Correlation Coefficient Calculator

Pearson's correlation coefficient r summarizes how closely paired data follow a straight line. It runs from −1 (a perfect downhill line) through 0 (no linear pattern) to +1 (a perfect uphill line), and it does not change when either variable is rescaled or shifted, so heights in centimeters and inches give the same r. Squaring it gives R², the share of the variation in Y that a straight line on X explains.

r describes a linear pattern only. A curve such as a perfect U can give r near 0, and a single extreme point can create or hide a correlation, so always look at the scatter plot. To predict Y from X, use the line of best fit calculator or the linear regression calculator; for a rank-based measure that tolerates outliers and curved monotonic trends, use the Spearman correlation calculator.

When the two variables are items of one questionnaire scale, the correlations between all pairs of items are what Cronbach's alpha summarizes: the Cronbach's alpha calculator reports alpha for the whole scale together with the correlation of each item with the rest.

The calculator adds up deviations from the means with compensated summation and evaluates the t and normal tail probabilities directly, so results stay accurate for data far from zero, such as years, and for r very close to ±1.

These bands are conventions, not tests. Cohen (1988) suggested 0.1, 0.3 and 0.5 as small, medium and large effects for behavioral research, and physics or engineering data routinely reach 0.99. The sign gives the direction: positive means Y tends to rise with X, negative means it tends to fall.

The p-value answers a narrower question than the size of r: if the true correlation ρ were 0, how often would a sample of this size give an |r| at least this large? It depends heavily on n, as the smallest significant |r| shows:

Values are two-sided critical values of |r| = t*/√(df + t*²). With only a handful of pairs, even a large r can be a chance result, and with thousands of pairs a trivial r becomes significant, so report r with its interval rather than the p-value alone.

Five students study X = 1, 2, 3, 4, 5 hours and earn quiz scores Y = 2, 4, 5, 4, 5. Load example fills in these numbers.

1. Means and sums of squares: x̄ = 3, ȳ = 4, Sxx = 10, Syy = 6, Sxy = 6.
2. Correlation: r = 6 ÷ √(10 × 6) = 6 ÷ 7.7460 = 0.7746, so R² = 0.6.
3. Covariance: 6 ÷ 5 = 1.2 for the population and 6 ÷ 4 = 1.5 for a sample.
4. Test: t = 0.7746 × √(3 ÷ 0.4) = 2.1213 on 3 degrees of freedom, so p = 0.124027.
5. Interval: z′ = atanh(0.7746) = 1.0317, SE = 1 ÷ √2 = 0.7071, z* = 1.96, so the 95% interval for ρ is tanh(1.0317 ± 1.3859) = (−0.3401, 0.9842).

Interpretation: r = 0.77 is a fairly strong positive linear pattern, and 60% of the variation in scores goes with study time. But with five students p = 0.124 is above 0.05 and the interval includes 0, so the data do not rule out no correlation at all; the table above shows that n = 5 needs |r| ≥ 0.8783.

• Two numeric variables measured on the same units. Each X must be paired with the Y of the same student, day or item; the lists need the same length.
• A linear pattern. Check the scatter plot for curves before relying on r.
• Few outliers. Pearson's r uses every distance from the mean, so one extreme point can move it a long way; the Spearman correlation is more resistant.
• Roughly bivariate normal data for the p-value and the Fisher z interval when n is small. The coefficient itself needs no normality.
• Restricted range. Looking at only part of the range of X, such as top students only, makes r smaller than in the full population.
• No causal claim. A strong r can come from a third variable or from chance (see correlation vs causation).

For the covariance on its own, with a choice of sample or population, use the covariance calculator; to split R² into SSR, SSE and SST, use the R-squared calculator.