AP Stats Correlation Coefficient Calculator

The correlation coefficient r measures how strong and in which direction a linear relationship is. It is always between −1 and 1: near 1 is a strong positive line, near −1 is a strong negative line, and near 0 is no linear relationship.

What Is the Correlation Coefficient?

The correlation coefficient, r, measures the strength and direction of the linear relationship between two quantitative variables. It is always between -1 and 1: values near 1 mean a strong positive linear pattern, values near -1 mean a strong negative one, and values near 0 mean no linear pattern. Correlation does not imply causation, a strong correlation can come from a lurking variable.

How to Use This Calculator

Enter x values in order, then the matching y value for each x, in the same order. The lists must have the same length and at least 3 pairs for a meaningful result. The scatter plot below the result shows your points and the least-squares line, so you can see whether a straight line fits at all.

How to Calculate the Correlation Coefficient

Formula: r = Σ(x − x̄)(y − ȳ) / √[ Σ(x − x̄)² × Σ(y − ȳ)² ].

Worked example: x = 1, 2, 3, 4, 5 (x̄ = 3) and y = 2, 4, 5, 4, 5 (ȳ = 4). The numerator is (−2)(−2) + (−1)(0) + (0)(1) + (1)(0) + (2)(1) = 4 + 0 + 0 + 0 + 2 = 6. The denominator is √[(4+1+0+1+4) × (4+0+1+0+1)] = √(10 × 6) = √60 ≈ 7.746. So r = 6 / 7.746 ≈ 0.775, a strong positive linear relationship, with r² ≈ 0.60 — about 60% of the variation in y is explained by the linear model.

Tips and Common Mistakes

• Order matters: pair the first x with the first y, the second with the second, and so on.
• r only measures linear association. A perfect curve can still have r near 0 — always look at the scatter plot.
• Outliers can change r a lot. One far point can make a weak relationship look strong, or vice versa.
• Correlation is not causation: a high r never proves that x causes y.
• Do not extrapolate: r describes the data you have, not values far outside its x-range.

Frequently Asked Questions

What counts as a strong correlation?

As a rough guide used in many AP Statistics classes: |r| below 0.3 is weak, 0.3 to 0.7 is moderate, and above 0.7 is strong. Always report the value, not just the label.

Can r be greater than 1?

No. r is always between −1 and 1. A result outside that range means a calculation error.

What does a negative r mean?

As x increases, y tends to decrease. For example, more time spent on a phone might be associated with fewer hours of sleep.

What is r-squared?

r² is the proportion of variation in y explained by the least-squares regression line on x. For r = 0.775, r² ≈ 0.60, or 60%.

Why do I need at least 3 pairs?

With only 2 points, a line always fits perfectly (r = ±1), so it tells you nothing about a real relationship.

Does correlation need units?

No — r has no units, which is why you can compare correlations across different kinds of data.

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