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Math & Conversions Correlation Calculator Correlation Coefficient Calculator (Pearson's $r$) Use this calculator to find the strength and direction of the linear relationship between two variables, X and Y. Enter your paired data points below. Each line should contain the X-value, followed by the Y-value, separated by a comma or space. Data Set X Data Set Y Make sure X and Y have the same number of data points. Use commas, spaces, or new lines to separate values. Calculate Correlation Results Pearson's $r$ Interpretation Coefficient of Determination ($R^2$) Step-by-Step Calculation Table Pearson's Correlation Coefficient ($r$) Formula Pearson's $r$ is calculated using the following formula, which is designed to measure the covariance of X and Y normalized by the standard deviations of X and Y: $$r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}$$ Where: $n$ is the number of paired observations (data points). $\sum x$ and $\sum y$ are the sums of the X and Y values. $\sum x^2$ and $\sum y^2$ are the sums of the squared X and squared Y values. $\sum xy$ is the sum of the products of X and Y for each pair. This formula always yields a value between $-1$ and $+1$. Interpreting the Correlation Coefficient ($r$) The value of $r$ tells you two things about the relationship between X and Y: Direction (Sign): **Positive ($r > 0$):** As X increases, Y tends to increase (a positive slope). **Negative ($r < 0$):** As X increases, Y tends to decrease (a negative slope). **Zero ($r \approx 0$):** No linear relationship exists. Strength (Magnitude): The closer the value is to $\pm 1$, the stronger the linear relationship. The closer it is to 0, the weaker the relationship. Strength Guidelines (Commonly Used) Magnitude of $|r|$ Strength 0.90 to 1.00 Very Strong 0.70 to 0.89 Strong 0.50 to 0.69 Moderate 0.30 to 0.49 Weak 0.00 to 0.29 Negligible Frequently Asked Questions (FAQ) What does a correlation coefficient measure? The correlation coefficient, typically Pearson's $r$, measures the strength and direction of the linear relationship between two quantitative variables. It is a value between -1 and +1. What is the difference between correlation and causation? Correlation indicates that two variables move together (e.g., ice cream sales and sunscreen sales both rise in summer). Causation means one variable directly causes a change in the other. Correlation does not imply causation (e.g., ice cream sales do not cause sunscreen sales to rise; the common cause is the heat). What is the Coefficient of Determination ($R^2$)? The Coefficient of Determination ($R^2$) is the square of the correlation coefficient ($r^2$). It represents the proportion of the variance in the dependent variable that is predictable from the independent variable. For example, if $r = 0.8$, $R^2 = 0.64$, meaning 64% of the variability in Y can be explained by the relationship with X. When should I use a different correlation method? Pearson's $r$ is best for data that is normally distributed and linear. If your relationship is monotonic (consistently increasing or decreasing but not necessarily linear), Spearman's rank correlation might be more appropriate. If your data involves binary or categorical variables, other measures (like Cramer's V or phi coefficient) are needed. Correlation Interpretation $r = +1.0$: Perfect Positive $r = -1.0$: Perfect Negative $r = 0.0$: No Linear Relationship $r \approx 0.8$: Strong Positive $r \approx -0.4$: Weak Negative Always consider the magnitude and the sign. Related Statistics Tools Correlation Coefficient (Alternative) Linear Regression Calculator Mean, Median, Mode Calculator Standard Deviation Calculator P-Value Calculator t-Test Calculator Covariance Calculator
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