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Pearson Correlation Calculator

Calculate the Pearson correlation coefficient (r) to measure the strength and direction of the linear relationship between two variables. Includes coefficient of determination (r-squared) for regression analysis and scientific research.

Reviewed by Christopher FloiedPublished Updated

This free online pearson correlation calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

Sample Size

5

Pearson r

0.996

R-squared

0.992

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Pearson Correlation Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Pearson Correlation Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Pearson Correlation Calculator when you need accurate results quickly without the risk of manual computation errors or unit conversion mistakes.
  • •Use it to verify calculations made by hand or in spreadsheets — an independent check can catch errors before they lead to costly decisions.
  • •Use it to explore how changing input parameters affects the output — a quick way to develop intuition and identify the most influential variables.
  • •Use it when collaborating with others to ensure everyone is working from the same numbers and applying the same assumptions.

About Pearson Correlation Calculator

The Pearson correlation calculator computes the most widely used measure of linear association between two continuous variables. The Pearson correlation coefficient r ranges from -1 (perfect negative linear relationship) through 0 (no linear relationship) to +1 (perfect positive linear relationship). The coefficient of determination r-squared tells you what proportion of the variance in one variable is explained by the linear relationship with the other. Pearson correlation is foundational in regression analysis, scientific research, psychology, economics, and any field that examines relationships between measured quantities. However, correlation does not imply causation, and Pearson's r only detects LINEAR relationships -- perfect curvilinear relationships can have r = 0.

The Math Behind It

The Pearson product-moment correlation coefficient was developed by Karl Pearson in the 1890s. It is defined as the covariance of two variables divided by the product of their standard deviations: r = Cov(X,Y) / (s_X * s_Y). This normalization ensures r always falls between -1 and 1 regardless of the units or scales of X and Y. The square of r, called R-squared or the coefficient of determination, represents the proportion of variance in Y that is linearly predicted by X. An r of 0.7 means R-squared = 0.49, so about 49% of the variance is explained. Pearson's r assumes both variables are continuous, approximately normally distributed, and that the relationship is linear. For non-linear relationships, Pearson's r underestimates the true association. The Spearman rank correlation coefficient is a non-parametric alternative that measures monotonic (not necessarily linear) relationships using ranks instead of raw values. Statistical significance of r is tested using a t-test: t = r * sqrt(n-2) / sqrt(1-r^2), with n-2 degrees of freedom. The critical value of r depends on sample size -- with n=10, r must exceed 0.632 for p < 0.05, while with n=100, r = 0.197 is significant. Fisher's z-transformation, z = 0.5 * ln((1+r)/(1-r)), normalizes the sampling distribution of r and is used for confidence intervals and comparing correlations between studies.

Formula Reference

Pearson Correlation Coefficient

r = sum((xi-x_mean)(yi-y_mean)) / sqrt(sum((xi-x_mean)^2) * sum((yi-y_mean)^2))

Variables: xi, yi = paired values; x_mean, y_mean = sample means; r ranges from -1 to 1

Coefficient of Determination

R^2 = r^2

Variables: Proportion of variance in Y explained by the linear relationship with X

Worked Examples

Example 1: Study hours vs exam scores

Hours: 2, 4, 6, 8, 10. Scores: 65, 75, 80, 88, 95.

Step 1:Mean of hours = 6, Mean of scores = 80.6.
Step 2:Sum of cross-products of deviations = (-4)(-15.6) + (-2)(-5.6) + (0)(-0.6) + (2)(7.4) + (4)(14.4) = 62.4 + 11.2 + 0 + 14.8 + 57.6 = 146.
Step 3:SS_x = 40, SS_y = 550.8.
Step 4:r = 146 / sqrt(40 * 550.8) = 146 / 148.4 = 0.9838.

r = 0.984, indicating a very strong positive linear relationship. R-squared = 0.968, meaning 96.8% of score variance is explained by study hours.

Example 2: No linear relationship

X: 1, 2, 3, 4, 5. Y: 1, 4, 1, 4, 1.

Step 1:Mean X = 3, Mean Y = 2.2.
Step 2:The alternating pattern creates no consistent linear trend.
Step 3:r ≈ -0.27.

r = -0.27, indicating no meaningful linear relationship. However, there is a clear pattern (alternating) that Pearson's r does not capture.

Common Mistakes & Tips

  • !Interpreting correlation as causation -- a strong correlation between X and Y does not mean X causes Y. Confounding variables, reverse causation, and coincidence are alternative explanations.
  • !Using Pearson's r for non-linear relationships -- a perfect quadratic or exponential relationship can have r = 0 if it is symmetric about the mean.
  • !Ignoring the effect of outliers -- a single extreme point can dramatically inflate or deflate the correlation coefficient.
  • !Treating small correlations as unimportant -- in large studies, r = 0.10 can be statistically significant and practically meaningful.

Related Concepts

Covariance Calculator

Covariance measures the unstandardized linear relationship between two variables. Pearson's r is the standardized version.

Linear Regression Calculator

Linear regression uses the correlation to fit a predictive line. R-squared from regression equals the square of Pearson's r.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Covariance Calculator

Frequently Asked Questions

What is a strong correlation?

Guidelines vary by field, but common benchmarks (Cohen 1988) are: |r| < 0.10 = negligible, 0.10-0.30 = small, 0.30-0.50 = medium, and > 0.50 = large. In physics and engineering, r > 0.95 is expected. In social sciences, r > 0.50 is considered strong. Always interpret correlation in the context of your field.

What is the difference between Pearson and Spearman correlation?

Pearson measures the strength of the LINEAR relationship using raw values. Spearman measures the strength of the MONOTONIC relationship using ranks. Use Spearman when data is ordinal, contains outliers, or the relationship is monotonic but not linear. For normally distributed data with a linear relationship, both give similar results.

Can correlation be exactly zero with a clear pattern in the data?

Yes. A perfect U-shaped (quadratic) relationship like Y = X^2 centered at zero will have r = 0 because positive and negative deviations cancel perfectly. This is why you should always plot your data -- Pearson's r only detects linear patterns, and non-linear relationships can be completely missed.