What is the correlation coefficient?

The Pearson correlation coefficient (r) measures the strength and direction of the linear relationship between two variables. It ranges from −1 to +1: r = +1 means a perfect positive linear relationship (as X increases, Y increases proportionally), r = −1 means a perfect negative linear relationship, and r = 0 means no linear relationship. For example, height and weight have a correlation of about r = 0.7, indicating a strong positive relationship — taller people tend to weigh more.

How do you calculate Pearson's r?

Pearson's r = [n·Σxy − Σx·Σy] / √[(n·Σx² − (Σx)²)(n·Σy² − (Σy)²)]. In words: compute the sum of the products of each (x, y) pair, subtract the product of the sums, then divide by the square root of the product of the sum-of-squares for x and y separately. Most statistical software and our calculator handle this automatically. With at least 2 data pairs entered in the X and Y fields, the calculator computes r, r², and the interpretation.

What does r = 0.8 mean?

An r of 0.8 indicates a strong positive linear relationship between the two variables. As X increases, Y tends to increase in a roughly linear pattern, and that pattern is consistent (not too scattered). r² = 0.64, meaning 64% of the variation in Y is explained by the linear relationship with X. Common interpretation guidelines: |r| ≥ 0.9 = very strong; 0.7–0.9 = strong; 0.5–0.7 = moderate; 0.3–0.5 = weak; < 0.3 = little to no linear relationship. These are rough guidelines — context matters.

r² (the coefficient of determination) is the square of the correlation coefficient. It represents the proportion of variance in Y that is explained by a linear relationship with X. For example, if r = 0.7, then r² = 0.49 — meaning 49% of the variation in Y is explained by X, and 51% is due to other factors. r² is always between 0 and 1 and is often reported as a percentage. In regression analysis, r² is the primary measure of how well the model fits the data.

What is the difference between correlation and causation?

Correlation measures statistical association — whether two variables tend to move together. Causation means one variable directly causes changes in the other. A high correlation does not imply causation. Classic examples: ice cream sales and drowning rates are highly correlated (both increase in summer), but ice cream does not cause drowning — temperature causes both. Establishing causation requires controlled experiments or rigorous causal inference methods, not just observational correlation.

What is a strong correlation?

A strong correlation is generally considered |r| ≥ 0.7, though this depends on the field. In physics or engineering, correlations below 0.95 might be considered weak. In social sciences, where human behavior is complex, r = 0.4 might be considered a meaningful result. The American Psychological Association uses these rough guidelines: small (r ≈ 0.10), medium (r ≈ 0.30), large (r ≈ 0.50). Always interpret r in the context of your field and the complexity of the variables involved.

What are real-world examples of correlation?

Height and weight: r ≈ 0.7 (strong positive). Hours studied and exam score: r ≈ 0.5–0.6 (moderate positive). Temperature and hot coffee sales: r ≈ −0.7 (strong negative). Ice cream sales and drowning rates: r ≈ 0.85 (strong positive — but this is a spurious correlation driven by summer temperature causing both). Always ask whether a third variable (a "confound") could explain the correlation before drawing conclusions.

What does R² of 0.75 mean?

An R² of 0.75 means that 75% of the variation in your Y variable is explained by its linear relationship with X. The remaining 25% of variation is due to other factors not captured in the model. In practical terms: if you are predicting house prices from square footage and R² = 0.75, your model explains three-quarters of the price variability — a solid fit. For R² below 0.5 (only 50% explained), predictions would be quite imprecise.

What is the difference between Pearson r and Spearman correlation?

Pearson r measures the strength of a linear relationship between two continuous variables and is sensitive to outliers. Spearman's ρ (rho) measures the monotonic relationship between ranked variables — it works on ordinal data and is robust to outliers and non-linear (but still monotonic) relationships. Use Spearman when your data is ranked, ordinal, or clearly non-normal. Both coefficients range from −1 to +1 with similar interpretation guidelines.

Correlation Coefficient Calculator

Calculates Pearson r and r² from any set of (x, y) data pairs — shows correlation strength and coefficient of determination.

