Kuder-Richardson formula

The Kuder-Richardson formula is a measure from multivariate statistics that determines the extent to which a group of test items can be viewed as a measurement of a single latent variable (not directly observable from outside ) . That is, it indicates the extent to which different items basically measure the same thing.

The measure is mainly used in the social sciences and in psychology , especially in test construction and evaluation. It is used to estimate the reliability of a psychometric instrument.

The formula is considered to be the forerunner of the measure of Cronbach's Alpha and differs only slightly from this measure. In contrast to Cronbach's Alpha, the Kuder-Richardson formula is applied to items with only two possible answers. One speaks here of dichotomous items.

The formula divides the scale into as many test halves as there are items. Correlations between these test halves are calculated accordingly and the values obtained are "extrapolated" to the respective length of the scale (see also Spearman-Brown formula ).

In SPSS, the formula is automatically used for dichotomous data when the "Coefficient Alpha" item is selected.

There are different formulas for the calculation, depending on the data given. So there is the KR-Formula 8, KR-Formula 20, KR-Formula 21.

KR Formula 8

${\ displaystyle r_ {tt} = {s_ {X} ^ {2} - \ sum _ {i = 1} ^ {n} p_ {i} q_ {i} \ over 2 * s_ {X} ^ {2} } + {\ sqrt {{\ sum _ {i = 1} ^ {n} r_ {it} ^ {2} + p_ {i} q_ {i} \ over s_ {X} ^ {2}} + \ left ({s_ {X} ^ {2} - \ sum _ {i = 1} ^ {n} p_ {i} q_ {i} \ over 2 * s_ {X} ^ {2}} \ right) ^ {2 }}}}$

KR Formula 20

${\ displaystyle r_ {tt} = {n \ over {n-1}} \ cdot \ left ({s_ {X} ^ {2} - \ sum _ {i = 1} ^ {n} p_ {i} q_ {i} \ over s_ {X} ^ {2}} \ right)}$

${\ displaystyle r_ {tt}}$ : Result of the KR formula, as an average correlation
${\ displaystyle n}$ : Number of questions
${\ displaystyle s_ {X} ^ {2}}$ : Variance of the test raw values
${\ displaystyle p}$ : Difficulty P / 100
${\ displaystyle q}$ : 1-p
${\ displaystyle r_ {it}}$ : Selectivity

Individual evidence

↑ ^a ^b M. Amelang, L. Schmidt-Atzert: Psychological diagnosis and intervention . Springer-Verlag, Heidelberg 2006, p. 146.

[Amelang-1] M. Amelang, L. Schmidt-Atzert: Psychological diagnosis and intervention . Springer-Verlag, Heidelberg 2006, p. 146.