Reliability, Kuder-Richardson Formula[Page 1418]

Formulas

The formula for KR20 is

$$\mathrm{KR}20=n/n-1\times \left[1-((pq)/\mathrm{Var})\right]$$

In this formula, KR20 is the output and is the score that serves as an estimation of the internal reliability of the measure. n is simply the number of items that the researcher has; p is the proportion of people passing the item. So if two-thirds of the people got the item correct, p would be 0.66; q is the proportion of people who failed the item. This is directly related to p, as p and q combined must equal 1. A researcher can determine q by taking 1 minus p. So if p is 0.66, q would be .34. The sum of p and q would be 1. Var is the variance of your entire test. The variance can be determined by taking the standard deviation and squaring it.

The formula for KR21 is:

$$\mathrm{KR}21=n/n-1\times \left[1-Mxn-Mn\times \mathrm{Var}\right]$$

In this formula, KR21 is the output and represents an estimation of the reliability of the test. n represents the number of items the researcher has; M stands for the mean score of the test. Var represents the variance of the entire test, which can be found by squaring the standard deviation.

Many statistics programs offer the ability to calculate these values, as calculating them by hand can be time consuming and can increase the possibility for calculation errors. For instance, the SPSS program can calculate KR-20 as a special case of Cronbach’s alpha. Since the KR-21 is used less frequently, it is not always included in statistical analysis programs.

When to Use

The KR-20 is used to analyze a measure that features dichotomous items. The most common use for the KR-20 is for the analysis of tests of knowledge or learning. These tests feature one correct answer for each item, meaning that even if the question presents the respondent with multiple options, only one of them would be considered correct and all the others would be incorrect. The fact that answers can be split into two categories, correct and incorrect, is what makes these type of questions dichotomous in nature, even when the item itself has more than two potential responses. The KR-20 can even be used to analyze fill-in-the-blank questions, where there are no potential responses offered to respondents. The KR-20 is used only when all items being analyzed are dichotomous. It is almost meant to be employed when the questions vary in difficulty. The KR-20 cannot be used if the test allows for some responses to earn partial credit, as this would mean that the item is no longer dichotomous since responses could be grouped as correct, incorrect, or partially correct.

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