Cramér's V

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In statistics, Cramér's V (sometimes referred to as Cramér's phi and denoted as φc) is a measure of association between two nominal variables, giving a value between 0 and +1 (inclusive). It is based on Pearson's chi-squared statistic and was published by Harald Cramér in 1946.[1]

Usage and interpretation[edit]

φc is the intercorrelation of two discrete variables[2] and may be used with variables having two or more levels. φc is a symmetrical measure: it does not matter which variable we place in the columns and which in the rows. Also, the order of rows/columns doesn't matter, so φc may be used with nominal data types or higher (notably, ordered or numerical).

Cramér's V may also be applied to goodness of fit chi-squared models when there is a 1 × k table (in this case r = 1). In this case k is taken as the number of optional outcomes and it functions as a measure of tendency towards a single outcome.[citation needed]

Cramér's V varies from 0 (corresponding to no association between the variables) to 1 (complete association) and can reach 1 only when each variable is completely determined by the other. It may be viewed as the association between two variables as a percentage of their maximum possible variation.

φc2 is the mean square canonical correlation between the variables.[citation needed]

In the case of a 2 × 2 contingency table Cramér's V is equal to the absolute value of Phi coefficient.

Note that as chi-squared values tend to increase with the number of cells, the greater the difference between r (rows) and c (columns), the more likely φc will tend to 1 without strong evidence of a meaningful correlation.[3]


Let a sample of size n of the simultaneously distributed variables and for be given by the frequencies

number of times the values were observed.

The chi-squared statistic then is:

where is the number of times the value is observed and is the number of times the value is observed.

Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the minimum dimension minus 1:


  • is the phi coefficient.
  • is derived from Pearson's chi-squared test
  • is the grand total of observations and
  • being the number of columns.
  • being the number of rows.

The p-value for the significance of V is the same one that is calculated using the Pearson's chi-squared test.[citation needed]

The formula for the variance of Vc is known.[4]

In R, the function cramerV() from the package rcompanion[5] calculates V using the chisq.test function from the stats package. In contrast to the function cramersV() from the lsr[6] package, cramerV() also offers an option to correct for bias. It applies the correction described in the following section.

Bias correction[edit]

Cramér's V can be a heavily biased estimator of its population counterpart and will tend to overestimate the strength of association. A bias correction, using the above notation, is given by[7]






Then estimates the same population quantity as Cramér's V but with typically much smaller mean squared error. The rationale for the correction is that under independence, .[8]

See also[edit]

Other measures of correlation for nominal data:

Other related articles:


  1. ^ Cramér, Harald. 1946. Mathematical Methods of Statistics. Princeton: Princeton University Press, page 282 (Chapter 21. The two-dimensional case). ISBN 0-691-08004-6 (table of content Archived 2016-08-16 at the Wayback Machine)
  2. ^ Sheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press.
  3. ^ Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Routledge. p78-81.
  4. ^ Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32. (pages 15–16)
  5. ^ "Rcompanion: Functions to Support Extension Education Program Evaluation". 2019-01-03.
  6. ^ "Lsr: Companion to "Learning Statistics with R"". 2015-03-02.
  7. ^ Bergsma, Wicher (2013). "A bias correction for Cramér's V and Tschuprow's T". Journal of the Korean Statistical Society. 42 (3): 323–328. doi:10.1016/j.jkss.2012.10.002.
  8. ^ Bartlett, Maurice S. (1937). "Properties of Sufficiency and Statistical Tests". Proceedings of the Royal Society of London. Series A. 160 (901): 268–282. doi:10.1098/rspa.1937.0109. JSTOR 96803.

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