Line of Best Fit Statistics Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Given five data points: (1,3), (2,5), (3,6), (4,8), (5,11). Estimate the line of best fit by finding the slope using the first and last points, then adjust to pass through the centroid (xห‰,yห‰)(\bar{x}, \bar{y}).

Solution

  1. 1
    Step 1: Centroid: xห‰=1+2+3+4+55=3\bar{x} = \frac{1+2+3+4+5}{5} = 3, yห‰=3+5+6+8+115=335=6.6\bar{y} = \frac{3+5+6+8+11}{5} = \frac{33}{5} = 6.6.
  2. 2
    Step 2: Slope estimate from endpoints: m=11โˆ’35โˆ’1=84=2m = \frac{11-3}{5-1} = \frac{8}{4} = 2.
  3. 3
    Step 3: Line through centroid with slope 2: yโˆ’6.6=2(xโˆ’3)y - 6.6 = 2(x - 3), so y=2x+0.6y = 2x + 0.6.

Answer

Estimated line of best fit: y=2x+0.6y = 2x + 0.6.
The line of best fit always passes through the centroid (point of means) of the data. While the formal least-squares method gives the optimal line, estimating the slope from the data trend and forcing the line through the centroid provides a reasonable approximation.

About Line of Best Fit

The line of best fit (trend line) is the straight line that best represents the overall trend in a scatter plot by minimizing the sum of squared vertical distances between the line and all data points. Its equation enables predictions for new x-values.

Learn more about Line of Best Fit โ†’

More Line of Best Fit Examples

Example 1 easy

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Example 3 medium

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Example 4 hard

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