Knowledge Base / Glossary

Pearson correlation

A number from negative one to positive one that summarizes the direction and strength of a linear relationship between two numerical features.

Pearson correlation, usually written \(r\), summarizes the direction and strength of a linear relationship between two numerical features. Its value ranges from \(-1\) to \(+1\).

A value near \(+1\) means high values of one feature tend to accompany high values of the other in an almost straight-line pattern.
A value near \(-1\) means high values of one tend to accompany low values of the other.
A value near \(0\) means little straight-line relationship. A curved relationship can still be present.

Four paired observations sit around the feature means. Points in same-sign quadrants add positive products, while points in opposite-sign quadrants subtract from Pearson correlation. — Pearson correlation compares each paired observation with both feature means. A point above both means or below both means has deviations with the same sign, so it contributes a positive product. A point above one mean and below the other contributes a negative product. In this toy example the positive products dominate and r is about 0.83. The figure explains linear association only; it does not show causation or population uncertainty.

The calculation

Suppose \(n\) observations each have a paired value \(x_i\) from feature \(x\) and \(y_i\) from feature \(y\). Let \(\bar{x}\) and \(\bar{y}\) be the two feature means. Pearson correlation is

\[ r= \frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})} {\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2} \sqrt{\sum_{i=1}^{n}(y_i-\bar{y})^2}}. \]

The numerator adds products of paired deviations from the means. The denominator divides by the two features’ spread, making \(r\) independent of a simple change of positive scale such as millimeters to centimeters.

Worked example

Take four pairs:

observation	\(x\)	\(y\)
1	1	1
2	2	3
3	3	2
4	4	5

The means are \(\bar{x}=2.5\) and \(\bar{y}=2.75\). The numerator is \(5.5\), the two sums of squared deviations are \(5\) and \(8.75\), and therefore

\[ r=\frac{5.5}{\sqrt{5\times8.75}}\approx0.83. \]

The pairs show a fairly strong positive linear tendency, but they do not sit perfectly on one line. These four values are a calculation example, not enough evidence for a biological population claim.

Important boundaries

Correlation requires paired measurements on the same observations. Shuffling one feature’s rows changes the question and usually changes \(r\).
If either feature is constant, its spread is zero and \(r\) is undefined.
One extreme observation can have a large effect.
A high absolute correlation does not prove that one feature causes the other. Both may respond to a third factor.
A near-zero value does not rule out a curved association or group-specific relationships.
A sample correlation is not automatically a precise population estimate. Dependence, sampling design, and uncertainty still matter.

In principal component analysis, a correlation matrix can reveal features that carry overlapping linear information. PCA can summarize that shared variation, but it does not turn correlation into causation.