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Revision:Correlation

From The Student Room

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Correlation

Product Moment Correlation

The product moment correlation is a process which indicates the linear strength between the actual values of two sets of data.

It's formula is given by:

r = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

Where:
S_{xy} = \sum xy - \dfrac{\sum x \sum y}{n}

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

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

Example

Find the product moment correlation of the following set of data:

x y
11 17
9 14
7 8
10 15
4 7

First we must calculate the values of x2, y2 and xy.

Doing that gives:

x y x2 y2 xy
11 17 121 289 187
9 14 81 196 126
7 8 49 64 56
10 15 100 225 150
4 7 16 49 28

We then total up each giving, giving \sum x = 41 \sum y = 61 \sum x^2 = 367 \sum y^2 = 823 \sum xy = 547 n =5
We then put these into our formula, giving:
S_{xy} = 547 - \dfrac{41 \times 61}{5} = 46.8

S_{xx} = 367 - \dfrac{41^2}{5} = 30.8

S_{yy} = 823 - \dfrac{61^2}{5} = 78.8

r = \dfrac{46.8}{\sqrt{30.8\times 78.8}} = 0.954

0.954 is therefore our coefficient of x. It is worth that: -1 \le r \le 1 A coefficient close to 1 indicates strong positive correlation whilst a coefficient close to -1 indicates strong negative correlation. A coefficient close to 0 indicates little correlation.

Spearman's Rank

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