Pearson Correlation Calculator

Correlation answers a single question: when one variable goes up, does the other tend to go up too, down, or neither? The Pearson product-moment coefficient r, introduced by Karl Pearson in 1895 building on Francis Galton's regression studies, is the workhorse measure. It expresses the covariance of x and y in standardized units, so its value is bounded between −1 (perfect negative linear relationship) and +1 (perfect positive). Pearson r assumes linearity, approximate bivariate normality (for the significance test and confidence interval), and equal variance across the range of x. When those assumptions hold, r is the most efficient estimator available.

Spearman ρ (Charles Spearman 1904) sidesteps the linearity and normality assumptions by working on the ranks of the data instead of the raw values. It captures any monotonic relationship (including curved ones) and is robust to outliers because extreme values are pulled in to the next rank rather than dominating the sum. Use Spearman when your data are ordinal (Likert scales, severity grades), when a scatter plot reveals a clear monotonic but non-linear pattern, or when outliers would otherwise distort Pearson. Kendall τ (Maurice Kendall 1938; τ-b for tied data) counts the proportion of concordant minus discordant pairs and tends to give smaller numerical values than Spearman for the same data; it is preferred for small samples and when many ties are expected, and underpins partial-correlation methods in survival and ordinal regression contexts.

All three coefficients are reported here from the same paired data so you can triangulate. A large discrepancy between Pearson and Spearman is informative: it flags non-linearity or outlier influence. If Pearson r is 0.40 but Spearman ρ is 0.75, the underlying relationship is monotonic but not linear, and reporting Pearson alone understates the association. Conversely, if Pearson r is 0.80 but ρ is 0.20, a few extreme points are inflating the linear correlation and the true association across the bulk of the data is weak. Anscombe (1973) famously constructed four datasets with identical Pearson r = 0.82 that look completely different on a scatter plot, which is why APA, CONSORT, and the Cochrane Handbook all require a scatter plot alongside any reported correlation.

For inference, convert r into a t-statistic using t = r × √[(n − 2) / (1 − r²)] with df = n − 2. The two-tailed p-value tests H0: ρ = 0. For the confidence interval, use Fisher's z transformation: z = 0.5 × ln[(1 + r) / (1 − r)] is approximately normal with SE = 1 / √(n − 3), so a 95% CI on z back-transforms to a CI on r via tanh. This is the method Hedges and Olkin (1985) standardized for meta-analytic pooling of correlations, and it is what the calculator reports. Two reminders matter: never treat a non-significant correlation as evidence of no association in small samples (the CI will be wide and contain meaningful effects), and always report r, ρ, and the 95% CI together rather than only the p-value.

Correlations feed naturally into other tools. To convert r into Cohen's d or another effect size for meta-analytic pooling, use the correlation to effect size converter. To plan sample size before collecting data, use the sample size calculator. To run a full regression with diagnostics rather than a single correlation, use the linear regression calculator. For full study design, robust alternatives, and APA-formatted reporting from a PhD statistician, our statistical analysis service covers everything end-to-end.

A correlation table in APA format lists each variable as a numbered row with its mean and standard deviation, then gives the correlations in the remaining columns, with significance marked by asterisks that are defined in the table note. In the text, r is reported with its degrees of freedom (n minus 2), the exact p value, and the confidence interval, all without leading zeros. Using this calculator's default example data (n = 10 pairs), the write-up reads:

There was a strong positive correlation between variable X (M = 15.90, SD = 6.17) and variable Y (M = 18.60, SD = 6.50), r(8) = .996, p < .001, 95% CI [.98, 1.00]. The coefficient of determination r² = .99 indicates that 99.1% of the variance is shared.

The APA Report button in the results panel downloads a Word document containing this write-up, an APA-ruled correlations table with the M and SD columns and significance asterisks already placed, a plain-English interpretation of r and r², and a reporting checklist. It handles the formatting details that get flagged in review: dropping the leading zero from r and p, giving degrees of freedom rather than n, and defining every asterisk used in the table note.