AP Stats Linear Regression Calculator

Enter paired x and y values to get the least-squares regression line, slope, intercept, and correlation r. This AP Stats Linear Regression Calculator is made for bivariate data, scatter plots, and LinReg questions on the AP Statistics exam.

What Is Linear Regression?

Linear regression fits a straight line, y-hat = a + bx, that best describes how a response variable y changes with an explanatory variable x. The slope b tells you how much y is predicted to change when x increases by one unit, and the intercept a is the predicted value of y when x is 0. The line is chosen by least squares, which makes the sum of the squared residuals as small as possible.

How to Use This Calculator

1. Type your explanatory variable values in the X box and your response values in the Y box, in the same order, separated by commas. 2. Press Calculate Regression Line. 3. Read the equation, slope, intercept, and r, then check the scatter plot with the fitted line. The point (x̄, ŷ) always lies on the line, which is a useful check.

How to Calculate Linear Regression

Formulas: slope b = Σ(x − x̄)(y − ŷ) / Σ(x − x̄)², intercept a = ŷ − b·x̄, and the line is ŷ = a + b·x. Correlation r = Σ(x − x̄)(y − ŷ) / √[Σ(x − x̄)² Σ(y − ŷ)²].

Worked example: For x = 1, 2, 3, 4, 5 and y = 2, 4, 5, 4, 7, x̄ = 3 and ŷ = 4.4. The slope is b = 11 / 10 = 1.1, the intercept is a = 4.4 − 1.1(3) = 1.1, so ŷ = 1.1 + 1.1x, and r ≈ 0.90. Try these exact values in the calculator to see the same result.

Tips and Common Mistakes

Do not mix up x and y: swapping them gives a different line. Always interpret slope in context, for example "for each additional unit of x, y increases by about b units on average." Do not extrapolate far beyond your x-values, and do not say correlation proves causation. Check residuals: a curved residual plot means a straight line is not the best model, even if r looks strong.

Frequently Asked Questions

What does the slope mean in AP Statistics?

The slope is the predicted change in y for a one-unit increase in x, on average, for the data in the study.

What does the intercept mean?

The intercept is the predicted y-value when x = 0. It only makes sense in context if x = 0 is inside or near the data range.

What is the difference between r and r-squared?

r measures the strength and direction of a linear relationship, from −1 to 1. r-squared is the proportion of variation in y explained by the linear model in x.

Can I use this calculator for LinRegTTest?

This calculator gives the line, slope, intercept, and r. A LinRegTTest also tests whether the population slope is zero; use its p-value for that inference question.

Why is my r close to 1 but the model still wrong?

Outliers and influential points can inflate r, and a strong r does not rule out a curved pattern. Always look at the scatter plot and residual plot.

Does the TI-84 give the same line?

Yes. On a TI-84, use STAT, CALC, LinReg(a+bx) with the same lists. The slope, intercept, and r should match this calculator.

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