To obtain a measure of the relation between X and Y independent of units of measurements. Karl Pearson in 1890 developed a measure of relationship and it’s called Karl Pearson correlation coefficient.

The population correlation denoted as ρ and is called a product-moment correlation coefficient.

The correlation coefficient is a measure of the degree or extent of the linear relationship between two variables. The population correlation coefficient is denoted as ρ and the sample estimate is r.

## What is the purpose of the correlation coefficient?

In the case of regression analysis, we estimate the value of dependent Y for a given value of independent variable X. The mean sum of squares is minimum then the regression estimate is good otherwise not good. The coefficient will provide the idea of regression analysis carried out or not.

If ρ=0 means no need to carry out regression analysis. If ρ is high automatically sum of squares is minimum and Y can determine accurately thorough X. If ρ square is called a coefficient of determination.

## What is the limit of correlation?

The limit correlation ρ lies between -1 to +1 and this is the same as for sample estimate r.

If ρ=0 means no correlation was observed between X and Y.

If ρ>0 means positive correlation was observed between X and Y.

If ρ<0 means a negative correlation was observed between X and Y.

A positive correlation indicates X increases Y also increases and a negative correlation means X increases Y decreases or vice versa.

If ρ=0 indicates lack of linear association between two variables like X and Y. In other words the values are completely scattered.

## What are the major properties?

The correlation coefficient is a pure number and has no units.

The ranges from -1 to +1

If the correlation coefficient is zero means the regression line of Y on X, will be a line parallel to abscissa at the distance of the intercept.

## What is mean by probable error?

The probable error is used for finding the significance of the coefficient of correlation. If r<3*Probable error means sample coefficient r is definitely significant.

How to calculate probable error?

The formula for probable error is, Probable Error= 0.6745*((1-r*r)/sqrt(n))

Pearson correlation coefficient is appropriate only if the relation between X and Y is linear. If the non-linear relation correlation ratio is more appropriate. The correlation ratio ranges from 0 to 1.

## What is the intraclass correlation?

To measure the similarity among the individuals/judges/panelist within the classes or groups called an intraclass correlation coefficient.

## What is the biserial correlation?

If one variable is dichotomous and another variable is quantitative biserial correlation is more appropriate. The assumption for biserial correlation as follows.

Y is normally distributed

Linear association between X and Y.

## When do we need to use the Tetrachoric correlation?

Tetrachoric correlation is suitable when both the variables X and Y are dichotomous.

## What is mean by autocorrelation?

Autocorrelation is used in time series data analysis. It has been observed that the figures for consecutive periods are correlated.