Determine the effect of outliers on measures of centre

An outlier is a data value unusually far from the rest of a distribution, and its effect depends on both its distance and the measure used. An extreme value typically pulls the mean toward it, while the median is comparatively resistant because it depends on ordered position, and the mode is usually unaffected unless the outlier occurs repeatedly; this supports choosing and interpreting an appropriate measure of centre.

Detailed Explanation: Determine the effect of outliers on measures of centre

An outlier is a value that is much higher or lower than the other data values. To determine its effect, compare the mean, median, and mode before and after including the outlier.

Suppose the data set is:

4,\ 5,\ 5,\ 6,\ 7

Now add an unusually large value, $30$ :

4,\ 5,\ 5,\ 6,\ 7,\ 30

1. Find the measures before the outlier

Mean:

\frac{4+5+5+6+7}{5}=\frac{27}{5}=5.4

Median: The middle value is $5$ .
Mode: The value that appears most often is $5$ .

2. Find the measures after adding the outlier

Mean:

\frac{4+5+5+6+7+30}{6}=\frac{57}{6}=9.5

The mean increases from $5.4$ to $9.5$ . The outlier pulls the mean upward because the mean uses every value.

Median: There are six values, so use the middle two values, $5$ and $6$ :

\frac{5+6}{2}=5.5

The median changes only slightly, from $5$ to $5.5$ , because it depends on the middle position.

Mode: The mode is still $5$ , since $5$ appears twice and all other values appear once.

Conclusion

The outlier $30$ has a large effect on the mean, a small effect on the median, and no effect on the mode in this example. When data contain an outlier, the median may describe the typical value better than the mean.

Learn by doing: Determine the effect of outliers on measures of centre

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Measure of Center - Data Set to Mean or Median

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