Correlation Coefficient Calculator

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how to interpret Pearson correlation

How to Interpret the Pearson Correlation

This calculator classifies |r| into 5 fixed bands: below 0.3 is negligible, 0.3 to 0.7 is weak, 0.7 to 0.9 is moderate, 0.9 to 1.0 is strong, and exactly 1.0 is perfect. These are the bands, not some generic textbook scale, that decide the label shown next to the result.

This calculator's real bands, and why r² is not linear

An r of -0.86 between weekly exercise hours and resting heart rate falls into this tool's 'moderate' band (0.7 to 0.9), not the 'strong' one, because the strong cutoff only starts at 0.9. That differs from looser academic scales that sometimes call any |r| above 0.7 strong: the classification used here is more conservative, and it is worth checking the exact r value before repeating another source's label.
r² measures the fraction of one variable's variance explained by the other, and it grows non-linearly with r: in the study-hours-and-exam-score example, r = 0.995038 gives r² = 0.990100, meaning 99% of the score's variation is associated with hours studied. In the shoe-size-and-vocabulary-score pair, r = 0.109109 gives r² of only 0.011905, just over 1%: a 9-fold drop in r pushes r² down to less than 1% of its previous value, because r² is the square, not half.
A single outlying point can distort r drastically, because the formula sums the product of each point's deviation from the mean: one extreme value pulls the entire sum to one side. The tool also clamps r to the [-1, 1] interval internally, a guard against floating-point rounding itself, which would sometimes compute 1.0000000002. And a high correlation never proves causation: ice cream sales and drowning incidents can reach r = 0.999208 in the same summer without one causing the other, since heat is the shared cause behind both.

Examples

Strong positive

Input: horas de estudo × nota da prova
Expected output: r = 0,995038

r² = 0.990100: 99% of the score's variation tracks the hours studied, within this calculator's 'strong' band (0.9 to 1.0).

Moderate negative

Input: horas de exercício × frequência cardíaca
Expected output: r = -0,859749

Falls into this tool's 'moderate' band (0.7 to 0.9), not 'strong': the strong cutoff only starts at 0.9.

Negligible

Input: tamanho de calçado × nota de vocabulário
Expected output: r = 0,109109

r² = 0.011905, just over 1% of shared variance: below 0.3, the tool classifies it as negligible even without the value being exactly zero.

Full tool FAQ

What is the Pearson correlation coefficient?

Pearson's r measures the strength and direction of the linear relationship between two variables. It ranges from −1 to +1: values near ±1 indicate strong correlation; near 0 indicate weak or absent correlation.

What is the difference between r and r²?

Does correlation imply causation?

What are the interpretation thresholds for r?

Is Pearson's r affected by outliers?

How many data points does the tool support?

What does zero variance mean?

Is data sent to the server?

Frequently asked questions

Does r = 0 mean statistical independence?

Not necessarily. r measures only linear relationship; a perfect curve like y = x² can have r close to 0 even with x and y fully dependent on each other, because the relationship is not a straight line. The tool computes r with the Pearson formula, which does not see that kind of pattern.

Why does an r of 0.86 show up as 'moderate' and not 'strong'?

Because this calculator's bands set the 'strong' cutoff at 0.9, not at 0.7 or 0.8 like some looser references use. Between 0.7 and 0.9 the label is always 'moderate', so 0.86 lands in that band even though it is a relatively high r value.

Does a high correlation prove that one variable causes the other?

No. r measures only statistical association, never causation: the classic ice cream and drowning example reaches r = 0.999208 because both rise in summer, not because one triggers the other. Ruling causation in or out requires experimental design or controlling for variables, something the Pearson coefficient alone does not provide.

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how to interpret Pearson correlation

How to Interpret the Pearson Correlation

This calculator's real bands, and why r² is not linear

An r of -0.86 between weekly exercise hours and resting heart rate falls into this tool's 'moderate' band (0.7 to 0.9), not the 'strong' one, because the strong cutoff only starts at 0.9. That differs from looser academic scales that sometimes call any |r| above 0.7 strong: the classification used here is more conservative, and it is worth checking the exact r value before repeating another source's label.

r² measures the fraction of one variable's variance explained by the other, and it grows non-linearly with r: in the study-hours-and-exam-score example, r = 0.995038 gives r² = 0.990100, meaning 99% of the score's variation is associated with hours studied. In the shoe-size-and-vocabulary-score pair, r = 0.109109 gives r² of only 0.011905, just over 1%: a 9-fold drop in r pushes r² down to less than 1% of its previous value, because r² is the square, not half.

A single outlying point can distort r drastically, because the formula sums the product of each point's deviation from the mean: one extreme value pulls the entire sum to one side. The tool also clamps r to the [-1, 1] interval internally, a guard against floating-point rounding itself, which would sometimes compute 1.0000000002. And a high correlation never proves causation: ice cream sales and drowning incidents can reach r = 0.999208 in the same summer without one causing the other, since heat is the shared cause behind both.

Examples

Strong positive

Input: horas de estudo × nota da prova
Expected output: r = 0,995038

r² = 0.990100: 99% of the score's variation tracks the hours studied, within this calculator's 'strong' band (0.9 to 1.0).

Moderate negative

Input: horas de exercício × frequência cardíaca
Expected output: r = -0,859749

Falls into this tool's 'moderate' band (0.7 to 0.9), not 'strong': the strong cutoff only starts at 0.9.

Negligible

Input: tamanho de calçado × nota de vocabulário
Expected output: r = 0,109109

r² = 0.011905, just over 1% of shared variance: below 0.3, the tool classifies it as negligible even without the value being exactly zero.

Full tool FAQ

What is the Pearson correlation coefficient?

What is the difference between r and r²?

Does correlation imply causation?

What are the interpretation thresholds for r?

Is Pearson's r affected by outliers?

How many data points does the tool support?

What does zero variance mean?

Is data sent to the server?

Frequently asked questions