Kuder–Richardson formulas

Kuder–Richardson Formulas are statistical measures used in psychometrics to assess the internal consistency reliability of a test score for tests that are dichotomously scored. Developed by Maurice Kuder and M. W. Richardson in 1937, these formulas provide a means to quantify the reliability of binary-scored tests, such as true/false or yes/no assessments. The most commonly used Kuder–Richardson formulas are KR-20 and KR-21.

Overview

The concept of reliability in psychometric testing refers to the extent to which a test is consistent in its measurement of a construct. A high reliability indicates that the test scores are consistent over repeated applications of the test and thus, the test is considered to be measuring something consistently. The Kuder–Richardson formulas are specifically designed for tests where questions have only two possible outcomes. They are a special case of the more general coefficient alpha (Cronbach's Alpha) used for tests with multiple-choice questions.

Kuder–Richardson Formula 20 (KR-20)

The KR-20 formula is used for tests in which each item can be scored only as correct or incorrect. It is an estimate of the test's reliability based on the inter-item correlations. The formula is given by:

\[ KR-20 = \frac{K}{K-1} \left(1 - \frac{\sum p_i q_i}{\sigma^2_t}\right) \]

where:

• \(K\) is the number of items on the test,
• \(p_i\) is the proportion of examinees answering item \(i\) correctly,
• \(q_i = 1 - p_i\) (the proportion of examinees answering item \(i\) incorrectly),
• \(\sigma^2_t\) is the variance of the total test scores across examinees.

Kuder–Richardson Formula 21 (KR-21)

The KR-21 formula provides a simpler calculation for estimating test reliability, assuming that all items on the test have approximately the same difficulty level. The formula is expressed as:

\[ KR-21 = \frac{K}{K-1} \left(1 - \frac{K \bar{p} \bar{q}}{\sigma^2_t}\right) \]

where:

• \(K\) is the number of items,
• \(\bar{p}\) is the average proportion of correct answers for all items,
• \(\bar{q} = 1 - \bar{p}\) is the average proportion of incorrect answers,
• \(\sigma^2_t\) is the variance of the total test scores.

Applications and Limitations

Kuder–Richardson formulas are widely used in educational and psychological testing to evaluate the reliability of binary-scored tests. They are particularly useful in the development and validation of new tests, as well as in research settings where the reliability of a test needs to be established.

However, the application of KR-20 and KR-21 is limited to tests with dichotomous outcomes. They also assume that the test items are unidimensional, meaning they measure a single construct. In cases where a test includes items of varying difficulty or measures multiple constructs, other reliability coefficients, such as Cronbach's Alpha, may be more appropriate.

See Also

• Psychometrics
• Reliability (statistics)
• Cronbach's Alpha
• Item Response Theory