AP Stats Residual Calculator

A residual is how far an observed value misses its predicted value. Positive means the actual value was above the line; negative means it was below the line. Residuals show whether a regression model fits well — or systematically misses.

What Is a Residual?

A residual is the difference between an observed value and the value predicted by the regression line: residual = observed y - predicted y. A positive residual means the actual value was above the line, and a negative residual means it was below it. Plotting residuals against the explanatory variable shows whether a straight line is a good model, the residuals should scatter randomly with no pattern.

How to Use This Calculator

Choose your input mode. If you already have predicted values, enter observed y and predicted ŷ in the same order. If you have a regression equation instead, enter x, slope, and intercept, and the calculator finds ŷ = a + bx for each x first. The table then shows every residual, and the chart draws positive residuals above zero and negative ones below zero.

How to Calculate Residuals

Formula: residual = observed − predicted, or e = y − ŷ. With a model, ŷ = a + bx first.

Worked example: Observed y = 10, 12, 15, 18 and predicted ŷ = 9, 13, 14, 18. Residuals are 10 − 9 = 1, 12 − 13 = −1, 15 − 14 = 1, and 18 − 18 = 0. The first and third points sit above the line, the second sits below it, and the fourth sits exactly on it. Same example from a model: with x = 1, 2, 3, 4, slope 2.5 and intercept 7, predicted values are 9.5, 12, 14.5, 17, giving residuals 0.5, 0, 0.5, 1.

Tips and Common Mistakes

• Residual is observed minus predicted — reversing the subtraction flips every sign.
• A residual of 0 means the point lies exactly on the regression line, not that it was “ignored.”
• Large residuals are worth checking: they may be outliers, or a sign the model is wrong.
• A residual plot should look random. A curved pattern means a straight line is the wrong model.
• Do not average residuals to judge fit — for a least-squares line they sum to about 0 by construction. Look at their size and pattern instead.

Frequently Asked Questions

What does a positive residual mean?

The observed value was higher than predicted — the point sits above the regression line.

What does a negative residual mean?

The observed value was lower than predicted — the point sits below the regression line.

Can a residual be zero?

Yes. It means the observed value equals the predicted value and the point lies exactly on the line.

Do residuals always sum to zero?

For a least-squares regression line fitted to the same data, they sum to approximately zero (apart from rounding). That is a property of the fit, not proof the model is good.

What is a residual plot used for?

It plots residuals against x (or against predicted values). A random scatter supports a linear model; a pattern, curve, or funnel shape warns against it.

How are residuals different from errors?

A residual is the observed minus predicted value from your fitted model. The true “error” would use the unknown true population line, which you never observe.

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