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

  • covariance
  • regression
  • control
  • variable

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.
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