Generation of correlation coefficient

Correlation Coefficient for Quantitative Data

•  The strength of correlation is measured by correlation coefficient.

•  The range of correlation coefficient is from ‒1 to 1.

• If a correlation coefficient has a value zero then that means there is no correlation.

• If the correlation coefficient has exactly ‒1 value then it is called perfect negative correlation.

•  If the correlation coefficient has exactly 1 value then it is called perfect positive correlation.

• There are various types of correlation coefficients. Out of which, Pearson Product Moment Correlation is most popularly used type of method used for obtaining correlation coefficient.

•  The correlation coefficient is denoted by variable r.

r = [ n(Σxy) – (Σx) (Σy) ] /   √[n Σx2 ‒ (Ex)2] [n Σy2 ‒ (Σy)2]

where

n = Quantity of information

Σx = Total of the first variable value

Σy = Total of the second variable value

Σxy = Sum of the product of first and second value

Σx2 = Sum of the squares of the first value

Σy2 = Sum of the squares of the second value

For example ‒ Following are the values of x and y. We will find the correlation coefficient

We will obtain x2, y2 and xy values.

Here n is total no. of data items, it is 12.

Σx = Sum of x = 149

Σy = Sum of y = 214

Σx2 = 1909

Σy2 = 3876

Σxy = 2716

Now we will put these values in the formula –

r = [ n(Σxy) – (Σx) (Σy) ] /   √[n Σx2 ‒ (Ex)2] [n Σy2 ‒ (Σy)2]

r =  (12) (2716) — (149) (214)  /  √[(12) (1909) − (149)2] [(12) (3876) – (214)2]

r = 0.62

Thus correlation coefficient is nearer to + 1. Hence it is strong positive correlation.

1. Computational Formula for Correlation Coefficient

• Pearson's correlation coefficient (r) measures the strength and direction of a relationship between two variables.

■ If r = 1, perfect positive correlation.

■ Ifr= ‒1, perfect negative correlation.

■ If r = 0, no correlation.

•  If we repeatedly take samples from a population and calculate r each time, we get a sampling distribution of r.

•  This is the distribution of Pearson's correlation coefficients (r) calculated from repeated samples of the same size taken from a population.

•  This distribution helps us determine if an observed correlation is statistically significant.

•  For example ‒ Consider that there are 1000 students (population) whose math and science scores are noted. There are 30 students in each sample. Randomly 100 such samples are recorded. Then calculate correlation coefficient (r) between maths and science scores for each sample. When we plot these 100 correlation coefficients on a graph then we get the sampling distribution of r. The distribution of r is approximately normal.

•  Correlation shows how two quantitative variables move together.

• A scatter plot gives a visual idea of the relationship, while the correlation coefficient (r) measures its strength and direction numerically.

• For quantitative data, Pearson's formula is used to computer.