Elementary Statistics and Computer Application

Pearsons Correlation coefficient

The measures of the degree of relationship between two continuous variables is called correlation coefficient. It is denoted by r.( in case of sample )and (in case of population). The correlation coefficient r is known as Pearson’s correlation coefficient as it was discovered by Karl Pearson. It is also called as product moment correlation.
The correlation coefficient r is given as the ratio of covariance of the variables X and Y to the product of the standard deviation of X and Y.
Symbolically,

which can be simplified as

This correlation coefficient r is known as Pearson’s Correlation coefficient. The numerator is termed as sum of product of X and Y and abbreviated as SP(XY). In the denominator the first term is called sum off squares of X (i.e) SS(X) and second term is called sum of squares of Y (i.e) SS(Y)
.^..

The denominator in the above formula is always positive. The numerator may be positive or negative making r to be either positive or negative.
Assumptions in correlation analysis

1. Correlation coefficient r is used under certain assumptions, they are
2. The variables under study are continuous random variables and they are normally distributed
3. The relationship between the variables is linear
4. Each pair of observations is unconnected with other pair (independent)

Properties

1. The correlation coefficient value ranges between –1 and +1.
2. The correlation coefficient is not affected by change of origin or scale or both.
3. If r > 0 it denotes positive correlation

r< 0 it denotes negative correlation between the two variables x and y.
r = 0 then the two variables x and y are not linearly correlated.(i.e)two
variables are independent.
r = +1 then the correlation is perfect positive
r = -1 then the correlation is perfect negative.