Pearson Correlation

Overview

The Pearson product-moment correlation coefficient $r$ is the most basic correlation measure, expressing how strongly two quantities move together in a straight-line (linear) fashion, on a scale from $-1$ to $+1$.

• It approaches $+1$ when one increases proportionally as the other increases.
• It approaches $-1$ when one decreases proportionally as the other increases.
• It approaches $0$ when they do not move together (no correlation).

A larger absolute value $|r|$ means a stronger linear co-movement.

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Formula

For two series $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ with means $\bar{x}, \bar{y}$:

$$r = \operatorname{Corr}(x, y) = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_i (x_i - \bar{x})^2 , \sum_i (y_i - \bar{y})^2}}$$

It is the covariance of the two quantities divided by the product of their standard deviations. Subtracting the means $\bar{x}, \bar{y}$ from each value measures whether the deviations from the mean move in the same direction. Dividing by the product of the standard deviations cancels out scale, normalizing the value to $[-1, +1]$.

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Characteristics

Strengths:

• Scale-independent (normalized to $[-1, +1]$).
• Simple and fast to compute.

Limitations:

• Captures only linear co-movement. A non-linear (curved) relationship can yield a small $r$ even when the relationship is strong.
• Sensitive to outliers (uses raw values, so it is pulled by extreme points).

To robustly capture non-linear but monotonic relationships, apply Pearson correlation to the rank series rather than the raw values — see Spearman rank correlation.

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Where It Is Used in the App

• Scatter Matrix: the correlation coefficient of each variable pair (upper-triangle cells) is computed as the Pearson correlation of the raw values → Scatter Matrix
• Spearman rank correlation: Spearman is simply Pearson correlation applied to the rank series → Spearman rank correlation

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References

• Pearson, K. (1895). Note on regression and inheritance in the case of two parents. Proceedings of the Royal Society of London, 58, 240–242. https://doi.org/10.1098/rspl.1895.0041