The

**Spearman-Brown prediction formula**(also known as the

*Spearman-Brown*

*prophesy formula*) is a formula relating psychometric reliability to test length:

where is the predicted reliability; N is the number of "tests" combined (see below); and 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, adding N parallel forms of the current exam to 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 acheive a degree of reliability:

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 acheived 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.

[insert something about the history, a link to Charles Spearman, who was Brown? and including the original citation]