So far we have studied measures that describe a single variable: averages that summarise its central tendency and measures of dispersion that describe its variability. But economics is full of questions about the relationship between two variables: Does an increase in income raise consumption? Do higher prices reduce demand? Does more rainfall increase agricultural output? The statistical tool that studies the relationship between two or more variables is called correlation.
Correlation is a statistical measure that indicates the degree and direction of the relationship between two variables. If the two variables move in the same direction, the correlation is positive; if they move in opposite directions, it is negative; and if there is no systematic relationship, there is no correlation. Correlation is measured by the correlation coefficient, the most common being Karl Pearson's coefficient of correlation.
In this chapter we study the meaning and types of correlation, the scatter diagram, Karl Pearson's coefficient of correlation and its computation for raw data, rank correlation, and the important cautions about interpreting correlation, especially the crucial difference between correlation and causation.
Correlation is the statistical measure of the degree to which two variables move together. It answers the question: when one variable changes, does the other change in a predictable way? The three characteristics of correlation are direction (positive or negative), degree (perfect, high, moderate, low or zero) and form (linear or non-linear).
Positive correlation: Two variables are positively correlated when they move in the same direction - one increases as the other increases, and one decreases as the other decreases. Examples: family income and consumption expenditure; rainfall and agricultural output; temperature in summer and sales of cold drinks.
Negative correlation: Two variables are negatively correlated when they move in opposite directions - one increases as the other decreases. Examples: price of a commodity and its demand (other things equal); number of doctors in a region and the incidence of a curable disease; fuel efficiency and vehicle speed beyond the optimum.
Zero correlation: When there is no systematic relationship between the variables, e.g. height of a person and his income, the correlation is zero or negligible.
Linear and non-linear correlation: When the changes in the two variables have a constant proportional relationship, the correlation is linear; when the relationship is curvilinear, it is non-linear. Pearson's coefficient measures only linear correlation.
Perfect correlation: If the correlation coefficient is exactly +1 or -1, the correlation is said to be perfect positive or perfect negative. Perfect correlation means one variable is an exact function of the other.
The scatter diagram is the simplest way to visualise correlation. The values of one variable are plotted on the X-axis and the values of the other on the Y-axis; each pair of values gives one point on the graph. The pattern of the points reveals the nature of the relationship:
The closer the points cluster around the line, the stronger the correlation. A scatter diagram shows only the direction and rough degree of correlation; it does not give a numerical measure.
The most widely used measure of correlation is Karl Pearson's coefficient of correlation, denoted by r. It measures the degree of linear relationship between two variables and always lies between -1 and +1. It is based on the covariance of the two variables standardised by their standard deviations.
$$r = \frac{\sum (X - \bar{X})(Y - \bar{Y})}{\sqrt{\sum (X - \bar{X})^2 \times \sum (Y - \bar{Y})^2}}$$
where X and Y are the two variables and \bar{X}, \bar{Y} are their means.
Short-cut formula: Using the deviations from assumed means (dx = X - A and dy = Y - B):
$$r = \frac{N \sum dxdy - \sum dx \cdot \sum dy}{\sqrt{N \sum dx^2 - (\sum dx)^2} \cdot \sqrt{N \sum dy^2 - (\sum dy)^2}}$$
Properties of the coefficient of correlation:
For discrete and continuous series, the formula is applied with frequencies as weights:
$$r = \frac{N \sum fdxdy - \sum fdx \cdot \sum fdy}{\sqrt{N \sum fdx^2 - (\sum fdx)^2} \cdot \sqrt{N \sum fdy^2 - (\sum fdy)^2}}$$
where f is the joint frequency of each pair of values, and for continuous series the mid-values of the classes are used. The computations are usually arranged in a correlation table, and the step-deviation method may be applied when the class intervals are equal.
When the data are qualitative or can only be ranked - for example, the ranking of students by teachers, or the ranking of countries by different indices - Karl Pearson's method cannot be applied directly. In such cases we use Spearman's rank correlation coefficient, denoted by R. The observations are first ranked, and the ranks are then correlated.
$$R = 1 - \frac{6 \sum d^2}{N(N^2 - 1)}$$
where d is the difference between the ranks of the two variables for each observation, and N is the number of observations. When equal ranks are given to some items, a correction factor is applied for each group of tied ranks:
$$\text{Correction} = \frac{m(m^2 - 1)}{12}$$
for each group of m tied ranks, which is added to Sum d^2.
The rank correlation coefficient also lies between -1 and +1. A value of R = +1 indicates perfect agreement in the rankings, and R = -1 indicates perfect disagreement (reverse ranking).
The most important caution in the study of correlation is that correlation does not imply causation. Two variables may be correlated for reasons other than a causal link between them:
Hence a high correlation coefficient only indicates the degree of association, not that one variable causes the other. Establishing causation requires controlled experimentation and economic reasoning, not just correlation.
Correlation is a valuable tool in economics because it helps in prediction, business decisions, understanding the strength of relationships, and in estimating the value of one variable from another. However, its limitations must be noted: it measures only linear relationships, it may be unduly affected by a few extreme observations, it does not establish causation, and a spurious correlation may appear when a third variable influences both. Therefore correlation analysis must always be supplemented by careful economic reasoning.
| Type of Correlation | Direction | Example | r value |
|---|---|---|---|
| Perfect positive | Same direction, exact | X = 2Y | r = +1 |
| Positive | Same direction | Income and consumption | 0 < r < 1 |
| Negative | Opposite direction | Price and demand | -1 < r < 0 |
| Perfect negative | Opposite, exact | X = -2Y | r = -1 |
| Zero | No relationship | Height and income | r = 0 |
| Method | Formula | When Used |
|---|---|---|
| Scatter diagram | Graph of points | Visual idea of correlation |
| Karl Pearson's r | Covariance/(sdx x sdy) | Quantitative linear data |
| Spearman's rank R | 1 - 6Sum d^2/N(N^2-1) | Qualitative/ranked data |
This chapter introduced correlation, the statistical tool that measures the degree and direction of the relationship between two variables. We distinguished positive, negative and zero correlation, and linear from non-linear relationships, and used the scatter diagram to visualise them. We studied Karl Pearson's coefficient of correlation, which measures linear correlation on a scale from -1 to +1, and its computation by direct and short-cut methods for individual, discrete and continuous series. We examined Spearman's rank correlation for qualitative or ranked data and the treatment of tied ranks. Most importantly, we emphasised the distinction between correlation and causation, warning that a high coefficient of association never by itself proves that one variable causes another. Correlation analysis is widely used in economics for prediction and decision making, and its companion measures, index numbers, form the subject of the next chapter.