12.9 Outliers

A scatter plot should be checked for outliers. An outlier is a point that seems out of place when compared with the other points. Some of these points can affect the equation of the regression line.

Example 1

Should linear regression be used with this data set?

$x$ 1 3 8 2 1 3 2 2 3 1 $y$ 2 3 8 2 3 1 3 1 2 1

Show solution

A regression analysis for the data set was run on Excel.

If we test for a significant correlation:

$H_{0}: \rho = 0$
$H_{1}: \rho \neq 0$

The correlation is $r = 0.844$ and the p-value is 0.002, which is less than $\alpha$ = 0.05, so we would reject $H_{0}$ and conclude there is a significant relationship between $x$ and $y$ .

However, if look at the scatterplot in Figure 12-18, with the regression equation we can clearly see that the point $(8,8)$ is an outlier. The outlier is pulling the slope up towards the point $(8,8)$ .

Figure 12-18: Scatterplot of data with an outlier.

If we were to take out the outlier point $(8,8)$ and run the regression analysis again on the modified data set we get the following Excel output.

See Figure 12-19: note the correlation is now 0 and the p-value is 1, so there is no relationship at all between $x$ and $y$ .

Figure 12-19: Scatterplot of the same data as above with outlier removed.

This type of outlier is called a leverage point. Leverage points are positioned far away from the main cluster of data points on the $x$ -axis.

There is another type of outlier called an influential point. Influential points are positioned far away from the main cluster of data points on the $y$ -axis. There is an option in most software packages to get the “standardized” residuals. Standardized residuals are z-scores of the residuals. Any standardized residual that is not between $-2$ and $2$ may be an outlier. If it is not between $-3$ and $3$ then the point is an outlier. When this happens, the points are called influential points or influential observations.

Example 2

Use technology to compute the standardized residuals. Should linear regression be used with this data set?

$x$ 1 3 2 2 4 5 7 9 6 8 $y$ 1 3 10 2 4 5 7 9 6 8

Show solution

A regression analysis for the given data set was run on Excel, producing the following results:

The point $(2, 10)$ shown in Figure 12-20 is pulling the left side of the line up and away from the points that form a line. This influential point changes the $y$ -intercept and slope.

Figure 12-20:

Adapted from Mostly Harmless Statistics by Rachel Webb (Portland State University, https://mostlyharmlessstat.wixsite.com/webpage), © Rachel Webb, licensed under CC BY-SA 4.0. Changes were made. License: CC-BY-SA-4.0.