correlation
Psychology
Statistics
Sociology
A reciprocal, parallel or complementary relationship between two or more comparable objects.
Examples of correlation in the following topics:

Properties of Pearson's r
 State the relationship between the correlation of Y with X and the correlation of X with Y
 A correlation of 1 means a perfect negative linear relationship, a correlation of 0 means no linear relationship, and a correlation of 1 means a perfect positive linear relationship.
 Pearson's correlation is symmetric in the sense that the correlation of X with Y is the same as the correlation of Y with X.
 For example, the correlation of Weight with Height is the same as the correlation of Height with Weight.
 For instance, the correlation of Weight and Height does not depend on whether Height is measured in inches, feet, or even miles.

Correlation and Causation
 A correlation can be positive/direct or negative/inverse.
 Ice cream consumption is positively correlated with incidents of crime.
 It is important to not confound a correlation with a cause/effect relationship.
 This diagram illustrates the difference between correlation and causation, as ice cream consumption is correlated with crime, but both are dependent on temperature.
 Thus, the correlation between ice cream consumption and crime is spurious.

Rank Correlation
 To illustrate the nature of rank correlation, and its difference from linear correlation, consider the following four pairs of numbers $(x, y)$:
 This means that we have a perfect rank correlation and both Spearman's correlation coefficient and Kendall's correlation coefficient are 1.
 This graph shows a Spearman rank correlation of 1 and a Pearson correlation coefficient of 0.88.
 In contrast, this does not give a perfect Pearson correlation.
 Define rank correlation and illustrate how it differs from linear correlation.

Describing linear relationships with correlation
 We denote the correlation by R.
 Figure 7.10 shows eight plots and their corresponding correlations.
 The correlation is intended to quantify the strength of a linear trend.
 Sample scatterplots and their correlations.
 Sample scatterplots and their correlations.

Correlational Research
 The attributes of correlations include strength and direction.
 Direction refers to whether the correlation is positive or negative.
 A correlation of 0 indicates no relationship between the variables.
 It is extremely rare to find a perfect correlation between two variables, but the closer the correlation is to 1 or +1, the stronger the correlation is.
 Always remember that correlation does not imply causation.

Other Types of Correlation Coefficients
 Other types of correlation coefficients include intraclass correlation and the concordance correlation coefficient.
 Thus, if we are correlating $X$ and $Y$, where, say, $Y=2X+1$, the Pearson correlation between $X$ and $Y$ is 1: a perfect correlation.
 The concordance correlation coefficient is nearly identical to some of the measures called intraclass correlations.
 Comparisons of the concordance correlation coefficient with an "ordinary" intraclass correlation on different data sets will find only small differences between the two correlations.
 Distinguish the intraclass and concordance correlation coefficients from previously discussed correlation coefficients.

Values of the Pearson Correlation
 Give the symbols for Pearson's correlation in the sample and in the population
 The Pearson productmoment correlation coefficient is a measure of the strength of the linear relationship between two variables.
 It is referred to as Pearson's correlation or simply as the correlation coefficient.
 The symbol for Pearson's correlation is "$\rho$" when it is measured in the population and "r" when it is measured in a sample.
 Because we will be dealing almost exclusively with samples, we will use r to represent Pearson's correlation unless otherwise noted.

95% Critical Values of the Sample Correlation Coefficient Table

An Intuitive Approach to Relationships
 Correlation refers to any of a broad class of statistical relationships involving dependence.
 These are all examples of a statistical factor known as correlation.
 Correlation refers to any of a broad class of statistical relationships involving dependence.
 Familiar examples of dependent phenomena include the correlation between the physical statures of parents and their offspring and the correlation between the demand for a product and its price.
 This graph shows a positive correlation between world population and total carbon emissions.

Coefficient of Correlation
 The most common coefficient of correlation is known as the Pearson productmoment correlation coefficient, or Pearson's $r$.
 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.
 If $r=1$, there is perfect positive correlation.
 If $r=1$, there is perfect negative correlation.
 Put the summary statistics into the correlation coefficient formula and solve for $r$, the correlation coefficient.