Top

# Linear Correlation Coefficient Formula

Linear correlation coefficient explores the relation between two variables in a population. This coefficient tells how strong the variables are connected. Linear Correlation coefficient value can be negative or positive. Other name for this linear correlation coefficient is Pearson's correlation coefficient. It is denoted by 'r'. The limiting values of r are -1 and 1. If the value is nearly equal to 1,it has strong positive relation. If it is closer to -1, it has high negative relation. Zero value indicates that, there is no relation that exists between the variables. The formula for linear correlation coefficient is given by,

Where, n is the number of observations, xi and yi are the variables.

 Related Calculators Linear Correlation Coefficient Calculator Calculator for Correlation Coefficient Pearson Correlation Coefficient Binomial Coefficient Calculator

## Linear Correlation Coefficient Problems

Problems related to the topic are given below:

### Solved Examples

Question 1: Calculate the linear correlation coefficient for the following data.
x : 5, 10, 15, 20 and y: 4, 6, 8, 10 ?

Solution:

Given variables are,
x: 5, 10, 15, 20 and y: 4, 6, 8, 10; n = 4

 x y xy x2 y2 5 4 20 25 16 10 6 60 100 36 15 8 120 225 64 20 10 200 400 100 50 28 400 750 216

$r_{xy}$ = $\frac{n\sum_{i=1}^{n}x_{i}y_{i}-\sum_{i=1}^{n}x_{i}\sum_{i=1}^{n}y_{i}}{\sqrt{n\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i=1}^{n}x_{i})^{2}}\sqrt{n\sum_{i=1}^{n}y_{i}^{2}-(\sum_{i=1}^{n}y_{i})^{2}}}$

$r_{xy}$ = $\frac{4\times400-50\times28}{\sqrt{4\times750-50^{2}}\sqrt{4\times216-28^{2}}}$

$r_{xy}$ = $\frac{200}{\sqrt{500}\sqrt{80}}$

$r_{xy}$ = 1

Question 2: Calculate the linear correlation coefficient of the given variables.
x : 3, 5, 7, 9 and y : 2, 4. 6, 8 ?

Solution:

Given variables are,
x : 3, 5, 7, 9 and y : 2, 4, 6, 8; n = 4

 x y xy x2 y2 3 2 6 9 4 5 4 20 25 16 7 6 42 49 36 9 8 48 81 64 24 20 116 164 120

$r_{xy}$ = $\frac{n\sum_{i=1}^{n}x_{i}y_{i}-\sum_{i=1}^{n}x_{i}\sum_{i=1}^{n}y_{i}}{\sqrt{n\sum_{i=1}^{n}x_{i}^{2}-(\sum_{i=1}^{n}x_{i})^{2}}\sqrt{n\sum_{i=1}^{n}y_{i}^{2}-(\sum_{i=1}^{n}y_{i})^{2}}}$

$r_{xy}$ = $\frac{4\times116-24\times20}{\sqrt{4\times164-24^{2}}\sqrt{4\times120-20^{2}}}$

$r_{xy}$ = $\frac{-16}{\sqrt{80}\sqrt{80}}$

$r_{xy}$ = -0.2

*AP and SAT are registered trademarks of the College Board.