Least-Squares Regression Line Simulator (AP Statistics)

How to use the simulator

There are no sliders. Four preset buttons load data: Tight, Moderate and Weak (12 points each) and Outlier (10 points). Tight is loaded when the page opens.

To move a point, press within about 20 pixels of it and drag. The point turns gold while you hold it, and both coordinates stay between 0.1 and 9.9 on axes that run from 0 to 10.

Scatterplot. The navy line is the least-squares regression line (LSRL). Each point has a dashed vertical segment to the line: teal for a positive residual (point above the line) and coral for a negative one. A small gold dot marks (x̄, ȳ).
Residual plot. Below the scatterplot, each residual is plotted against x around a zero line. Its vertical scale resizes automatically, so read it for pattern rather than size.
Readouts. The equation ŷ = b₀ + b₁x to two decimals, then Slope b₁, Intercept b₀ and r² to three decimals, and SSR Σe², the sum of squared residuals, to two decimals. r² is shown as a decimal, so 0.841 means 84.1%.

The formula

The AP formula sheet writes the line as

\hat{y} = a + bx

; the simulator calls the intercept

b_0

and the slope

b_1

b = r\,\frac{s_y}{s_x} \qquad a = \bar{y} - b\bar{x}

\text{residual} = y - \hat{y}

The LSRL is the one line that minimizes $\sum (y - \hat{y})^2$ .
It always passes through $(\bar{x}, \bar{y})$ , which is why the gold dot sits on the line however you drag.
The residuals always add to 0.
The slope is the predicted change in y for a one-unit increase in x. The intercept is the predicted y when x = 0, which only means something if x = 0 is reasonable for the data.
$r^2$ is the proportion of variation in y accounted for by the linear model.

Worked example

Problem: The Moderate preset has 12 points with x̄ = 5.4333, s_x = 2.9586, ȳ = 5.2833, s_y = 2.0126 and r = 0.9169. Find the LSRL, interpret the slope and r², and find the residual for the point (5.1, 4.1).

Step 1: Slope.

b = 0.9169 \times \dfrac{2.0126}{2.9586} = 0.6237

Step 2: Intercept.

a = 5.2833 - 0.6237(5.4333) = 1.894

. The line is

\hat{y} = 1.894 + 0.624x

Step 3: Interpret. For each increase of 1 in x, the predicted y increases by about 0.624. Since r² = 0.9169² = 0.841, about 84.1% of the variation in y is accounted for by the linear model with x.

Step 4: Residual. At

x = 5.1

\hat{y} = 1.894 + 0.6237(5.1) = 5.075

. The residual is

4.1 - 5.075 = -0.975

: the point is 0.975 below the line, so the model overpredicts it.

Check it in the simulator: press Moderate. The equation reads ŷ = 1.89 + 0.62x, with b₁ = 0.624, b₀ = 1.894, r² = 0.841 and SSR = 7.10. The point at x = 5.1 has a coral segment.

Extension with the Outlier preset: nine points lie almost on a line and one sits far below at (5, 0.5). The readout shows ŷ = 0.83 + 0.86x, r² = 0.651 and SSR = 23.73. Drag that point up to about y = 5.6, into the pattern. The slope stays 0.858, the intercept rises to about 1.34, r² goes to almost 1 and SSR falls close to 0. The outlier sits at x = 5, which is x̄, so it pulls the whole line down without tilting it. Drag an end point such as (9, 9) instead and the slope changes a lot, because points far from x̄ have more leverage.

Common mistakes on the AP exam

Reversing the residual. Residual = actual − predicted (y − ŷ). A negative residual means the line overpredicted.
Interpreting the slope as certain. Say "the predicted y increases by 0.624 for each 1-unit increase in x", and use the variable names from the problem.
Extrapolating. Predictions far outside the range of the x data are unreliable, and that includes the intercept when x = 0 is far from the data.
Trusting r² alone. A high r² does not show that a line is the right model. A curved pattern in the residual plot means a linear model is not appropriate.
Treating every outlier as influential. An influential point is one whose removal changes the slope or intercept noticeably. A point with an extreme x value has high leverage; an outlier near x̄ may change r² a lot but barely move the slope.
Swapping x and y. Regressing y on x does not give the same line as regressing x on y. Predict the response from the explanatory variable.

When the AP exam uses this

This is topic 5.5, Least-Squares Regression, in Unit 5 (Regression Analysis). Expect to write the equation from summary statistics or computer output, interpret the slope, intercept and r² in context, compute and interpret a residual, and judge a residual plot. Questions often give regression output from software and ask you to read the coefficients off it.

Least-Squares Regression Line Simulator

How to use the simulator

The formula

Worked example

Common mistakes on the AP exam

When the AP exam uses this

More AP Statistics simulators