Statistical Sampling and Regression: Convariance andCorrelationCovariance and correlation describe how two variables are related. • Variables are positively related if they move in the same direction. • Variables are inversely related if they move in opposite directions.Both covariance and correlation indicate whether variables are positively orinversely related. Correlation also tells you the degree to which the variables tendto move together.You are probably already familiar with statements about covariance andcorrelation that appear in the news almost daily. For example, you might hearthat as economic growth increases, stock market returns tend to increase as well.These variables are said to be positively related because they move in the samedirection. You may also hear that as world oil production increases, gasolineprices fall. These variables are said to be negatively, or inversely, related becausethey move in opposite directions.The relationship between two variables can be illustrated in a graph. In theexamples below, the graph on the left illustrates how the positive relationshipbetween economic growth and market returns might appear. The graph indicatesthat as economic growth increases, stock market returns also increase. The graphon the right is an example of how the inverse relationship between oil productionand gasoline prices might appear. It illustrates that as oil production increases,gas prices fall.To determine the actual relationships of these variables, you would use theformulas for covariance and correlation.CovarianceCovariance indicates how two variables are related. A positive covariance meansthe variables are positively related, while a negative covariance means thevariables are inversely related. The formula for calculating covariance of sampledata is shown below.
x = the independent variabley = the dependent variablen = number of data points in the sample = the mean of the independent variable x = the mean of the dependent variable yTo understand how covariance is used, consider the table below, which describesthe rate of economic growth (xi) and the rate of return on the S&P 500 (yi).Using the covariance formula, you can determine whether economic growth andS&P 500 returns have a positive or inverse relationship. Before you compute thecovariance, calculate the mean of x and y. (The Summary Measures topic of theDiscrete Probability Distributions section explains the mean formula in detail.)
Now you can identify the variables for the covariance formula as follows.x = 2.1, 2.5, 4.0, and 3.6 (economic growth)y = 8, 12, 14, and 10 (S&P 500 returns) = 3.1 = 11Substitute these values into the covariance formula to determine the relationshipbetween economic growth and S&P 500 returns.
The covariance between the returns of the S&P 500 and economic growth is 1.53.Since the covariance is positive, the variables are positively related—they movetogether in the same direction.CorrelationCorrelation is another way to determine how two variables are related. In additionto telling you whether variables are positively or inversely related, correlation alsotells you the degree to which the variables tend to move together.As stated above, covariance measures variables that have different units ofmeasurement. Using covariance, you could determine whether units wereincreasing or decreasing, but it was impossible to measure the degree to whichthe variables moved together because covariance does not use one standard unitof measurement. To measure the degree to which variables move together, youmust use correlation.Correlation standardizes the measure of interdependence between two variablesand, consequently, tells you how closely the two variables move. The correlationmeasurement, called a correlation coefficient, will always take on a value between1 and – 1: • If the correlation coefficient is one, the variables have a perfect positive correlation. This means that if one variable moves a given amount, the second moves proportionally in the same direction. A positive correlation coefficient less than one indicates a less than perfect positive correlation, with the strength of the correlation growing as the number approaches one. • If correlation coefficient is zero, no relationship exists between the variables. If one variable moves, you can make no predictions about the movement of the other variable; they are uncorrelated. • If correlation coefficient is –1, the variables are perfectly negatively correlated (or inversely correlated) and move in opposition to each other. If one variable increases, the other variable decreases proportionally. A negative correlation coefficient greater than –1 indicates a less than perfect negative correlation, with the strength of the correlation growing as the number approaches –1.Test your understanding of how correlations might look graphically. In the boxbelow, choose one of the three sets of purple points and drag it to the correlationcoefficient it illustrates: 1, –1, or 0. If your choice is correct, an explanation of thecorrelation will appear. Remember to close the Instructions box before you begin.
This interactive tool illustrates the theoretical extremes of the idea of correlationcoefficients between two variables: 1, –1, or 0. These figures serve only toprovide an idea of the boundaries on correlations. In practice, most variables willnot be perfectly correlated, but they will instead take on a fractional correlationcoefficient between 1 and –1.To calculate the correlation coefficient for two variables, you would use thecorrelation formula, shown below.r(x,y) = correlation of the variables x and yCOV(x, y) = covariance of the variables x and ysx = sample standard deviation of the random variable xsy = sample standard deviation of the random variable yEarlier in this discussion, you saw how the covariance of S&P 500 returns andeconomic growth was calculated using data from the following table. Now considerhow their correlation is measured.
To calculate correlation, you must know the covariance for the two variables andthe standard deviations of each variable. From the earlier example, you know thatthe covariance of S&P 500 returns and economic growth was calculated to be1.53. Now you need to determine the standard deviation of each of the variables.You would calculate the standard deviation of the S&P 500 returns and theeconomic growth from the above example as follows. (For a more detailedexplanation of calculating standard deviation, refer to the Summary Measurestopic of the Discrete Probability Distributions section of the course.)
Using the information from above, you know thatCOV(x,y) = 1.53sx = 0.90sy = 2.58Now you can calculate the correlation coefficient by substituting the numbersabove into the correlation formula, as shown below.
A correlation coefficient of .66 tells you two important things: • Because the correlation coefficient is a positive number, returns on the S&P 500 and economic growth are postively related. • Because .66 is relatively far from indicating no correlation, the strength of the correlation between returns on the S&P 500 and economic growth is strong.Both covariance and correlation identified that the variables are positively related.By standardizing measures, correlation is also able to measure the degree towhich the variables tend to move together.In business, covariance and correlation are used frequently to analyze marketreturns for anything from an individual stock to a market composite. In addition,marketing executives use covariance and correlation to understand theinterdependence between consumer behavior and the consumption of theirproducts.1. If there is a positive relationship between the scores of job incumbents on a jobknowledge test and actual job performance, which of the following graphs wouldmost likely be an accurate representation of this situation?
Solution 12. In each of the graphs, are job performance and test performance shown to bepositively related, inversely related, or unrelated? Solution 23. Given the following return information, what is the covariance between thereturn of Stock A and the return of the market index? Solution 34. Using the table and your calculations from above, calculate the correlation of
Stock As returns and the return of the market index.