Co-efficient of Correlation and Spearman's Rank Difference Method

According to -

R.C. Lathrop - Correlation indicates a joint relationship between two variables.
Munn - Correlation is a statistical measure of the degree of association between two variables.
Guilford - A coefficient of correlation is a single number that tells us to what extent two things are related to and to what extent variations in the one leads to the variations in the other.

Examples of correlation -

Types of correlation

When two variables change in the same direction then their correlation is positive correlation.
E.g.-
- Effect of motivation on learning
- Relationship between IQ and achievement

When two variables changes in the opposite direction then their correlation is negative correlation.
E.g.- Writing speed and accuracy

When changes in one variable have no effect on the other variable then their correlation is zero correlation.
e.g.- Social background of a student and his performance in the examination.

When a scattered diagram shows correlation as a straight line then their correlation is linear correlation.
Positive, Negative and Zero correlation are a part of this type of crelation.

The relation between two variables is positive upto a pointand then it becomes negative.

It is a kind of ratio which expresses the extent to which changes in one variable are accomplained with the changes in other variable.
It involves no units and varies from -1 (perfect negative correlation) to +1 (perfect positive correlation). A coefficient of correlation equal to zero means no correlation between the variables.

When there are relatively few measures in the series, the rank difference method of calculating correlation is generally employed.
As the name indicates, the size of the correlation coefficient depends upon the sum of the differences in the ranks between each pair of measures (A pair meaning one individual's measures in the two variables).

ρ = 1 - (6 × ∑d²) / [N (N² - 1)]
where, ρ = Coefficient of correlation
d = Difference between ranks of two variables
N = Total number of measures

Let's understand it with the help of an example -

ρ = 1 - (6 × ∑d²) / [N (N² - 1)]
ρ = 1 - (6 × 92) / [11 × (11² - 1)]
ρ = 1 - (6 × 92) / (11 × 120)
ρ = 1 - 23 / 55
ρ = 1 - 0.42
ρ = 0.58

As the value of ρ = 0.58, we can say that the series have positive moderate correlation.

Let's consider another example -

ρ = 1 - (6 × ∑d²) / [N (N² - 1)]
ρ = 1 - (6 × 74.5) / [12 × (12² - 1]
ρ = 1 - (6 × 74.5) / (12 × 143)
ρ = 1 - 149 / 572
ρ = 1 - 0.262
ρ = 0.738

As the value of ρ = 0.738, we can say that the series have positive high correlation.

In these types of cases we found the average of the ranks of the individuals.

If two students with equal marks are at the rank 3 and 4, in this case we take the average of the ranks 3 and 4, i.e., (3+4)/2 = 3.5, thus we take both students at a rank of 3.5 and the next student is taken at 5th rank.
If three students with equal marks are at the rank 4, 5 and 6, in this case we take their rank as (4+5+6)/3 = 5 and the next student is taken at 7th rank.