Spearman-Brown Formula

The Spearman-Brown formula is used to calculate the reliability of a psychological test after changing its length (length = number of items). This requires the reliability (Rel) of the original test. The Spearman-Brown formula is as follows: ${\displaystyle l}$

${\displaystyle \operatorname {Rel} (k\cdot l)={\frac {k\cdot \operatorname {Rel} }{1+(k-1)\cdot \operatorname {Rel} }}}$

where is the factor by which the test was extended (or shortened); Rel is the reliability. Rel (k · l) is the new reliability after changing the test length. The formula can be used, for example, to calculate how much longer a test would have to be in order to increase its reliability to a certain value. ${\displaystyle k}$

application

If a test is extended by the original length, there is also an increase in the reliability of the test for mathematical reasons. This increase in reliability can be captured by the Spearman-Brown formula.

Let it be a test that is divided into two equally long, parallel halves and . The reliability Rel of the two test halves is usually calculated by their correlation or . The reliability of the overall test is now given by the following expression: ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle r(Y,Z)}$ ${\displaystyle r(Z,Y)}$ ${\displaystyle X}$

${\displaystyle \operatorname {Rel} (X)={\frac {2\cdot \operatorname {r} (Y,Z)}{1+\operatorname {r} (Y,Z)}}}$

If the reliability of the overall test is calculated in this way, one speaks of split-half reliability or test halving reliability.

literature

• Moosbrugger, H. and Kelava, A .: Test theory and questionnaire construction , Springer Medizin Verlag, Heidelberg 2007, ISBN 978-3-540-71634-1