Suppose you want to fit a line to three data points. If no line passes exactly through your points, what’s the best compromise you could make?
Chebyshev suggested the best thing to do is find the minmax line, the line that minimizes the maximum error. That is, for each candidate line, find the vertical distance from each point to the line, and take the largest of these three distances as the measure of how well the line fits. The best line is the that minimizes this measure.
Note that this is not the same line you would get if you did a linear regression. Customary linear regression minimizes the average squared vertical distance from each point to the line. There are reasons this is the standard approach when you have at least a moderate amount of data, but when you only have three data points, it makes sense to use the minmax approach.
We can say several things about the minmax line. For one thing, there exists such a line and it is unique. Also, the vertical distance from each data point to the line will be the same. Either two points will be over the line and one under, or two will be under and one over.
Suppose your three points are (x1, y1), (x2, y2), and (x3, y3), with x1 < x2 < x3. Then the slope will be
and the intercept will be
I made an online calculator to find the best line for three points.