Last updated: 2026-05-22

Definition

The Pearson correlation coefficient (r) is a dimensionless measure of the strength and direction of the linear relationship between two numerical variables, ranging from −1 (a perfect negative linear relationship) through 0 (no linear relationship) to +1 (a perfect positive linear relationship). From two paired datasets of equal length, this calculator returns r, the coefficient of determination (r²), and a classification of the strength.

Formula

$r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2 \cdot \sum_{i=1}^{n}(y_i - \bar{y})^2}}$

The numerator is the sum of cross-products of deviations — it measures whether X and Y tend to deviate from their respective means in the same direction (positive product) or opposite directions (negative product). The denominator normalizes by the total spread in each variable, so r always lands between −1 and +1 regardless of the scale of X or Y.

Worked example

Using the default dataset: X = [1, 2, 3, 4, 5] and Y = [2, 4, 5, 4, 5].

Step 1 — Compute the means.

$\bar{x} = \frac{1+2+3+4+5}{5} = 3 \qquad \bar{y} = \frac{2+4+5+4+5}{5} = 4$

Step 2 — Compute deviations and cross-products.

i	xᵢ	xᵢ − x̄	yᵢ	yᵢ − ȳ	(xᵢ−x̄)(yᵢ−ȳ)	(xᵢ−x̄)²	(yᵢ−ȳ)²
1	1	−2	2	−2	4	4	4
2	2	−1	4	0	0	1	0
3	3	0	5	+1	0	0	1
4	4	+1	4	0	0	1	0
5	5	+2	5	+1	2	4	1
		Σ			6	10	6

Step 3 — Apply the formula.

$r = \frac{6}{\sqrt{10 \times 6}} = \frac{6}{\sqrt{60}} \approx 0.7746$

$r^2 = (0.7746)^2 \approx 0.60$

Interpretation: r ≈ 0.77 — a strong positive correlation. About 60% of the variance in Y is explained by the linear relationship with X.

Interpreting r

r range	Interpretation
0.7 ≤ r ≤ 1.0	Strong positive (r = 1: perfect)
0.3 ≤ r < 0.7	Moderate positive
−0.3 < r < 0.3	Weak or no linear relationship
−0.7 < r ≤ −0.3	Moderate negative
−1.0 ≤ r ≤ −0.7	Strong negative (r = −1: perfect)

These thresholds are guidelines, not rules. In epidemiology, r = 0.3 can be practically meaningful; in engineering, r < 0.99 might be unacceptably noisy.

r vs r²

r and r² answer different questions:

• r tells you the direction and strength of the linear association. A value of −0.9 is just as strong as +0.9; the sign tells you which way the relationship runs.
• r² (coefficient of determination) tells you the proportion of variance in Y explained by X. It ranges from 0 to 1 and has no sign. r² = 0.64 means 64% of the scatter in Y can be attributed to the linear trend with X.

In short, r answers "how tight is the linear fit?" and r² answers "how much of the variance in Y does X account for?".

Correlation and causation

A high r only means X and Y move together linearly — it says nothing about why. Common examples:

• Ice cream sales and drowning rates both peak in summer (common cause: hot weather).
• Shoe size and reading ability are positively correlated in children (common cause: age).
• Countries with more TVs per household have higher life expectancies (both reflect wealth).

Establishing causation requires a controlled experiment, ruling out confounders, and demonstrating that the cause precedes the effect in time.

Pearson versus Spearman

The Pearson r assumes:

1. Both variables are continuous (interval or ratio scale).
2. The relationship is approximately linear.
3. No extreme outliers dominate the sum.

When these assumptions break down, the Spearman rank correlation is the appropriate alternative: it converts both datasets to ranks and computes the Pearson r on the ranks. Spearman handles ordinal data, monotone-but-curved relationships, and datasets with outliers much better than Pearson. As a rough guide, a scatter plot showing a consistent pattern that is clearly curved, or one or two points sitting far from the rest, points to Spearman rather than Pearson.

Frequently Asked Questions (FAQ)

What does r = 0.7 mean?

An r of 0.7 indicates a strong positive linear relationship between X and Y. As X increases, Y tends to increase too, and the relationship is fairly tight around a straight line.

In practice, |r| ≥ 0.7 is commonly treated as strong, 0.3 ≤ |r| < 0.7 as moderate, and |r| < 0.3 as weak — though the exact thresholds depend on the field. In the social sciences, r = 0.5 is often considered strong; in physics, r < 0.99 may indicate significant noise.

Does a high correlation prove causation?

No. A high r means X and Y move together linearly, but it does not explain why. A third variable (a confounder) might drive both. Ice cream sales and drowning rates correlate strongly — both rise in summer — but ice cream does not cause drowning. Establishing causation requires a controlled experiment, temporal ordering (cause precedes effect), and ruling out confounders. Correlation is evidence of a relationship, not its mechanism.

What is the difference between r and r²?

r (Pearson r) measures the direction and strength of the linear relationship, ranging from −1 (perfect negative) to +1 (perfect positive). r² (the coefficient of determination) tells you the proportion of variance in Y explained by X.

For example, r = 0.8 gives r² = 0.64, meaning 64% of the variation in Y is accounted for by the linear relationship with X. r² is always non-negative and easier to interpret as a percentage, but it loses sign information — r² alone does not reveal whether the relationship is positive or negative.

When should I use Spearman instead of Pearson?

Use the Spearman rank correlation when your data is ordinal (ranked categories), when the relationship between X and Y is monotone but not linear, or when the data contains significant outliers. Spearman converts raw values to ranks first, making it robust to extreme values and non-linear but consistently increasing (or decreasing) patterns.

Pearson is optimal when both variables are continuous, approximately normally distributed, and the underlying relationship is linear. As a rule of thumb: if a scatter plot shows a curved but consistent pattern, or if a few outliers are driving the r value, try Spearman.