correlation

(noun)

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

Related Terms

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.

  • 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.

  • 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.

  • 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 product-moment 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 product-moment 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.

  • The Correlation Coefficient r

    • If r = 1, there is perfect positive correlation.
    • If r = − 1, there is perfect negative correlation.
    • NOTE : Strong correlation does not suggest that x causes y or y causes x.
    • We say "correlation does not imply causation."
    • (a) A scatter plot showing data with a positive correlation. 0 < r < 1 (b) A scatter plot showing data with a negative correlation. − 1 < r < 0 (c) A scatter plot showing data with zero correlation. r=0

  • Hypothesis Tests with the Pearson Correlation

    • We need to look at both the value of the correlation coefficient $r$ and the sample size $n$, together.
    • We decide this based on the sample correlation coefficient $r$ and the sample size $n$.
    • If the test concludes that the correlation coefficient is significantly different from 0, we say that the correlation coefficient is "significant."
    • If the test concludes that the correlation coefficient is not significantly different from 0 (it is close to 0), we say that correlation coefficient is "not significant. "
    • There IS a significant linear relationship (correlation) between $x$ and $y$ in the population.