How to Use the Correlation Coefficient Calculator

This correlation coefficient calculator finds Pearson r and r² from any dataset — enter your X values in the left text area and your corresponding Y values in the right text area, one value per line or separated by commas. The number of X and Y values must be equal. The calculator instantly computes Pearson r, r², the means of X and Y, and a plain-language interpretation of the correlation strength. With at least 2 pairs of values, all results appear automatically as you type.

Both positive and negative values are supported, and there is no limit on sample size. If all X values are identical or all Y values are identical, r is undefined (division by zero in the formula) and the calculator will display an error. For descriptive statistics about your datasets, see our standard deviation calculator. Once you know the correlation is strong, use our linear regression calculator to find the best-fit equation and make predictions.

The Pearson Correlation Formula

The Pearson correlation coefficient measures the linear association between two variables:

r = [n·Σxy − Σx·Σy] / √[(n·Σx² − (Σx)²)(n·Σy² − (Σy)²)]

An equivalent formula using means is:

r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √[Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²]

Worked Example

Given five (x, y) pairs: (1, 2), (2, 4), (3, 5), (4, 4), (5, 5):

n = 5, Σx = 15, Σy = 20, Σxy = 66, Σx² = 55, Σy² = 86
Numerator: 5 × 66 − 15 × 20 = 330 − 300 = 30
Denominator: √[(5 × 55 − 225)(5 × 86 − 400)] = √[(275 − 225)(430 − 400)] = √[50 × 30] = √1500 ≈ 38.73
r = 30 / 38.73 ≈ 0.7746

r ≈ 0.77 indicates a strong positive linear relationship, and r² ≈ 0.60 means 60% of the variation in Y is explained by X.

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Interpreting the Correlation Coefficient

The value of r tells you two things simultaneously: the direction and the strength of the linear relationship.

Direction: Positive r means Y tends to increase as X increases; negative r means Y tends to decrease as X increases.
Strength: |r| close to 1 means the data points fall nearly on a straight line; |r| close to 0 means the data is scattered with no linear pattern.

General guidelines for |r| (these vary by field):

0.90 – 1.00: Very strong relationship
0.70 – 0.89: Strong relationship
0.50 – 0.69: Moderate relationship
0.30 – 0.49: Weak relationship
0.00 – 0.29: Negligible or no linear relationship

Note that Pearson r only measures linear relationships. Two variables can have a strong curved (non-linear) relationship and still show r ≈ 0. Always plot your data in a scatter plot before relying on r as the sole measure of association.

The Coefficient of Determination (r²)

Squaring the correlation coefficient gives r², the coefficient of determination. This has a direct interpretation: it is the proportion of variance in Y that is explained by a linear relationship with X.

r = 0.9 → r² = 0.81 → 81% of variance in Y is explained by X
r = 0.7 → r² = 0.49 → 49% explained
r = 0.5 → r² = 0.25 → 25% explained
r = 0.3 → r² = 0.09 → only 9% explained

The remaining percentage (1 − r²) is attributable to other variables, measurement error, or random variation. r² is the primary output of simple linear regression and is always reported alongside the regression equation in scientific literature.

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Limitations and Common Mistakes

Pearson r has several important limitations to keep in mind:

Only linear relationships: A perfect U-shaped relationship between X and Y would give r ≈ 0, even though a clear pattern exists. Use a scatter plot to check for non-linearity.
Sensitive to outliers: A single extreme data point can dramatically change r. Always check your data for errors before interpreting the correlation.
Correlation ≠ causation: A high r tells you the variables move together, not that one causes the other. See the FAQ below for details.
Not meaningful for non-numeric data: Pearson r requires continuous (or at least interval-scale) data. For ranked data, use Spearman's ρ instead.
Sample size matters: With n = 5, even r = 0.8 may not be statistically significant. With n = 1000, even r = 0.1 may be statistically significant. Always report the sample size alongside r.

For comparing variability within each dataset, our variance calculator shows the individual spread of X and Y values.

Sources & References

Pearson's Correlation — NIST/SEMATECH e-Handbook of Statistical Methods
Correlation Coefficients — Khan Academy — Khan Academy

Correlation Coefficient Calculator