The most common coefficient of correlation is known as the Pearson product-moment correlation coefficient, or Pearson's r. It is a measure of the linear correlation (dependence) between two variables X and Y, giving a value between +1 and −1. It is widely used in the sciences as a measure of the strength of linear dependence between two variables. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.
Pearson's correlation coefficient between two variables is defined as the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
Pearson's correlation coefficient when applied to a population is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient.
Pearson's correlation coefficient when applied to a sample is commonly represented by the letter r and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. The formula for r is as follows: .
An equivalent expression gives the correlation coefficient as the mean of the products of the standard scores. Based on a sample of paired data (X_{i}, Yi), the sample Pearson correlation coefficient is shown in .
Mathematical Properties
- The value of r is always between -1 and +1: -1≤r≤1.
- The size of the correlation r indicates the strength of the linear relationship between x and y. Values of r close to -1 or to +1 indicate a stronger linear relationship between x and y.
- If r=0 there is absolutely no linear relationship between x and y (no linear correlation).
- A positive value of r means that when x increases, y tends to increase and when x decreases, y tends to decrease (positive correlation).
- A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation).
- If r=1, there is perfect positive correlation. If r=-1, there is perfect negative correlation. In both these cases, all of the original data points lie on a straight line. Of course, in the real world, this will not generally happen.
- The Pearson correlation coefficient is symmetric.
Another key mathematical property of the Pearson correlation coefficient is that it is invariant to separate changes in location and scale in the two variables. That is, we may transform X to a + bX and transform Y to c + dY, where a, b, c, and d are constants, without changing the correlation coefficient. This fact holds for both the population and sample Pearson correlation coefficients.
Example
Consider the following example data set of scores on a third exam and scores on a final exam: .
To find the correlation of this data we need the summary statistics; means, standard deviations, sample size, and the sum of the product of x and y.
To find (xy), multiply the x and y in each ordered pair together then sum these products. For this problem, Σ(xy) = 122,500. To find the correlation coefficient we need the mean of x, the mean of y, the standard deviation of x and the standard deviation of y.
x = 69.1818, y = 160.4545, s_{x }= 2.85721, s_{y }= 20.8008, Σ(xy) = 122,500
Put the summary statistics into the correlation coefficient formula and solve for r, the correlation coefficient.