Many lines could fit a scatterplot. The least-squares regression line is the one that makes the total of the squared residuals as small as possible. Drag points, try to beat it with your own line, compute the slope and intercept from summary statistics, and read the values that matter in computer output.
The least-squares regression line is the one line that makes the sum of the squared residuals as small as possible. Drag points, compare your own line, and see every square.
Read Coef for the intercept (Constant row) and slope (variable row), plus S and R-Sq. The gray SE Coef, T, and P columns are used for inference about the slope, which is not part of the revised AP Statistics course.
Of all possible lines, the least-squares regression line (LSRL) makes Σ(y − ŷ)² as small as possible. Squaring keeps positive and negative residuals from canceling and penalizes big misses.
b = r · sy ÷ sₓ and a = ȳ − b·x̄. These are on the AP formula sheet. The slope has the same sign as r.
Substituting x̄ into the LSRL always gives ȳ. Moving one point changes x̄ and ȳ, so the whole line shifts.
r² is the fraction of the variation in y that is accounted for by the linear relationship with x. It is between 0 and 1. Interpret it as a percent, in context.
s, the standard deviation of the residuals, gives the typical distance between actual y-values and the line's predictions, in y's units.
The LSRL minimizes vertical distances, so it depends on which variable is the response. Predicting x from y uses a different line, even though r is the same.
Where to find each value in typical computer output, and how to interpret it.
| Value | Where it appears | Interpretation template |
|---|---|---|
| Intercept a | Coef column, Constant row | When x = 0, the predicted y is a (only if x = 0 makes sense). |
| Slope b | Coef column, row named for x | For each 1-unit increase in x, the predicted y changes by b. |
| r² | R-Sq (often as a percent) | About __% of the variation in y is accounted for by the linear relationship with x. |
| s | S | Actual y-values typically differ from predicted values by about s. |
| r | Not shown directly: r = ±√r², with the sign of the slope | Strength and direction of the linear association. |
| SE Coef, T, P | Other columns | Used for inference about the slope, which is not in the revised course. |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 5.5. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 5 terminology guide.