Statistics Definitions > Rank-Biserial Correlation

## What is Rank-Biserial Correlation?

Rank-biserial correlation (rank biserial *r *or *r*_{rb}) is used when you want to find a correlation between dichotomous (binary) nominal data and ordinal (ranked) data. It is a special case of **Somers’ D.**

## Formula

The formula (assuming there are no tied ranks) is:

**r _{rb} = 2 * (Y_{1} – Y_{0}) / n.**

**Where**:

- n = number of data pairs in the sample,
- Y
_{0}= Y score means for data pairs with x = 0, - Y
_{1}= Y score means for data pairs with x = 1.

For example, let’s say you had the following data:

Dichotomous variable: 1,1,1,0,1

Ordinal variable: 3,1,5,4,2

- Y
_{0}= 4 (only one ordinal variable is paired with 0). - Y
_{1}= 3+1+5+2/4 = 11/4 = 2.75 - n = 5

Giving a rank-biserial correlation coefficient of: 2 * (2.75 – 4)/6 = -0.21.

## Running the Test

- In
**SAS**: Run the %BISERIAL macro. - In
**SPSS**: Click Analyze → Correlate → Bivariate. Add your variables, deselect Pearson (the default) and click Spearman.* Click OK.

*Glass(1966) noted that the rank biserial correlation is appropriate to estimate Spearman correlation, so I’m assuming this works both ways. You should note though, that it is an estimate.

## Matched Pairs Rank Biserial

Kerby (2014) suggested the following for calculating matched pairs rank biserial correlation:

- Run a Wilcoxon signed ranks test.
- Add the sum of negative ranks and the sum of positive ranks (from the output). This equals the total sum of ranks.
- Divide the sum of negative ranks by the total sum of ranks to get a proportion.
- Divide the sum of positive ranks by the total sum of ranks to get a proportion.
- Find the difference between the two proportions. This is the matched pairs rank biserial.

**References**:

Glass, G. V. (1966). Note on rank biserial correlation. Educational and Psychological Measurement, 26, 623-631. Retrieved Jan 1, 2017 from here: http://journals.sagepub.com/doi/pdf/10.1177/001316446602600307

Kerby, D. S. (2014). The simple difference formula: an approach to teaching nonparametric correlation. Innovative Teaching, 3(1), Article-1.

Willson, V. L. (1976). Critical values of the rank-biserial correlation coefficient. Educational and Psychological Measurement, Vol. 36, pp. 297-300.