Correlation Coefficient Calculator

The Pearson correlation coefficient is the standardized version of covariance, defined as:

$$r = \frac{\text{Cov}(X, Y)}{s_X \cdot s_Y} = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2} \cdot \sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}$$

The coefficient of determination is:

$$R^2 = r^2$$

Where:

• Cov(X, Y) is the sample covariance
• s_X, s_Y are the sample standard deviations of X and Y
• R² represents the proportion of variance in Y explained by X (or vice versa)

Interpreting the Pearson correlation coefficient:

• r = +1: Perfect positive linear relationship
• 0.7 ≤ r < 1: Strong positive correlation
• 0.3 ≤ r < 0.7: Moderate positive correlation
• 0 < r < 0.3: Weak positive correlation
• r = 0: No linear correlation
• r < 0: Negative correlation (same strength categories, mirrored)

The Pearson correlation was developed by Karl Pearson in the 1890s, building on earlier work by Francis Galton on regression toward the mean. It remains the most widely used measure of association in the sciences.

Important assumptions and limitations:

1. Linearity: Pearson's r only captures linear relationships. A perfect quadratic relationship would not yield r = 1.
2. Outlier sensitivity: A single outlier can dramatically change the correlation. Always plot your data first.
3. Scale invariance: r is unchanged by linear transformations of the variables (adding, multiplying by constants).
4. Correlation ≠ Causation: A strong correlation does not prove that changes in one variable cause changes in the other.

R² has a particularly intuitive interpretation: if R² = 0.81, then 81% of the variance in one variable is linearly predictable from the other. The remaining 19% is due to other factors or non-linear relationships. In regression analysis, R² is the primary measure of model fit.

For non-normal data or ordinal variables, consider Spearman's rank correlation or Kendall's tau, which are based on ranks rather than raw values and are more robust to outliers and non-linearity.

Pearson r close to +1 or -1 indicates a strong linear relationship; close to 0 means weak or no linear relationship. R² tells you the proportion of variance explained: R² = 0.64 means 64% of Y's variability is explained by X. The sample covariance provides the un-standardized measure of joint variability.