To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)
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,
 Linear Correlation Coefficient
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

Back to Top
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.