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Correlation Coefficient Calculator

Calculate Pearson correlation coefficient (r), Spearman rank correlation (ρ), or Kendall’s tau (τ) from raw paired data or from summary stats. Get p-value (one- or two-tailed), confidence interval (Pearson via Fisher z), r², covariance, and the simple linear regression line.

Pearson r Spearman ρ Kendall τ p-value & CI
Note: Correlation describes association, not causation. P-values and confidence intervals rely on assumptions and can be sensitive to outliers.

Calculate Correlation

Choose a method and input format. You can paste data, enter pairs in a table, or compute Pearson r from summary totals.

Pearson r measures linear association.
Row X Y Remove
Spearman’s ρ is Pearson correlation computed on ranks. It is commonly used for ordinal data or for monotonic relationships. This tool uses average ranks for ties. The p-value uses a t-approximation (best for moderate/large n).
Good when outliers or non-linearity affect Pearson.
Handles ties via average rank.
Kendall’s τ measures concordance of pairs (how often rankings agree). It is useful for ordinal data and is often more conservative than Spearman. This tool computes τ-b (tie-adjusted). The p-value uses a normal approximation (best for larger n).
τ-b adjusts for ties in X and/or Y.
O(n²) pairs; very large n may be slow.
Use this mode when you don’t have raw data but you do have totals: n, Σx, Σy, Σx², Σy², Σxy. The calculator computes Pearson r, regression slope/intercept, and the correlation test p-value.

What a Correlation Coefficient Measures

A Correlation Coefficient Calculator helps you quantify how two variables move together. Instead of relying on visual inspection alone (“it looks like they trend together”), a correlation coefficient reduces the relationship to a single number. Depending on which coefficient you choose, correlation can describe a linear relationship, a monotonic relationship (consistently increasing or decreasing), or rank agreement.

Correlation coefficients generally range from −1 to +1. A value near +1 means that higher X values tend to go with higher Y values. A value near −1 means that higher X values tend to go with lower Y values. A value near 0 means little or no association in the sense measured by that coefficient.

Pearson Correlation Coefficient r

Pearson’s r is the most common correlation coefficient. It measures the strength and direction of a linear relationship. If your data follow an approximately straight-line pattern (even with scatter), Pearson correlation is usually appropriate. Pearson r is based on covariance and standard deviations:

r = cov(X, Y) / (sX · sY)

Where cov(X, Y) is the sample covariance and sX, sY are the sample standard deviations. Pearson r is sensitive to outliers because a single extreme point can strongly affect the covariance and the fitted line. If you suspect outliers or non-linear patterns, you may want to compare Pearson with Spearman ρ or Kendall τ.

Spearman Rank Correlation ρ

Spearman’s rho (ρ) is Pearson correlation computed on ranked data. It measures whether two variables have a monotonic relationship (as one increases, the other tends to increase or decrease consistently). Spearman is often used when:

  • Your variables are ordinal (ranked categories).
  • The relationship is non-linear but still monotonic (curved trend that never reverses direction).
  • Outliers distort Pearson but not the rank ordering as much.

This calculator assigns average ranks when there are ties, then computes correlation on those ranks. The displayed p-value is an approximation (common in practice) and becomes more reliable as sample size increases.

Kendall’s Tau τ

Kendall’s tau (τ) focuses on how often pairs of observations agree in rank. Consider any two observations (i and j). If both X and Y move in the same direction, the pair is concordant. If they move in opposite directions, the pair is discordant. Kendall τ compares these counts.

This tool computes τ-b, which adjusts for ties in X and/or Y. Kendall τ is frequently preferred for:

  • Small datasets where rank agreement is more interpretable than linear correlation.
  • Ordinal scales with many ties.
  • Situations where you want a conservative, robust measure of association.

Interpreting Correlation Strength

There is no universal rule for “strong” versus “weak” correlation because context matters. In some scientific fields, r = 0.30 might be meaningful; in others, it may be considered small. That said, many people use rough guidelines based on absolute value |r|:

|Correlation| (absolute value) Common interpretation Practical note
0.00 to 0.19 Very weak Often hard to distinguish from noise without large samples.
0.20 to 0.39 Weak May still matter if the outcome is important or costs are low.
0.40 to 0.59 Moderate Common in social/behavioral data where many factors contribute.
0.60 to 0.79 Strong Often suggests meaningful association, but check outliers and non-linearity.
0.80 to 1.00 Very strong May indicate close coupling, shared drivers, or even duplicated measurement.

