Spearman-Brown prediction formula

The Spearman-Brown prediction formula (also known as the "Spearman-Brown" "prophecy formula") is a formula relating psychometric reliability to test length:

:$\{\; ho\}^*\_\{xx\text{'}\}=frac\{N\{\; ho\}\_\{xx\text{'}\{1+(N-1)\{\; ho\}\_\{xx\text{'}$

where $\{\; ho\}^*\_\{xx\text{'}\}$ is the predicted reliability; "N" is the number of "tests" combined (see below); and $\{\; ho\}\_\{xx\text{'}\}$ is the reliability of the current "test". The formula predicts the reliability of a new test composed by replicating the current test "N" times (or, equivalently, creating a test with "N" parallel forms of the current exam). Thus "N" = 2 implies doubling the exam length by adding items with the same properties as those in the current exam. Values of "N" less than one may be used to predict the effect of shortening a test.

The formula can also be rearranged to predict the number of replications required to achieve a degree of reliability:

:$N=frac$ ho}^*_{xx'}(1-{ ho}_{xx'})} ho}_{xx'}(1-{ ho}^*_{xx'})}

This formula is commonly used by psychometricians to predict the reliability of a test after changing the test length. This relationship is particularly vital to the split-half and related methods of estimating reliability.

The formula is also helpful in understanding the nonlinear relationship between test reliability and test length.

If the longer/shorter test is not parallel to the current test, then the prediction will not be strictly accurate. For example, if a highly reliable test was lengthened by adding many poor items then the achieved reliability will probably be much lower than that predicted by this formula.

Item response theory "item information" provides a much more precise means of predicting changes in the quality of measurement by adding or removing individual items.