{Range}	\displaystyle =	\displaystyle 109-25	Substitute the values
	\displaystyle =	\displaystyle 84	Evaluate the subtraction

Which data value is an outlier?

Worked Solution

Create a strategy

Choose the value that is much greater or much smaller than the rest of the data set.

Apply the idea

\text{Outlier}=109

Find the mean, median, mode, and range after removing the outlier.

Worked Solution

Apply the idea

\displaystyle \text{Mean}	\displaystyle =	\displaystyle \dfrac{25+30+30+30+40+46+47+50+53}{9}	Substitute all the scores
	\displaystyle =	\displaystyle \dfrac{351}{9}	Evaluate the addition
	\displaystyle =	\displaystyle 39	Evaluate the division

To find the median, order the scores: 25,\,30,\,30,\,30,\,40,\,46,\,47,\,50,\,53

The middle score is 40.

\text{Median}=40

To find the mode, choose the score which occurs most often.

\text{Mode}=30

To find the range:

\displaystyle \text{Range}	\displaystyle =	\displaystyle 53-25	Substitute the values
	\displaystyle =	\displaystyle 28	Evaluate the subtraction

Let A be the original data set and B be the data set without the outlier.

Complete the table using the symbols >,< and = to compare the statistics before and after removing the outlier.

	\text{With outlier}		\text{Without\ outlier}
Mean:	A	⬚	B
Median:	A	⬚	B
Mode:	A	⬚	B

Worked Solution

Create a strategy

Compare the statistics in part (a) and in part (c).

Apply the idea

Statistics from parts (a) and (c):

	With outlier	Without outlier
Mean	46	39
Median	43	40
Mode	30	30
Range	84	28

Comparison table:

	\text{With outlier}		\text{Without\ outlier}
Mean:	A	>	B
Median:	A	>	B
Mode:	A	=	B
Range:	A	>	B

Idea summary

Removing outliers will have the following effects on the summary statistics:

A really low outlier	A really high outlier
The range will decrease	The range will decrease
The median might increase	The median might decrease
The mean will increase	The mean will decrease
The mode will not change	The mode will not change

Outcomes

ACMSP207

Investigate the effect of individual data values, including outliers, on the mean and median

10.05 Outliers

Ideas

Outliers

Examples

Example 1

Effects of outliers

Examples

Example 2

Outcomes

ACMSP207

What is Mathspace

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