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Coefficient of Correlation
The correlation coefficient is a measure of the linear dependence between two variables X and Y, giving a value between +1 and −1.
Learning Objectives

List the mathematical properties of the correlation coefficient.

Compute Pearson's productmoment correlation coefficient.
Key Points
 The correlation coefficient 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.
 Pearson's correlation coefficient when applied to a sample is commonly represented by the letter r.
 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.
Terms

correlation
One of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship.

covariance
A measure of how much two random variables change together.
Full Text
The most common coefficient of correlation is known as the Pearson productmoment 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 meanadjusted random variables; hence the modifier productmoment 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: .
Correlation Coefficient
This is the formula for calculating the coefficient of correlation.
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 .
Alternate Correlation Coefficient
This equivalent expression gives the correlation coefficient as the mean of the products of the standard scores.
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: .
Example
This table shows an 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.
Key Term Reference
 Pearson's correlation coefficient
 Appears in these related concepts: Other Types of Correlation Coefficients and Hypothesis Tests with the Pearson Correlation
 correlation coefficient
 Appears in these related concepts: Coefficient of Determination, Inferences of Correlation and Regression, and Overview of How to Assess StandAlone Risk
 datum
 Appears in these related concepts: Inferential Statistics, Misleading Graphs, and Mean: The Average
 deviation
 Appears in these related concepts: Outliers, Graphs of Qualitative Data, and Variance
 line
 Appears in these related concepts: Plotting Lines, Varieties of Line, and Qualities of Line
 mean
 Appears in these related concepts: Summation Notation, The Mean Value Theorem, Rolle's Theorem, and Monotonicity, and Understanding Statistics
 population
 Appears in these related concepts: The Purpose of Statistics, What Is a Confidence Interval?, and Collecting and Measuring Data
 sample
 Appears in these related concepts: Applications of Statistics, Sampling, and Defining the Sample and Collecting Data
 standard deviation
 Appears in these related concepts: Typical Shapes, Mean, Variance, and Standard Deviation of the Binomial Distribution, and Basic Descriptive Statistics
 standard score
 Appears in these related concepts: Change of Scale, Variation and Prediction Intervals, and When Does the ZTest Apply?
 statistics
 Appears in these related concepts: What Is Statistics?, Communicating Statistics, and Population Demography
 variable
 Appears in these related concepts: Related Rates, Math Review, and Psychology and the Scientific Method: From Theory to Conclusion
Sources
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Cite This Source
Source: Boundless. “Coefficient of Correlation.” Boundless Statistics. Boundless, 21 Jul. 2015. Retrieved 23 Jul. 2015 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/correlationandregression11/correlation44/coefficientofcorrelation2082660/