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r=∑i=1n(xi−xˉ)(yi−yˉ)∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2r = \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}}r=∑i=1n​(xi​−xˉ)2​∑i=1n​(yi​−yˉ​)2​∑i=1n​(xi​−xˉ)(yi​−yˉ​)​

∑i=1n(xi−xˉ)(yi−yˉ)\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})∑i=1n​(xi​−xˉ)(yi​−yˉ​)

,

∑i=1n(xi−xˉ)2\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}∑i=1n​(xi​−xˉ)2​

,

∑i=1n(yi−yˉ)2\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}∑i=1n​(yi​−yˉ​)2​

Pearson Correlation Coefficient

Click on formula components below to explore their properties

Full Formula Properties

Category: Statistics

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Baby Fast Definition

This number tells us how strongly two things move together in a straight line pattern.

Pearson's r quantifies how tightly points cluster around a straight line, acting as a standardized ruler for association strength. It transforms raw paired data into a bounded measure that remains consistent regardless of measurement units.

Role:

Measures linear relationship strength between two variables

Domain:

Real numbers between -1 and 1

Binding:

Links paired data points to a single correlation value

Variance:

Changes as the pattern of paired data changes

Geometric:

Cosine of the angle between centered data vectors

Invariant:

Invariant to linear scaling of variables

Limits:

Approaches -1 or +1 for perfect linear relationships, 0 for no linear relationship

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