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The Kuder Richardson Coefficient of reliability (KR 20) is used to test the reliability of binary measurements
such as exam questions, to see if the items within the instruments obtained the same binary (no/yes, right/wrong) results
over a population of testing subjects.
The formula for the coefficient can easily be obtained from Wikipedia on the Internet. Please Note that the KR 20 was first described in 1937. Hoyt in 1940 modified the formula so that it can be applied to measurements that are not binary. Hoyt's modification eventually was popularised and is now known as Cronbach's Alpha. Cronbach's Alpha, when applied to binary data, will therefore produce the same result as KR20. Cronbach's Alpha is now much preferred, and will be discussed in its own page. Example The example data is artificially created to demonstrate the procedures, and not real. It purports to be a study of 4 no/yes questions in an exam. The questions are tested on 5 students. Please note This data set is deliberately small to make the demonstration easier to interpret. In a real study, many more questions (many tens) and students (serveral hundreds) would be required to ensure the results are reproducible.
We have 4 multiple choice questions (T1 to T4), administered to 5 students, as shown in the table on th.e left.
The number 0 is used to represent the wrong answer, and 1 the correct answer. The data set to be analysed is therefore as shown in the table to the right, and the results are KR 20 = 0.75 The interpretation of the KR 20 value is similar to that of Kappa. A KR 20 of <0.2 is considered poor agreement, 0.210.4 fair, 0.410.6 moderate, 0.610.8 strong, and more than 0.8 near complete agreement. The original descriptions of KR 20 provided no test of statistical significance or confidence interval, although these can be obtained using the Cronbach's Alpha algorithm. ReferencesKuder, G. F. ; M. W. Richardson (1937)The theory of the estimation of test reliability. Psychometrika, 2: 15160 https://en.wikipedia.org/wiki/Kuder%E2%80%93Richardson_formulas Kuder Richardson Coefficient on Wikipedia
the following is the algorithm for Kuder Richardson Coefficient of Reliability
dat = (" 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 1 ") mx = read.table(textConnection(dat),header=FALSE) rows = nrow(mx) cols = ncol(mx) SumPQ = 0 # sum of pq for each item or col for(j in 1:cols) { p = 0 for(i in 1:rows) { if(mx[i,j]==1) { p = p + 1 } } p = p / rows SumPQ = SumPQ + p * (1.0  p) } ex = 0 exx = 0 for(i in 1:rows) { p = 0 for(j in 1:cols) { if(mx[i,j]==1) { p = p + 1 } } ex = ex + p exx = exx + p * p } v = (exx  ex*ex/rows) / (rows) # variance between subjects kr_20 = (1.0 * cols / (cols  1.0)) * ((v  SumPQ) / v) kr_20 # Kuder Richardson Coefficient of ReliabilityThe result is > kr_20 # Kuder Richardson Coefficient of Reliability [1] 0.7536232
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