Linear Regression and Correlation

Scatterplot, correlation r, r², the least-squares line, and predictions.

Inputs

ŷ = 51.4783 + 3.4674x · r = 0.9967

There is a strong positive linear relationship between hours studied and exam score. Each 1-unit increase in hours studied goes with a predicted change of 3.4674 in exam score, and 99.3% of the variation in exam score is explained by the line. At x = 7, the predicted exam score is 75.75.

Results

Slope b	3.4674
y-intercept a	51.4783
Correlation r	0.9967
r²	0.9934
Standard error of estimate s_e	1.1107
n	8
Test of ρ = 0: t, p-value	29.9428, 9.2033 × 10⁻⁸
Significant at α = 0.05?	Yes
ŷ at x = 7	75.75

On the TI-84

LinReg(a+bx) L1, L2, Y1

LinRegTTest L1, L2, 0, ≠

Show steps

Means. x̄ = 7, ȳ = 75.75.
Sums of squares. S_xx = Σ(x − x̄)² = 92, S_yy = Σ(y − ȳ)² = 1113.5, S_xy = Σ(x − x̄)(y − ȳ) = 319.
Slope. b = S_xy ÷ S_xx = 319 ÷ 92 = 3.4674
Intercept. a = ȳ − b·x̄ = 75.75 − (3.4674)(7) = 51.4783
Correlation. r = S_xy ÷ √(S_xx S_yy) = 0.9967, so r² = 0.9934.
Prediction. ŷ = 51.4783 + 3.4674(7) = 75.75

What the numbers mean

r (correlation) measures the direction and strength of a linear relationship, from −1 to 1. Values near ±1 are strong; values near 0 are weak.
r² is the fraction of the variation in y that the line explains.
Slope b: the predicted change in y for each 1-unit increase in x.
Intercept a: the predicted y when x = 0 (only meaningful if x = 0 makes sense).

Cautions

Always look at the scatterplot first. A curved pattern or an outlier can make r misleading. Don't predict far outside the range of your x values (extrapolation). And correlation is not causation: two variables can move together because of a third variable.

Linear Regression and Correlation

Inputs

Results

Results

On the TI-84

About this calculator

What the numbers mean

Cautions