When should I use correlation analysis?

Use it to measure the strength and direction of a linear relationship between two continuous variables. For example, the relationship between study hours and exam scores.

What's the difference between Pearson and Spearman correlation?

Pearson measures linear relationships and assumes normality. Spearman is rank-based, can detect non-linear monotonic relationships, and doesn't require normality.

How do I interpret the correlation coefficient r?

|r| < 0.3 is weak, 0.3-0.7 is moderate, and 0.7+ is strong. Positive r means both variables increase together; negative r means one increases as the other decreases.

Does correlation imply causation?

No. Correlation only shows that two variables tend to change together. It does not prove that one variable causes the other. A third variable may be involved.

Correlation Calculator

Calculate Pearson r or Spearman rho between two variables. Results include scatter plot, p-value, and APA-formatted output.

What is Correlation?

Correlation is a statistical measure that quantifies the strength and direction of the relationship between two variables. The correlation coefficient ranges from -1 (perfect negative relationship) to +1 (perfect positive relationship), with 0 indicating no linear relationship. Correlation analysis is one of the most widely used techniques in psychology, education, medicine, economics, and the social sciences.

The concept of correlation was pioneered by Sir Francis Galton in the 1880s during his studies of heredity and regression toward the mean. His work was formalized by Karl Pearson, who developed the product-moment correlation coefficient (Pearson's r) in 1896, providing the mathematical foundation still used today. In 1904, Charles Spearman introduced the rank-order correlation coefficient (Spearman's rho), a non-parametric alternative designed for ordinal data and monotonic relationships. Together, these two measures form the backbone of modern bivariate correlation analysis.

Pearson Correlation (r)

Pearson's r measures the strength of the linear relationship between two continuous variables. It is calculated as the covariance of the two variables divided by the product of their standard deviations. Use Pearson when both variables are measured on interval or ratio scales, the relationship is approximately linear, and the data are roughly normally distributed.

Spearman Correlation (rho)

Spearman's rho (r_s) is a non-parametric measure that assesses the monotonic relationship between two variables using their ranks rather than raw values. Use Spearman when data are ordinal (e.g., Likert scales), when the relationship is monotonic but not necessarily linear, or when outliers are a concern. Because it operates on ranks, Spearman's rho is more robust to extreme values than Pearson's r.

Worked Example: Study Hours vs. Exam Score

A psychology professor collects data from 10 students to examine whether weekly study hours predict exam performance. Each student reports their average weekly study hours, and their final exam score (out of 100) is recorded.

Study Hours (X)

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

M = 11.00, SD = 6.06

Exam Score (Y)

52, 58, 61, 68, 72, 78, 81, 85, 90, 95

M = 74.00, SD = 14.23

Scatter Plot Description

Plotting these 10 data points reveals a clear upward trend: as study hours increase from 2 to 20, exam scores rise from 52 to 95. The points cluster tightly around an upward-sloping regression line, indicating a strong positive linear relationship with minimal scatter.

Results

r(8) = .87, p < .001, 95% CI [.53, .97]

There is a strong positive correlation between weekly study hours and exam scores. Students who studied more hours per week tended to score substantially higher on the final exam. The coefficient of determination (r² = .76) indicates that study hours account for approximately 76% of the variance in exam scores.

Pearson vs. Spearman: When to Use Each

Choosing the correct correlation method depends on your data type, distribution, and the nature of the relationship you expect. Here is a side-by-side comparison to guide your decision:

Feature	Pearson r	Spearman r_s
Type	Parametric	Non-parametric
Data level	Interval / Ratio	Ordinal / Interval / Ratio
Relationship detected	Linear only	Any monotonic relationship
Normality required	Yes (bivariate normality)	No
Sensitive to outliers	Yes, highly	More robust
Best for	Continuous, normally distributed data	Ranked data, non-normal distributions, ordinal scales
Example use case	Height vs. weight	Customer satisfaction (1-5) vs. purchase frequency

Interpreting Correlation Strength

The absolute value of the correlation coefficient indicates the strength of the relationship. While context matters and different fields have different norms, the following guidelines (based on Evans, 1996) provide a general framework:

\|r\| Value	Strength	Interpretation
.00 – .19	Very Weak	Negligible relationship; practically no predictive value
.20 – .39	Weak	Small but potentially meaningful relationship
.40 – .59	Moderate	Noticeable relationship with meaningful predictive power
.60 – .79	Strong	Substantial relationship; good predictive accuracy
.80 – 1.00	Very Strong	Near-perfect relationship; excellent predictive accuracy

Note: These thresholds apply equally to positive and negative correlations. An r = -.85 is just as strong as r = +.85; only the direction differs.