What r² Means

For Pearson correlation, r² is the squared correlation. In simple linear regression, r² is the proportion of variance in Y explained by X using a straight-line model. For example, r = 0.70 implies r² = 0.49, meaning about 49% of the variability in Y is explained by a linear relationship with X. The remaining variability comes from other factors, randomness, measurement error, or non-linear structure.

Correlation p-value and Hypothesis Testing

This calculator reports a p-value for correlation by testing the null hypothesis that the population correlation is zero. For Pearson correlation, a common test statistic is:

t = r · √((n − 2) / (1 − r²)),   df = n − 2

You can choose a two-tailed test (the usual default) or a one-tailed test if you have a justified directional hypothesis. A small p-value indicates that the observed correlation is unlikely under the null model—assuming independence and the usual model conditions. Importantly, a “significant” p-value does not guarantee the relationship is large or useful; that’s why reporting r and a confidence interval is valuable.

Confidence Interval for Pearson r

For Pearson r, this Correlation Coefficient Calculator provides a confidence interval using Fisher’s z transform. Fisher’s transform stabilizes variance of the correlation estimate:

z = atanh(r) = 0.5 · ln((1 + r)/(1 − r)),   SE = 1/√(n − 3)

The interval is computed in z space and then transformed back with tanh(). This approach is widely used and performs well for moderate n, but like all intervals it depends on assumptions. If your data are extremely non-normal, contain strong outliers, or the relationship is non-linear, interpret the interval cautiously.

Regression Line from Correlation

When you compute Pearson r from raw data or summary totals, the calculator also reports the simple linear regression line:

ŷ = b0 + b1x

The slope b1 reflects how much Y changes per unit change in X (in your measurement units), while b0 is the intercept. Correlation and regression are closely related in the two-variable case, but regression focuses on prediction (Y from X) while correlation focuses on symmetric association.

Common Pitfalls: Correlation vs Causation

One of the most important interpretive rules is: correlation does not imply causation. Two variables can correlate for several reasons:

  • Confounding: a third variable influences both X and Y.
  • Reverse causality: Y influences X rather than X influencing Y.
  • Shared trend or seasonality: both move together over time even without a direct link.
  • Selection bias: the data you observe are not representative of the true population.

Use correlation as a descriptive and diagnostic tool, not as proof of cause. If causality is the goal, consider experimental design, causal inference methods, or carefully controlled observational studies.

How to Use This Correlation Coefficient Calculator

Quick workflow

  1. Choose Pearson r for linear numeric relationships, Spearman ρ for monotonic or ordinal data, or Kendall τ for rank agreement.
  2. Enter data as pairs or lists, or use the summary mode if you only have totals.
  3. Pick α and one- or two-tailed testing based on your hypothesis.
  4. Review r (or ρ/τ), r² (Pearson), the p-value, and the confidence interval.
  5. If results are surprising, check for outliers, non-linearity, or data entry mistakes.

FAQ

Correlation Coefficient Calculator FAQs

Answers about Pearson r, Spearman ρ, Kendall τ, p-values, confidence intervals, and interpretation.

A correlation coefficient is a number that summarizes the strength and direction of the relationship between two variables. Common types include Pearson r (linear), Spearman ρ (rank/monotonic), and Kendall τ (rank concordance).

Pearson measures linear association using the original values. Spearman uses ranks and is better for monotonic relationships or ordinal data. Kendall τ compares concordant vs discordant pairs and is robust for ranks and small samples.

No. A strong correlation can exist without a causal relationship. Confounding variables, reverse causality, or shared trends can create correlation without direct cause.

The p-value tests the null hypothesis that the true correlation is zero. A small p-value suggests the observed correlation is unlikely under the null, given the model assumptions.

r² (the coefficient of determination) is the proportion of variance in Y explained by a linear relationship with X (in simple linear regression). It is r squared for Pearson correlation.

For Pearson r, a common confidence interval uses Fisher’s z transform: z = atanh(r), with standard error 1/sqrt(n−3), then transforms back with tanh().

Yes. If you know n, Σx, Σy, Σx², Σy², and Σxy, you can compute Pearson r and the regression slope/intercept without the raw data.

Correlation is undefined if X or Y has zero variance (all values equal). This calculator will show an error or “undefined” if that occurs.

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