For non-constant paired data, the Pearson product-moment correlation coefficient r measures the direction and strength of linear association. r=((x_i- x)(y_i- y))/(sqrt((x_i- x)2 (y_i- y)2)) r=(x_iy_i- (x_i)(y_i))/(n)sqrt((x_i2-((x_i)2)/(n))(y_i2-((y_i)2)/(n))) **Feature** | **Interpretation** Direction | r>0 positive; r<0 negative Strength | -1 <= r 1; larger |r| means stronger linear association Units | r is dimensionless Limitation | Correlation does not establish causation and does not measure general non-linear association. Affine transformations If a 0 and c 0, then r(aX+b,cY+d)=sgn(ac),r(X,Y). Thus translations and positive rescalings preserve r; reflecting exactly one variable reverses its sign; |r| is preserved. If either transformed variable is constant, its standard deviation is zero and Pearson’s r is undefined.

For non-constant paired data, the Pearson product-moment correlation coefficient \(r\) measures the direction and strength of linear association.

\[r=\frac{\sum (x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum (x_i-\bar x)^2\sum (y_i-\bar y)^2}}\]

Centred form

\[r=\frac{\sum x_iy_i-\frac{(\sum x_i)(\sum y_i)}{n}}{\sqrt{\left(\sum x_i^2-\frac{(\sum x_i)^2}{n}\right)\left(\sum y_i^2-\frac{(\sum y_i)^2}{n}\right)}}\]

Computational form — MF27

If \(a\ne0\) and \(c\ne0\), then \(r(aX+b,cY+d)=\operatorname{sgn}(ac)\,r(X,Y)\). Thus translations and positive rescalings preserve \(r\); reflecting exactly one variable reverses its sign; \(|r|\) is preserved.

Exact rule

If either transformed variable is constant, its standard deviation is zero and Pearson’s \(r\) is undefined.

Pearson Correlation Coefficient

Pearson Correlation Coefficient

Feature	Interpretation
Direction	\(r>0\) positive; \(r<0\) negative
Strength	\(-1\leq r\leq1\); larger \(\|r\|\) means stronger linear association
Units	\(r\) is dimensionless
Limitation	Correlation does not establish causation and does not measure general non-linear association.