Assumptions of Correlation Analysis

Before interpreting your correlation results, verify that these assumptions are met:

1. Linearity

Pearson's r assumes a linear relationship between the two variables. Always inspect a scatter plot first. If the relationship is curved (e.g., U-shaped or logarithmic), Pearson's r will underestimate the true strength of the association. In such cases, consider Spearman's rho or a non-linear transformation.

2. Bivariate Normality (Pearson only)

Pearson's r assumes that both variables are approximately normally distributed. This assumption is important primarily for significance testing and confidence intervals. With sample sizes above 30, the test is reasonably robust to moderate violations. For non-normal data, use Spearman's rho instead.

3. Continuous Data

Both variables should be measured on a continuous scale (interval or ratio) for Pearson's r. If either variable is ordinal (e.g., Likert-type ratings, class rank), use Spearman's rho, which operates on ranks and does not require continuous measurement.

4. No Significant Outliers

Outliers can dramatically inflate or deflate Pearson's r. A single extreme data point can shift the correlation from near zero to strong (or vice versa). Always visualize your data with a scatter plot to identify outliers. If outliers are present, consider removing them with justification or switching to Spearman's rho.

Correlation Does Not Imply Causation

One of the most important principles in statistics is that correlation does not equal causation. A strong correlation between two variables means they tend to move together, but it does not prove that one variable causes the other to change.

There are three possible explanations for any observed correlation:

Direct causation: X actually causes Y (or Y causes X).
Reverse causation: The causal direction is opposite to what you assumed.
Third variable (confound): An unmeasured variable Z causes both X and Y, creating a spurious correlation.

Classic example: Ice cream sales and drowning deaths are strongly positively correlated. Does ice cream cause drowning? Of course not. The confounding variable is temperature -- hot weather increases both ice cream consumption and swimming activity, leading to more drownings. Without controlling for temperature, you would incorrectly conclude a causal link between ice cream and drowning.

To establish causation, you need a well-designed experimental study with random assignment, or use advanced techniques like instrumental variables, regression discontinuity, or difference-in-differences.

How to Report Correlation in APA Format

According to APA 7th edition guidelines, correlation results should include the correlation coefficient, degrees of freedom (N - 2), the p-value, and ideally the 95% confidence interval. Here are templates with actual numbers you can adapt:

Pearson Correlation

A Pearson correlation was computed to assess the relationship between weekly study hours and exam scores. There was a strong positive correlation between the two variables, r(8) = .87, p < .001, 95% CI [.53, .97]. Students who spent more hours studying per week tended to achieve higher exam scores.

Spearman Correlation

A Spearman rank-order correlation was computed to assess the relationship between customer satisfaction ratings and repurchase frequency. There was a moderate positive correlation, r_s(48) = .52, p < .001. Customers who reported higher satisfaction levels tended to repurchase more frequently.

Note: Report correlation coefficients to two decimal places without a leading zero (e.g., .87 not 0.87). Report p-values to three decimal places, except use p < .001 when the value is below .001. Degrees of freedom for correlation are N - 2.

Common Mistakes to Avoid

Confusing correlation with causation: A significant correlation only indicates that two variables are related; it does not mean one causes the other. Always consider confounding variables and avoid causal language (e.g., say "associated with" rather than "caused by").
Ignoring outliers: A single outlier can dramatically change Pearson's r. For example, one extreme data point can turn a weak correlation into a strong one (or vice versa). Always inspect scatter plots before reporting results.
Restricted range: If your sample covers only a narrow range of one variable, the observed correlation will be attenuated (weakened). For example, correlating GPA and GRE scores among admitted graduate students (already high on both) will underestimate the true population correlation.
Using Pearson with non-linear data: Pearson's r only captures linear relationships. If the scatter plot shows a clear curve (e.g., quadratic, logarithmic), Pearson's r will underestimate the true association. Use Spearman's rho or transform the data.
Reporting p = .000: Statistical software sometimes displays p = .000. Always report this as p < .001. A p-value is never exactly zero.

Calculation Accuracy

StatMate's correlation calculations have been validated against R's cor.test() function. We compute Pearson's r using the standard product-moment formula and Spearman's rho using ranked values. Significance testing uses the t-distribution with N - 2 degrees of freedom. The 95% confidence interval for Pearson's r is computed via Fisher's z-transformation. All results match R output to at least 4 decimal places.

Try Other Calculators

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Descriptive

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One-Sample T

Test against a known value

Mann-Whitney U

Non-parametric group comparison

Wilcoxon

Non-parametric paired test

Regression

Model X-Y relationships

Multiple Regression

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Cronbach's Alpha

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Logistic Regression

Binary outcome prediction

Factor Analysis

Explore latent factor structure

Kruskal-Wallis

Non-parametric 3+ group comparison

Repeated Measures

Within-subjects ANOVA

Two-Way ANOVA

Factorial design analysis

Friedman Test

Non-parametric repeated measures

Fisher's Exact

Exact test for 2×2 tables

McNemar Test

Paired nominal data test