coefficient of variation

(noun)

The ratio of the standard deviation to the mean.

Related Terms

Examples of coefficient of variation in the following topics:

  • Coefficient of Determination

    • The coefficient of determination provides a measure of how well observed outcomes are replicated by a model.
    • The coefficient of determination (denoted $r^2$) is a statistic used in the context of statistical models.
    • The coefficient of determination is actually the square of the correlation coefficient.
    • Therefore, the coefficient of determination is $r^2 = 0.6631^2 = 0.4397$.
    • Interpret the properties of the coefficient of determination in regard to correlation.

  • The Coefficient of Determination

    • r2 is called the coefficient of determination. r2 is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form. r2 has an interpretation in the context of the data:
    • r2 , when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
    • 1-r2 , when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line.
    • Approximately 44% of the variation (0.4397 is approximately 0.44) in the final exam grades can be ex- plained by the variation in the grades on the third exam, using the best fit regression line.
    • Therefore approximately 56% of the variation (1 - 0.44 = 0.56) in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line.

  • 95% Critical Values of the Sample Correlation Coefficient Table

  • Rank Correlation

    • A rank correlation coefficient measures the degree of similarity between two rankings and can be used to assess the significance of the relation between them.
    • A rank correlation coefficient can measure that relationship, and the measure of significance of the rank correlation coefficient can show whether the measured relationship is small enough to be likely to be a coincidence.
    • They are best seen as measures of a different type of association rather than as alternative measure of the population correlation coefficient.
    • However, in the extreme case of perfect rank correlation, when the two coefficients are both equal (being both $+1$ or both $-1$), this is not in general so, and values of the two coefficients cannot meaningfully be compared.
    • This graph shows a Spearman rank correlation of 1 and a Pearson correlation coefficient of 0.88.

  • 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 between two variables is defined as the covariance of the two variables divided by the product of their standard deviations.
    • An equivalent expression gives the correlation coefficient as the mean of the products of the standard scores.
    • Based on a sample of paired data $(X_i, Y_i)$, the sample Pearson correlation coefficient is shown in:
    • 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 $y$.

  • Qualitative Variable Models

    • A dummy independent variable (also called a dummy explanatory variable), which for some observation has a value of 0 will cause that variable's coefficient to have no role in influencing the dependent variable, while when the dummy takes on a value 1 its coefficient acts to alter the intercept.
    • The intercept (the value of the dependent variable if all other explanatory variables hypothetically took on the value zero) would be the constant term for males but would be the constant term plus the coefficient of the gender dummy in the case of females.
    • Analysis of variance (ANOVA) models are a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups).
    • This type of ANOVA modelcan have differing numbers of qualitative variables.
    • Graph showing the regression results of the ANOVA model example: Average annual salaries of public school teachers in 3 regions of a country.

  • Hypothesis Tests with the Pearson Correlation

    • Pearson's correlation coefficient, $r$, tells us about the strength of the linear relationship between $x$ and $y$ points on a regression plot.
    • We need to look at both the value of the correlation coefficient $r$ and the sample size $n$, together.
    • The 95% critical values of the sample correlation coefficient table shown in gives us a good idea of whether the computed value of $r$ is significant or not.
    • Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied.
    • Use a hypothesis test in order to determine the significance of Pearson's correlation coefficient.

  • Estimating and Making Inferences About the Slope

    • Values of $b_1$, et cetera, (the "partial regression coefficients") and the intercept are found so that they minimize the squared deviations between the expected and observed values of $Y$.
    • The magnitude of the partial regression coefficient depends on the unit used for each variable.
    • Where $b'_1$ is the standard partial regression coefficient of $y$ on $X_1$.
    • The magnitude of the standard partial regression coefficients tells you something about the relative importance of different variables; $X$ variables with bigger standard partial regression coefficients have a stronger relationship with the $Y$ variable.
    • Discuss how partial regression coefficients (slopes) allow us to predict the value of $Y$ given measured $X$ values.

  • Testing the Significance of the Correlation Coefficient

    • The correlation coefficient, r, tells us about the strength of the linear relationship between x and y.
    • We need to look at both the value of the correlation coefficient r and the sample size n, together.
    • We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.
    • The sample correlation coefficient, r, is our estimate of the unknown population correlation coefficient.
    • The 95% Critical Values of the Sample Correlation Coefficient Table (Section 12.10) at the end of this chapter (before the Summary (Section 12.11)) may be used to give you a good idea of whether the computed value of r is significant or not.

  • Comparing Two Populations: Independent Samples

    • Nonparametric methods for testing the independence of samples include Spearman's rank correlation coefficient, the Kendall tau rank correlation coefficient, the Kruskal–Wallis one-way analysis of variance, and the Walk–Wolfowitz runs test.
    • Spearman's rank correlation coefficient, often denoted by the Greek letter $\rho$ (rho), is a nonparametric measure of statistical dependence between two variables.
    • The Kendall $\tau$ coefficient is defined as:
    • If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value $-1$.
    • consists of 6 runs, 3 of which consist of $+$ and the others of $-$.