# Total proportion selected on two tests with known correlation

© Paul Cooijmans

## Explanation

When one selects the top proportion *q* on each of two tests, one is in total selecting more than *q* of the population because of the fact that tests in practice correlate less than unity. Were the correlation *r* between the tests 0, one would in total be selecting 2*q*, and for correlations greater than 0, the square of the correlation must be subtracted from this factor 2. This is so because the square of two variables' correlation is equal to the proportion of the variance that is shared between the two variables. Therefore:

*q*_{total} = ( 2 - *r*^{2} ) × *q*