Coefficient of Correlation

The coefficient of correlation, usually written r and also called the Pearson correlation coefficient, is a statistic that measures the strength and direction of the linear relationship between two variables. It ranges from −1 to +1. A value of +1 means the two variables move together in a perfect straight line; −1 means they move in exactly opposite directions; and 0 means there is no linear relationship.

In accounting and finance, the coefficient of correlation is used to test whether a proposed cost driver, such as machine hours, really moves with a cost in cost accounting, such as maintenance expense, and to measure how investments move relative to each other. It is closely related to the coefficient of determination, which in a simple regression equals r squared, and to the broader idea of correlation.

Think of two friends walking through a park. If they always step together, side by side, their paths are perfectly correlated. If one always moves left whenever the other moves right, they are perfectly negatively correlated. If they wander about with no connection to each other, there is no correlation.

The coefficient of correlation puts a single number on how closely two things move together. The closer it is to +1 or −1, the more confidently you can use one to predict the other.

For paired observations of two variables, x and y, the coefficient is:

• $x$ is the independent variable, such as machine hours.
• $y$ is the dependent variable, such as maintenance cost.
• $\bar{x}$ and $\bar{y}$ are the averages of each variable.

A plant records machine hours (in hundreds) and maintenance cost (in thousands of dollars) for four months: (2, 3), (4, 5), (6, 6) and (8, 10). The averages are 5 hours and 6 thousand dollars. The deviations from the averages are (−3, −3), (−1, −1), (1, 0) and (3, 4). The sum of the products of the deviations is 9 + 1 + 0 + 12 = 22. The sum of squared x deviations is 9 + 1 + 1 + 9 = 20, and of squared y deviations 9 + 1 + 0 + 16 = 26. So r = 22 ÷ √(20 × 26) = 22 ÷ √520 ≈ 22 ÷ 22.80 ≈ 0.96.

A coefficient of about 0.96 indicates a very strong positive relationship. Squaring it gives about 0.93, meaning roughly 93% of the variation in maintenance cost is associated with machine hours, which supports using machine hours as the cost driver.

• Cost estimation. Before fitting a cost equation with regression analysis, accountants check that the chosen activity is strongly correlated with the cost. Weak correlation suggests a different driver is needed, and it also warns that simpler methods such as the high-low method may give misleading results.
• Overhead allocation. Activity-based costing relies on drivers that correlate well with the costs in each pool.
• Budgeting and forecasting. Correlated relationships, such as sales and advertising, help in preparing budgets.
• Portfolio management. Combining assets whose returns have low or negative correlation reduces overall risk, the principle behind diversification, and correlation with the market underlies measures such as beta.

Correlation does not prove causation. Two costs may both rise in winter without one causing the other, a trap well known in cost behavior studies. The coefficient measures only linear relationships, so a strong curved relationship can produce a low r. It is also sensitive to outliers: one unusual month, such as a plant shutdown, can raise or lower r sharply. With few observations, a high value may arise by chance, so accountants should look at a scatter plot, consider whether the relationship makes economic sense and test it on new data. Finally, a relationship observed within one relevant range of activity may not hold outside it.