![]() Recall that recalculation of the least-squares regression line and summary statistics, following deletion of an outlier, may be used to determine whether an outlier is also an influential point. Numerically and graphically, we have identified point (65, 175) as an outlier. How Does the Outlier Affect the Best-Fit Line? Note that when the graph does not give a clear enough picture, you can use the numerical comparisons to identify outliers. On a computer, enlarging the graph may help on a small calculator screen, zooming in may make the graph clearer. Sometimes a point is so close to the lines used to flag outliers on the graph that it is difficult to tell whether the point is between or outside the lines. The outlier is the student who had a grade of 65 on the third exam and 175 on the final exam. On the calculator screen, it is barely outside these lines, but it is considered an outlier because it is more than two standard deviations away from the best-fit line. You will see that the only point that is not between Y2 and Y3 is the point (65, 175). Graph the scatter plot with the best-fit line in equation Y1, then enter the two extra lines as Y2 and Y3 in the Y= equation editor. Y2 and Y3 have the same slope as the line of best fit. Where ŷ = –173.5 + 4.83 x is the line of best fit. You would generally need to use only one of these methods. The graphical procedure is shown first, followed by the numerical calculations. With regard to the TI-83, 83+, or 84+ calculators, the graphical approach is easier. Or, we can do this numerically by calculating each residual and comparing it with twice the standard deviation. Any data points outside this extra pair of lines are flagged as potential outliers. We can do this visually in the scatter plot by drawing an extra pair of lines that are two standard deviations above and below the best-fit line. The standard deviation used is the standard deviation of the residuals or errors. As a rough rule of thumb, we can flag as an outlier any point that is located farther than two standard deviations above or below the best-fit line. However, we would like some guideline regarding how far away a point needs to be to be considered an outlier. We could guess at outliers by looking at a graph of the scatter plot and best-fit line. A graph showing both regression lines helps determine how removing an outlier affects the fit of the model. The new regression will show how omitting the outlier will affect the correlation among the variables, as well as the fit of the line. Regression analysis can determine if an outlier is, indeed, an influential point. Computers and many calculators can be used to identify outliers and influential points. Sometimes, it is difficult to discern a significant change in slope, so you need to look at how the strength of the linear relationship has changed. You also want to examine how the correlation coefficient, r, has changed. To begin to identify an influential point, you can remove it from the data set and determine whether the slope of the regression line is changed significantly. These points may have a big effect on the slope of the regression line. Influential points are observed data points that are far from the other observed data points in the horizontal direction. The key is to examine carefully what causes a data point to be an outlier.īesides outliers, a sample may contain one or a few points that are called influential points. Other times, an outlier may hold valuable information about the population under study and should remain included in the data. Sometimes, they should not be included in the analysis of the data, like if it is possible that an outlier is a result of incorrect data. They have large errors, where the error or residual is not very close to the best-fit line. ![]() Outliers are observed data points that are far from the least-squares line. ![]() In some data sets, there are values (observed data points) called outliers. ![]()
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