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Influential Point — Definition, Formula & Examples

An influential point is a data point that has a large effect on a regression line. If you remove it and refit, the slope or intercept changes noticeably.

In regression, an observation is influential if deleting it from the data set substantially changes the fitted model, such as its slope, intercept, or correlation coefficient. Influential points usually have high leverage, meaning their xx-value lies far from xˉ\bar{x}. They also tend not to follow the pattern of the other data.

Worked Example

Problem: The points (1,2)(1, 2), (2,4)(2, 4), (3,6)(3, 6), (4,8)(4, 8) lie exactly on a line. A fifth point, (10,0)(10, 0), is added. Find the new least-squares regression line and decide whether (10,0)(10, 0) is influential.
Step 1: Without the fifth point, every yy-value is twice its xx-value. So the regression line fits the four points exactly, with slope 2 and intercept 0.
y^=2x\hat{y} = 2x
Step 2: Include (10,0)(10, 0). Find the means. Then find the sum of products of deviations and the sum of squared xx-deviations. The xx-deviations are −3,−2,−1,0,6-3, -2, -1, 0, 6 and the yy-deviations are −2,0,2,4,−4-2, 0, 2, 4, -4.
xˉ=205=4,yˉ=205=4,Sxy=−20,Sxx=50\bar{x} = \frac{20}{5} = 4, \quad \bar{y} = \frac{20}{5} = 4, \quad S_{xy} = -20, \quad S_{xx} = 50
Step 3: Compute the slope b=Sxy/Sxxb = S_{xy}/S_{xx}. Then compute the intercept a=yˉ−bxˉa = \bar{y} - b\bar{x}.
b=−2050=−0.4,a=4−(−0.4)(4)=5.6b = \frac{-20}{50} = -0.4, \quad a = 4 - (-0.4)(4) = 5.6
Answer: The new line is y^=5.6−0.4x\hat{y} = 5.6 - 0.4x. One point flipped the slope from 22 to −0.4-0.4, so (10,0)(10, 0) is highly influential.

Why It Matters

In AP Statistics, you are expected to check a scatterplot for influential points before you interpret a slope or r2r^2. A single unusual measurement can reverse a trend in real data, such as one extreme city in a pollution study. Comparing the fit with and without the point shows whether your conclusion depends on one observation.

Common Mistakes

Mistake: Judging influence by the size of the residual and assuming a point close to the line cannot be influential.
Correction: An influential point drags the line toward itself, so its residual is often small. In the example, the new line predicts 1.61.6 at x=10x = 10, a residual of only −1.6-1.6. To test for influence, refit the line without the point and compare the slopes.

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