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In statistics (classical test theory), Cronbach's α {displaystyle alpha } (alpha) is a name used for tau-equivalent reliability ( ρ T {displaystyle ho _{T}} ) as an (lowerbound) estimate of the reliability of a psychometric test. Synonymous terms are: coefficient alpha, Guttman's λ 3 {displaystyle lambda _{3}} , Hoyt method and KR-20. In statistics (classical test theory), Cronbach's α {displaystyle alpha } (alpha) is a name used for tau-equivalent reliability ( ρ T {displaystyle ho _{T}} ) as an (lowerbound) estimate of the reliability of a psychometric test. Synonymous terms are: coefficient alpha, Guttman's λ 3 {displaystyle lambda _{3}} , Hoyt method and KR-20. It has been proposed that α {displaystyle alpha } can be viewed as the expected correlation of two tests that measure the same construct. By using this definition, it is implicitly assumed that the average correlation of a set of items is an accurate estimate of the average correlation of all items that pertain to a certain construct. Cronbach's α {displaystyle alpha } is a function of the number of items in a test, the average covariance between item-pairs, and the variance of the total score. It was first named alpha by Lee Cronbach in 1951, as he had intended to continue with further coefficients. The measure can be viewed as an extension of the Kuder–Richardson Formula 20 (KR-20), which is an equivalent measure for dichotomous items. Alpha is not robust against missing data. Several other Greek letters have been used by later researchers to designate other measures used in a similar context. Somewhat related is the average variance extracted (AVE). This article discusses the use of α {displaystyle alpha } in psychology, but Cronbach's alpha statistic is widely used in the social sciences, business, nursing, and other disciplines. The term item is used throughout this article, but items could be anything—questions, raters, indicators- for all of which, one might ask, to what extent they 'measure the same thing.' Items that are manipulated are commonly referred to as variables. It has been argued that the term 'Cronbach's α {displaystyle alpha } ' be abandoned in favour of 'tau-equivalent reliability' ( ρ T {displaystyle ho _{T}} ); and that in many cases an alternative approach, congeneric reliability, should be used to calculate reliability instead. Scale purification, i.e. 'the process of eliminating items from multi-item scales' (Wieland et al., 2017) can influence Cronbach's alpha. A framework presented by Wieland et al. (2017) highlights that both statistical and judgmental criteria need to be taken under consideration when making scale purification decision. Suppose that we measure a quantity which is a sum of K {displaystyle K} components (K-items or testlets): X = Y 1 + Y 2 + ⋯ + Y K {displaystyle X=Y_{1}+Y_{2}+cdots +Y_{K}} . Cronbach's α {displaystyle alpha } is defined as where σ X 2 {displaystyle sigma _{X}^{2}} is the variance of the observed total test scores, and σ Y i 2 {displaystyle sigma _{Y_{i}}^{2}} the variance of component i for the current sample of persons.

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