# Karl Pearsonâ€™s (Product Moment) Correlation Coefficient

Karl Pearsonâ€™s correlation coefficient quantitatively measures degree of relationship between two variables*x*and

*y*.

**Definition**: A ratio between the co-variance between two variables to the product of their standard deviations is called Karl Pearsonâ€™s correlation coefficient.

# Covariance

When*n*pairs of observations (

*x*

_{1},

*y*

_{1}), (

*x*

_{2},

*y*

_{2}), (

*x*

_{3},

*y*

_{3}) â€¦ (

*x*,

_{n}*y*) are given between two variables

_{n}*x*and

*y*, then the covariance of

*x*and

*y*, cov(

*x*,

*y*) and is defined as:

We can also express the covariance between the variables *x* and *y* as

**Note:** Variance is always positive, whereas covariance may be positive, negative or zero.

- Using the above values of cov(
*x*,*y*),*S*and_{x}*S*, we can express the correlation coefficient as_{y}

Example

If

*r*= 0.25,*S*= 6,_{x}*S*= 8, then find cov(_{y}*x*,*y*).Solution

â‡’ cov(x, y) =12

Example

Compute the correlation co-efficient between

*x*and*y*for the given data.X |
1 | 2 | 3 | 4 | 5 |

y |
10 | 20 | 30 | 40 | 50 |

Solution

- For grouped bivariate data, Karl Pearsonâ€™s correlation co-efficient can be written as

**Note:**

- The ratio of explained variance to total variance is known as
*coefficient of determination*.^{2} - The ratio of unexplained variance to total variance is known as
*coefficient of non-determination.*^{2} - The square root of the coefficient of determination is known as the
*coefficient of alienation.*

# Properties of Coefficient of Correlation

- The value of the coefficient of correlation is independent of the origin
*i.e.*, increasing or decreasing the values of*x*_{i}and*y*_{i}by some non-zero constant will not affect the value of the coefficient of correlation - The value of the coefficient of correlation is independent of the scale
*i.e.*, multiplying or dividing the values of*x*_{i}and*y*_{i}by some non-zero constant will not affect the value of the coefficient of correlation*a*â€™ on*x*and â€˜*b*â€™ on*y*and a change of scale of â€˜*c*â€™ on*x*and â€˜*d*â€™ on*y*, then we have*u*_{i}and*v*_{i}, we have - The value of the correlation coefficient lies between â€“1 and +1
- Correlation coefficient is unit free
*rXY*= {(*b*.*d*)/|*b*||*d*|}*r**UV*

*Merits*

- It gives us the direction as well as the degree of correlation between the variables.
- It helps in estimating the value of a dependent variable from the value of the independent variables.

*Demerits*

- Computation of Karl Pearsonâ€™s correlation coefficient consumes a lot of time.
- It is affected by the values of extreme observations.