Provide Information Regarding Statistics & Econometrics : Correlation & Statistical Measure

Correlation

Statistical Measure

Regression is a statistical technique that quantifies the relationship between a response variable and predictor(s) using a probabilistic model. In contrast, correlation measures the strength and direction of a linear relationship between two variables, and both variables are assumed to be independent.

The regression model is a one-way cause-and-effect model that explains variation in response variables caused by predictors. In regression analysis, the response variable is random and influenced by a fixed variable called a predictor. The correlation is a two-way caused phenomenon that quantifies the strength of the linear relationship between two independent variables that are treated symmetrically.

Correlation

Correlation is the phenomenon in which two variables vary simultaneously, either in the same direction or in opposite directions.

Examples:

i. As a vehicle's speed increases, so does its fuel consumption.

ii. The sale of ice cream increases as the temperature increases.

iii. The demand for a product decreases in the market as its price increases.

There are two types of correlation:

1. Positive correlation

If the two variables move simultaneously in the same direction.

For example, the sale of a product increases in the market as advertising increases.

2. Negative correlation

If the two variables move simultaneously in the opposite direction

For example, the demand for a product decreases as the price increases.

Coefficient of correlation

A numerical quantity that measures the strength and direction of a linear relationship between two variables denoted by ρ (population) and by r (sample).

Properties of correlation coefficient

The sample correlation coefficient has the following properties:

i. The sample correlation coefficient (r) is symmetrical with respect to variables X and Y.

ii. The sample correlation coefficient (r) lies between -1 and +1.

iii. The sample correlation coefficient is independent of change of origin and scale.

iv. In the case of a bivariate population where X and Y are random variables. Then r is the geometric mean of the two regression coefficients.

Practice Question

Calculate the product moment correlation coefficient between X and Y from the following data:

Solution:

Question:

The coefficient of correlation is independent of the change of origin and scale.

Question:

The correlation between X and Y is r. Show that the correlation between aX and bY is +r, r, -r according as a and b have the same sign or different signs.

Solution: