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Coefficient of Correlation
The correlation coefficient is a measure of the linear dependence between two variables
Learning Objective

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

covariance
A measure of how much two random variables change together.

correlation
One of the several measures of the linear statistical relationship between two random variables, indicating both the strength and direction of the relationship.
Full Text
The most common coefficient of correlation is known as the Pearson productmoment correlation coefficient, or Pearson's
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
Pearson's correlation coefficient when applied to a sample is commonly represented by the letter
An equivalent expression gives the correlation coefficient as the mean of the products of the standard scores. Based on a sample of paired data
Mathematical Properties
 The value of
$r$ is always between$1$ and$+1$ :$1\leq r \leq 1$ .  The size of the correlation
$r$ indicates the strength of the linear relationship between$x$ and$y$ . Values of$r$ close to$1$ or$+1$ 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
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
To find (
Put the summary statistics into the correlation coefficient formula and solve for
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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: Controlling for a Variable, Type I and II Errors, and Lab 1: Confidence Interval (Home Costs)
 deviation
 Appears in these related concepts: Variance, Basic properties of point estimates, and Introduction to Bivariate Data
 line
 Appears in these related concepts: Plotting Lines, Line, and Qualities of Line
 mean
 Appears in these related concepts: Mean, Variance, and Standard Deviation of the Binomial Distribution, Understanding Statistics, and Averages
 population
 Appears in these related concepts: Applications of Statistics, The Functionalist Perspective on Deviance, and Quorum Sensing
 sample
 Appears in these related concepts: Identifying Product Benefits, Surveys, and Basic Inferential Statistics
 standard deviation
 Appears in these related concepts: Using the Normal Curve, Interpreting the Standard Deviation, and Variance
 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: What is a Linear Function?, Math Review, and Introduction to Variables
Sources
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Cite This Source
Source: Boundless. “Coefficient of Correlation.” Boundless Statistics. Boundless, 26 May. 2016. Retrieved 28 Jul. 2016 from https://www.boundless.com/statistics/textbooks/boundlessstatisticstextbook/correlationandregression11/correlation44/coefficientofcorrelation2082660/