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Pearson Correlation Formula

Correlation is the relationship between two variables. Correlation coefficient is the measurement of correlation. It indicates how well the two set of data are interconnected. Pearson correlation coefficient measures the linear dependence of two variables upon each other. It is also referred as Pearson product-moment correlation coefficient. The value of Pearson correlation coefficient lies between -1 to +1. If the coefficient of correlation is zero, then there is no correlation between given two variables. On the other hand, the perfectly positive correlation has a value of +1, while a perfectly negative correlation has a value of -1. 

The graphical representation of positive, negative and no correlation is shown below:

Pearson correlation coefficient for sample data is denoted by "r". The formula for Pearson correlation coefficient r is given by:

Where,

r = Pearson correlation coefficient

x = Values in first set of data

y = Values in second set of data

n = Total number of values.

Pearson Correlation Problems

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The few problems based on Pearson correlation coefficient are listed below:

Solved Examples

Question 1: Marks obtained by 5 students in algebra and trigonometry as given below:

Algebra | 15 | 16  | 12  | 10  | 8 |

Trigonometry  | 18  | 11 | 10  | 20 | 17 |

Calculate the Pearson correlation coefficient.

Solution:

Construct the following table:

x  | y | x2  | y2  | xy  |

15  | 18  | 225  | 324 | 270  |

16  | 11 | 256  | 121  | 176  |

12  | 10  | 144  | 100  | 120  |

10  | 20  | 100  | 400  | 200  |

8  | 17  | 64  | 289  | 136  |

∑x = 61  | ∑y = 76  | ∑x2 = 789  | ∑y2 = 1234 | ∑xy = 902  |

Formula for Pearson correlation coefficient is given by:

r = n(∑xy)−(∑x)(∑y)[n∑x2−(∑x)2][n∑y2−(∑y)2]√

r = 5×902−61×76[5×789−(61)2][5×1234−(76)2]√

r = -0.424

Question 2: Following the values for x and y:

x  | 5 | 6 | 4 | 2 |

y    | 7    | 3    | 9    | 8   |

Evaluate...