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Outliers

An outlier is a value in a data set that is very different from the other values. That is, outliers are values unusually far from the middle.

In most cases, outliers have influence on mean , but not on the median , or mode . Therefore, the outliers are important in their effect on the mean.

There is no rule to identify the outliers. But some books refer to a value as an outlier if it is more than $1.5$ times the value of the interquartile range beyond the quartiles .

Also plotting the data on a number line as a dot plot will help in identifying the outliers.

Example:

Find the outliers of the data set. Also find the mean of the data set including the outliers and excluding the outliers.

$15, 75, 20, 35, 25, 85, 30, 30, 15, 25, 30$

First arrange the data set in order.

$15, 15, 20, 25, 25, 30, 30, 30, 35, 75, 85$

Plot the data on a number line as a dot plot.

Math diagram

The values $75$ and $85$ are far off the middle. So, these two values are outliers for the given data set.

Find the mean, median and mode of the data including the outliers:

$Mean = \frac{Sum of the data values}{Number of data values}$

$= \frac{15 + 15 + 20 + 25 + 25 + 30 + 30 + 30 + 35 + 75 + 85}{11}$

$= 35$

Find the mean of the data excluding the outliers:

$Mean = \frac{Sum of the data values}{Number of data values}$

$= \frac{15 + 15 + 20 + 25 + 25 + 30 + 30 + 30 + 35}{9}$

$= 25$

The mean of the given data set is $35$ when outliers are included, but it is $25$ when outliers are excluded.