The following calculations are needed when
finding Pearson's product moment correlation coefficient or

S_{xx} = \displaystyle \sum x^2 - \frac{(\sum x)^2}{n}

S_{yy} = \displaystyle \sum y^2 - \frac{(\sum y)^2}{n}

S_{xy} = \displaystyle \sum xy - \frac{(\sum x)(\sum y)}{n}

**pmcc**:S_{xx} = \displaystyle \sum x^2 - \frac{(\sum x)^2}{n}

S_{yy} = \displaystyle \sum y^2 - \frac{(\sum y)^2}{n}

S_{xy} = \displaystyle \sum xy - \frac{(\sum x)(\sum y)}{n}

## Summary/Background

The pmcc can be found using:

r = \displaystyle \frac{S_{xy} }{\sqrt{S_{xx} S_{yy} } }

r = \displaystyle \frac{S_{xy} }{\sqrt{S_{xx} S_{yy} } }

Karl Pearson FRS (27 March 1857 – 27 April 1936) established the discipline of mathematical statistics. A sesquicentenary conference was held in London on 23 March 2007, to celebrate the 150th anniversary of his birth.

In 1911 he founded the world's first university statistics department at University College London. He was a proponent of eugenics, and a protege and biographer of Sir Francis Galton. He was also a socialist. Pearson was instrumental in the development of regression and correlation theory. One of his classic data sets (originally collected by Galton) involves the regression of sons' height upon that of their fathers'. Pearson built a 3-dimensional model of this data set (which remains in the care of the Statistical Science Department) to illustrate the ideas. The Pearson

In 1911 he founded the world's first university statistics department at University College London. He was a proponent of eugenics, and a protege and biographer of Sir Francis Galton. He was also a socialist. Pearson was instrumental in the development of regression and correlation theory. One of his classic data sets (originally collected by Galton) involves the regression of sons' height upon that of their fathers'. Pearson built a 3-dimensional model of this data set (which remains in the care of the Statistical Science Department) to illustrate the ideas. The Pearson

**product-moment correlation coefficient**is named after him, and it was the first important "effect size" to be introduced into statistics.## Software/Applets used on this page

## Glossary

### coefficient

The constant value in an expression, for example in 3x the coefficient of x is 3.

### correlation

The degree of relationship between two random variables, in the range -1 to 1.

### moment

The turning effect of a force whose value depends on the magnitude of the force and where it is applied.

### pmcc

The product moment correlation coefficient

### product moment correlation coefficient

Often abbreviated to PMCC; a measure of correlation.

### regression

A statistical model, whereby the expected value of a variable y is expressed in terms of known values of x.

### union

The union of two sets A and B is the set containing all the elements of A and B.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic |
---|---|---|---|

AQA A-Level (UK - Pre-2017) | S1 | Correlation and Regression | Correlation |

AQA A2 Maths 2017 | Statistics | Correlation and Regression | Product Moment Correlation Coefficient |

AQA AS Further Maths 2017 | Statistics | Extra | Product Moment Correlation Coefficient |

AQA AS/A2 Further Maths 2017 | Statistics | Extra | Product Moment Correlation Coefficient |

AQA AS/A2 Maths 2017 | Statistics | Correlation and Regression | Product Moment Correlation Coefficient |

CCEA A-Level (NI) | S2 | Correlation and Regression | Correlation |

L Statistics - Data | L2 Bivariate Data | Correlation | |

Edexcel A-Level (UK - Pre-2017) | S1 | Correlation and Regression | Correlation |

Edexcel A2 Maths 2017 | Statistics | Regression and Correlation | Product Moment Correlation Coefficient |

Edexcel AS Further Maths 2017 | Further Statistics 2 | Correlation | Product Moment Correlation Coefficient |

Edexcel AS/A2 Further Maths 2017 | Further Statistics 2 | Correlation | Product Moment Correlation Coefficient |

Edexcel AS/A2 Maths 2017 | Statistics | Regression and Correlation | Product Moment Correlation Coefficient |

I.B. Higher Level | 7 | Correlation and Regression | Correlation |

OCR A-Level (UK - Pre-2017) | S1 | Correlation and Regression | Correlation |

OCR A2 Maths 2017 | Statistics | Using and Testing Correlation Coefficient | Product Moment Correlation Coefficient |

OCR AS Further Maths 2017 | Statistics | Correlation | Product Moment Correlation Coefficient |

OCR MEI A2 Maths 2017 | Statistics | Using and Testing Correlation Coefficient | Product Moment Correlation Coefficient |

OCR MEI AS Further Maths 2017 | Statistics A | Correlation | Product Moment Correlation Coefficient |

OCR-MEI A-Level (UK - Pre-2017) | S2 | Correlation and Regression | Correlation |

Pre-U A-Level (UK) | Prob | Correlation and Regression | Correlation |

Scottish (Highers + Advanced) | HS | Correlation and Regression | Correlation |

Scottish Highers | S | Correlation and Regression | Correlation |

Universal (all site questions) | C | Correlation and Regression | Correlation |