Regression and correlation Updated November 2026

Residuals

A residual is the gap between an observed value and the value the regression predicts: e = y - y-hat. Positive residuals sit above the line, negative below. Least squares residuals always sum to zero, and their pattern in a residual plot tells you whether the linear model fits.

Syntax

e = y - y-hat = y - (a + b * x)

Part	What it is
`y`	the observed value
`y-hat`	the predicted value from the fitted line

Examples

Line y-hat = 2.2 + 0.6x; point (3, 5)
y-hat = 2.2 + 1.8 = 4.0; e = 5 - 4.0

Result: 1.0 (the point is above the line)

All five points x: 1..5, y: 2, 4, 5, 4, 5
predicted: 2.8, 3.4, 4.0, 4.6, 5.2
residuals: -0.8, 0.6, 1.0, -0.6, -0.2

Result: sum = 0, SSE = 2.4

Excel: =B1 - FORECAST.LINEAR(A1, $B$1:$B$5, $A$1:$A$5), filled down

Residual standard error
s = sqrt(SSE / (n - 2)) = sqrt(2.4 / 3)

Result: 0.894

Excel: =STEYX(B1:B5, A1:A5)

Common mistakes

Subtracting in the wrong order (y-hat - y); the convention is observed minus predicted, so a point above the line has a positive residual.
Ignoring a curved or fanning pattern in the residual plot; residuals should look like random noise around zero, any pattern means the model is missing something.
Calling a large residual an error in the data; it may be a real unusual case, check it before deleting it.

Practice Residuals

These drills check your answer on live data, free, no account needed.

Residual of one observationData on monthly ad spend and sales at regional stores (x = Ad spend ($k), y = Sales ($k)): (9, 87.8), (10, 98.7), (6, 81.3), (12, 99.1), (15, 111.9), (6, 78.6). Fit the least squares line. What is the residual for observation 4, at x = 12? Round to 2 decimal places.
Predicted value from a fitted lineData on monthly ad spend and sales at regional stores (x = Ad spend ($k), y = Sales ($k)): (9, 87.8), (10, 98.7), (6, 81.3), (12, 99.1), (15, 111.9), (6, 78.6). Fit the least squares line and predict y when x = 11. Round to 2 decimal places.

Questions

What should a residual plot look like?

A shapeless horizontal band around zero. A curve means the relationship is not linear; a funnel means the spread changes with x (heteroscedasticity); a trend over time means autocorrelation.

Why do residuals sum to zero?

Because the least squares line with an intercept passes through the point of means; the positive and negative gaps balance exactly.

What is a standardized residual?

The residual divided by its standard error, so it reads like a z-score. Values beyond about 2 or 3 flag points the model fits poorly.

Linear regressionHow to fit a regression line y = a + bx: slope b = Sxy / Sxx, intercept a = y-bar - b x-bar, a worked example, prediction and Excel SLOPE and INTERCEPT.
R-squaredR-squared is the share of variation in y explained by the regression, 1 - SSE / SST. Worked example with exact numbers, link to r, Excel RSQ and its limits.
OutliersHow to detect outliers with the 1.5 IQR fence rule and the z-score rule, worked examples with exact fences, and what to do with outliers once found.
Standard errorThe standard error of the mean is s / sqrt(n); for a proportion it is sqrt(p(1 - p) / n). Worked examples, SE versus standard deviation and the Excel formula.