Introduction to Correlation

In previous chapters, you've learned how to summarise a single dataset. Now, we will explore how to examine the relationship between two variables. This is the essence of correlation analysis.

In our daily lives, we often observe relationships between different phenomena. For example:

• As the summer temperature rises, the number of visitors to hill stations increases, and so do ice-cream sales.
• As the supply of tomatoes increases in the local market, their price drops.

Correlation analysis provides a systematic means to examine such relationships. It helps answer questions like:

• Is there a relationship between two variables?
• If the value of one variable changes, does the value of the other also change?
• Do both variables move in the same direction or in opposite directions?
• How strong is the relationship?

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What Does Correlation Measure?

Correlation studies and measures the direction and intensity of the relationship between variables. It's crucial to understand that correlation measures covariation, not causation. The presence of a correlation between two variables, X and Y, simply means that when X changes, Y also changes in a definite way. It does not mean that a change in X causes a change in Y.

For instance, a brisk sale of ice-creams might be correlated with a higher number of deaths by drowning. This doesn't mean eating ice cream causes drowning. Instead, a third variable—rising temperature—leads to both increased ice-cream sales and more people going swimming, which in turn might increase the number of drowning incidents.

For simplicity, we will assume that the correlation, if it exists, is linear, meaning the relationship can be represented by a straight line on a graph.

Types of Correlation

Correlation is commonly classified based on the direction of the relationship between the two variables.

1. Positive Correlation

The correlation is said to be positive when the two variables move together in the same direction. If one variable increases, the other variable also increases. If one decreases, the other also decreases.

Examples:

• As income rises, consumption also rises.
• As temperature rises, the sale of ice-cream also rises.

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2. Negative Correlation

The correlation is negative when the two variables move in opposite directions. If one variable increases, the other variable decreases, and vice versa.

Examples:

• When the price of apples falls, the demand for them increases.
• As you spend more time studying, the chances of failing decline.

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3. No Correlation

When there is no discernible relationship between the movements of two variables, it is a case of no correlation. Changes in one variable do not lead to any definite change in the other.

Example: The relationship between the size of your shoes and the amount of money in your pocket is likely a coincidence with no real correlation.

Techniques for Measuring Correlation

There are three important tools used to study and measure correlation:

1. Scatter Diagrams: A graphical method to visually assess the nature of the relationship.
2. Karl Pearson’s Coefficient of Correlation: A numerical measure of the linear relationship between two variables.
3. Spearman’s Rank Correlation: A measure of the linear association between the ranks of variables, particularly useful for qualitative data.

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Scatter Diagram

A scatter diagram is a graphical technique where the values of two variables are plotted as points on a graph paper. This visual representation provides a good idea of the nature and strength of the relationship without calculating any numerical value.

The pattern of the scattered points reveals the relationship:

• If the points cluster around an upward rising line, it indicates a positive correlation (Fig. 6.1).
• If the points cluster around a downward sloping line, it indicates a negative correlation (Fig. 6.2).
• If the points are spread randomly with no discernible pattern, it indicates no correlation (Fig. 6.3).
• If all the points lie exactly on an upward or downward sloping line, it is a case of perfect positive or perfect negative correlation, respectively (Fig. 6.4 and Fig. 6.5).
• The relationship can also be non-linear if the points form a curve (Fig. 6.6 and Fig. 6.7).

Karl Pearson’s Coefficient of Correlation

Also known as the product-moment correlation coefficient, this is a precise numerical measure of the degree of linear relationship between two variables, X and Y. It is denoted by 'r'. It is advisable to first examine a scatter diagram to ensure the relationship is linear before calculating this coefficient.

Formula for Karl Pearson’s Coefficient (r)

The correlation coefficient is the covariance of X and Y divided by the product of their standard deviations.

$ r = \frac{Cov(X,Y)}{\sigma_x \sigma_y} $

Where:

• $Cov(X,Y) = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{N} = \frac{\sum xy}{N}$
• $\sigma_x$ is the standard deviation of X.
• $\sigma_y$ is the standard deviation of Y.

The formula can be expressed in various forms for calculation:

$ r = \frac{\sum xy}{\sqrt{\sum x^2 \sum y^2}} \quad \text{where } x = X - \bar{X} \text{ and } y = Y - \bar{Y} $

Or, for direct calculation from raw data:

$ r = \frac{N \sum XY - (\sum X)(\sum Y)}{\sqrt{[N \sum X^2 - (\sum X)^2][N \sum Y^2 - (\sum Y)^2]}} $

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Properties of the Correlation Coefficient (r)

• No Unit: 'r' is a pure number and has no unit.
• Direction of Relationship:
  • A negative value of r indicates an inverse relationship (variables move in opposite directions).
  • A positive value of r indicates a direct relationship (variables move in the same direction).
• Range of Value: The value of r always lies between -1 and +1 (i.e., $-1 \le r \le +1$).
• Interpretation of Value:
  • If $r = 0$, the variables are uncorrelated (no linear relationship).
  • If $r = +1$ or $r = -1$, the correlation is perfect.
  • A value of r close to +1 or -1 indicates a strong linear relationship.
  • A value of r close to 0 indicates a weak linear relationship.
• Unaffected by Change of Origin and Scale: The value of r is not affected by adding/subtracting a constant from the variables (change of origin) or by multiplying/dividing them by a constant (change of scale). This property is used in the step-deviation method to simplify calculations.

Spearman’s Rank Correlation

Developed by the British psychologist C.E. Spearman, this method measures the linear association between the ranks assigned to individual items, not their actual values. It is denoted by $r_k$.

When to Use Rank Correlation

Spearman’s rank correlation is particularly useful in the following situations:

1. When the data is qualitative and cannot be measured numerically but can be ranked. For example, ranking individuals based on attributes like honesty, beauty, or intelligence.
2. When the relationship between variables is clearly non-linear, but the direction of the relationship is consistent.
3. When the data contains extreme values (outliers), as rank correlation is not affected by them, unlike Pearson’s correlation.

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Formula for Spearman’s Rank Correlation ($r_k$)

The formula for calculating the rank correlation coefficient is:

$ r_k = 1 - \frac{6 \sum D^2}{n(n^2 - 1)} $

Where:

• $D$ = The difference between the ranks assigned to a pair of observations ($R_X - R_Y$).
• $n$ = The number of pairs of observations.

Calculation of Rank Correlation

The process involves three main scenarios:

1. When Ranks are Given: Simply calculate the differences (D) between the ranks for each pair, square them ($D^2$), sum them up ($\sum D^2$), and apply the formula.
2. When Ranks are Not Given: First, assign ranks to the values of each variable separately. The highest value is typically given rank 1, the next highest rank 2, and so on. Then, proceed as above.
3. When Ranks are Repeated (Tied Ranks): If two or more items have the same value, a common rank is assigned to them. This common rank is the average of the ranks they would have occupied if they were slightly different. For example, if two items are tied for the 3rd and 4th positions, both are assigned the rank $ (3+4)/2 = 3.5 $.

When ranks are repeated, a correction factor is added to the $\sum D^2$ term in the formula:

$ r_k = 1 - \frac{6 \left( \sum D^2 + \frac{m_1(m_1^2 - 1)}{12} + \frac{m_2(m_2^2 - 1)}{12} + \dots \right)}{n(n^2 - 1)} $

Where $m_1, m_2, \dots$ are the number of times each rank is repeated.

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