AP Stats R-Squared Calculator
R-squared (r²) is the proportion of the variation in y that is explained by the linear relationship with x. If r² = 0.64, about 64% of the variation in y is explained by the least-squares line — and 36% is left in the residuals.
R-Squared Calculator
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What Is R-Squared?
R-squared is the proportion of the variation in the response variable that is explained by the regression model. An r-squared of 0.72 means the model explains about 72% of the variation in y. R-squared is the square of the correlation coefficient r for a simple linear regression, and it is always between 0 and 1.
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Example: r = 0.80, so r-squared = 0.64 — about 64% of the variation in y is explained.
How to Use This Calculator
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Step 1: Open the AP Stats R-Squared Calculator calculator at the top and enter the values it asks for.
1. Enter the x list and the y list, in the same order, separated by commas. 2. Make sure both lists have the same count and at least 3 pairs. 3. Press Calculate. 4. Read r, r², the regression equation, and the scatter plot with the least-squares line.
How to Calculate R-Squared
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First find the correlation r: r = Σ(x − x̄)(y − ȳ) / √[Σ(x − x̄)² × Σ(y − ȳ)²]. Then square it: r² = r × r. The slope is b = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² and the intercept is a = ȳ − b × x̄.
Worked example: x = 0, 1, 2, 3, 4 and y = 1, 3, 2, 5, 4. The means are x̄ = 2 and ȳ = 3. Then Σ(x − x̄)(y − ȳ) = 8, Σ(x − x̄)² = 10, and Σ(y − ȳ)² = 10, so r = 8 / √(10 × 10) = 0.80. Therefore r² = 0.80² = 0.64: about 64% of the variation in y is explained by the linear relationship with x. The line is ŷ = 1.4 + 0.8x.
Tips and Common Mistakes
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Mistake: Saying R-squared is the correlation.
- R-squared is always between 0 and 1, even when r is negative, because squaring removes the sign. Use r itself to report direction.
- A high r² does not prove causation. A strong linear pattern can still come from a lurking variable.
- Do not report r² as a percentage of points on the line. It is a share of variation, not a share of points.
- Check the scatter plot and residuals. A curved pattern can have a misleading r² if you only fit a straight line.
- Outliers can inflate or deflate r², so always look at the plot before trusting the number.
Frequently Asked Questions
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What does r² = 0 mean?
It means the least-squares line explains none of the variation in y beyond the mean. The line is flat, and knowing x does not help predict y linearly.
What does r² = 1 mean?
It means all points fall exactly on the least-squares line, so the line explains 100% of the variation in y. This is rare with real data.
Is r² the same as r?
No. r measures the strength and direction of a linear relationship, from −1 to 1. r² is r squared, from 0 to 1, and measures the proportion of variation explained.
Can r² be negative?
No, for a least-squares line r² cannot be negative, because it is a square. A negative value usually means a calculation or software error.
Does a higher r² always mean a better model?
Not always. Adding variables or fitting a curve can raise r² without making predictions better, and a high r² with a curved pattern or influential outliers can still be misleading.
How do I interpret r² in a sentence?
For example: “About 64% of the variation in y is explained by the least-squares regression line using x.” Always name both variables in context.
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R-Squared Calculator → Linear Regression Calculator → Correlation Coefficient Calculator