# Spearman Brown test

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The **Spearman Brown test**, **Spearman-Brown prediction formula** (also known as the *Spearman-Brown*
*prophecy 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 achieve 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 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.

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## References

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