Chapter 6: Correlation

Correlation is a statistical technique used to measure the relationship or association between two variables. If a change in one variable results in a corresponding change in the other series, they are said to be correlated.

Properties of Correlation

• Correlation measures the degree and direction of relationship.
• The coefficient of correlation (\(r\)) always lies between -1 and +1 (\(-1 \le r \le 1\)).
• If \(r = 1\), there is a perfect positive correlation; if \(r = -1\), there is a perfect negative correlation; if \(r = 0\), there is no correlation.

Scatter Diagram

A scatter diagram is a simple visual tool to determine if two variables are related. It involves plotting pairs of observations as points on a graph.

Positive Correlation Scatter Plot

When values of both variables move in the same direction.

Measuring Correlation

Karl Pearson’s Method

Karl Pearson developed a mathematical method for measuring the linear relationship between two continuous variables. $$ r = \frac{\sum xy}{\sqrt{\sum x^2 \times \sum y^2}} $$ Where:

• \(x = X - \bar{X}\) (deviation of X from its mean)
• \(y = Y - \bar{Y}\) (deviation of Y from its mean)

Spearman’s Rank Correlation

When data cannot be measured quantitatively but can be ranked, Spearman’s rank correlation coefficient (\(R\)) is used. This is especially useful for qualitative data like intelligence or beauty.

For Non-Repeated Ranks: $$ R = 1 - \frac{6 \sum D^2}{N^3 - N} $$ Where \(D\) is the difference between ranks of the corresponding pairs of variables (\(R_1 - R_2\)) and \(N\) is the number of pairs.

For Repeated Ranks: An adjustment/correction factor \(\frac{1}{12}(m^3 - m)\) is added to \(\sum D^2\) for every tied rank, where \(m\) is the number of times a rank is repeated.

Competency-Based Questions

Scenario Analysis

Q1. Two judges independently rank 10 contestants in a singing competition. You calculate Spearman’s rank correlation coefficient and find \(R = -0.85\). (a) Interpret this result. (b) Does this signify that the judges have highly similar or highly dissimilar tastes?

Answer: (a) An \(R = -0.85\) indicates a strong negative correlation between the rankings given by the two judges. (b) This signifies that the judges have highly dissimilar tastes. A contestant ranked high by one judge is very likely ranked low by the other judge, and vice versa.

Knowledge Question

Q2. Identify the method of correlation analysis appropriate for determining the relationship between the honesty and punctuality of 15 employees. (a) Karl Pearson’s method (b) Spearman’s Rank method (c) Scatter Diagram (d) Time Series

Answer: (b) Spearman’s Rank method. Honesty and punctuality are qualitative attributes that cannot be measured in exact units but can be ranked in order of preference.