Cramér’s V Coefficient

The Cramér’s V (also known as Cramér’s φ) is one of a number of correlation statistics developed to measure the strength of association between two nominal variables. Cramér’s V is a nonparametric statistic used in cross-tabulated table data. These data are usually measured at the nominal level, although some researchers will use Cramér’s V with ordinal data or collapsed (grouped) interval or ration data. Although an italicized capital V is most often used as the symbol for the statistic (V), the lowercase Greek letter φ with a subscripted c may also be used as follows: φc. This entry further describes the Cramér’s V and discusses its assumptions, calculation, and interpretation. It concludes with an example of the use of the Cramér’s V.

The V is a nonparametric inferential statistic used to measure correlation (also known as effect or effect size) for cross-tabulated tables when the variables have more than two levels. It is the effect size statistic of choice for tables greater than 2 × 2 (read two-by-two). Typical significance statistics for those tables include the chi-square and the maximum likelihood chi-square. The data in columns and rows should be nominal, although the V is frequently used with ordinal variables and collapsed interval/ratio data. Unlike the contingency coefficient, the V can be used when there are an unequal number of rows and columns. For example, the researcher should choose the V when the table has two columns and three rows.

The Cramér’s V was developed by Carl Harald Cramér, a Swedish mathematician known for his work on analytic number theory and probability theory. Based on Karl Pearson’s chi-square statistic, the V was developed to measure the size of the effect for significant chi-square tables.

The V is a correlation statistic, and as such, it measures the strength of an association between two variables. The V statistic provides two items of information:

• First, it answers the question, “Do these two variables covary?” That is, does one variable change when the other changes? (i.e., are the two variables independent?)
• Second, the size of the V describes the strength of the association. As the V approaches one level, the association is stronger. In a perfect correlation, for every one level of rise in one variable, the other variable would change exactly one level. The value of a V statistic can range only from 0 to +1.0; it cannot be a negative number. (Given that the calculation requires the square root of a number, the result cannot be negative with the standard formula.)

Many statistical computer programs (e.g., STATA, SPSS, and SAS) compute the V statistic as an option to accompany the output of the chi-square statistic, and the significance of V is the same as the significance of the chi-square.