Comparing Actual Values to Model Values to Judge Fit (F 701)

A model is only useful if it matches reality, and residuals are how you check that match point by point. At any x-value, take the actual measured value and subtract what the model predicted: actual minus predicted. If the model predicted 31 and the real value was 27, the residual is 27 - 31 = -4. A residual of zero means a perfect hit at that point. The sign tells you which way the model missed: positive means the model undershot, negative means it overshot.

To judge a whole model rather than one point, look at all its residuals together. Since positives and negatives can cancel if you just add them, square each residual first, then add. That sum of squares can never hide a miss, so a smaller sum always means a tighter fit. Compare a model with errors 3, -2, 4 (squares 9, 4, 16, sum 29) against one with errors 1, 6, -5 (squares 1, 36, 25, sum 62). The first model fits better even though one of its individual misses, 4, is close to the second model's smallest miss.

This is why a curve can look deceptively good and still be wrong. A line drawn through only the first and last data point will always have zero residual at those two ends, but that tells you nothing about the points between. If every middle point sits above the line, every one of those residuals is positive, meaning the model is wrong in the same direction every time. That one-sided pattern is also the signature of fitting a straight line to data that actually curves: the residuals won't scatter randomly, they'll swing from positive to negative to positive again, exposing the wrong shape.