著者

柏木 繁男

出版者

日本教育心理学会

雑誌

教育心理学研究 (ISSN:00215015)

巻号頁・発行日

vol.16, no.4, pp.13-25, 1968-12

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One of the purposes of this paper is to apply the theory of the failure distribution of system reliability (e. g., cf. Barlow et al ; 1967) to the analysis of the frequency distribution of tes scores. Assuming that test scores, x s, correlate positively with a underlying ability, we define as follows : R(x)=1-F(x) and R(0)=1, (1) where F(x)=・^x_0 f(t)dt. And further we define λ(x)=F'(x)/(1-F(X))=f(x)/(1-F(x))=-R'(x)/R(x), (2) which is called the ratio of success. From (2) we get R(x)=exp{-・^x_0λ(t)dt} F(x)=1-exp{-・^x_0λ(t)dt} (3) (Davis ; 1952, McGill et al ; 1965). Another one of the purposes is to recommend for the use of "Weibull" (1952) distribution in order to analyze the frequency distribution of test scores. The distribution has often been used in the studies of system reliability because of its wide applicability. We define the ration of success of this distribution as follows : λ(x)=m/α(x-γ)^<m-1>, x&ge;γ ; 0, x&le;γ. (12) Then, R(x)=exp{-(x-γ)^m/α} {F(x)=1-exp{-(x-γ)^m/α} (10) are obtained. Here, m, α, and γ are shape, scale, and location parameters. Tha shape parameter m of weibull distribution plays an important role in discriminating the degree of the difficulties and validities of psychological tests. We investigate several data by Lord (1952) and it is shown that the shape parameter m is a suprisingly useful and powerful measure inpsychological testing (cf. Fig. 1). Finally, it should be noted the follwoing. That is, our proposal which is based upon a differential equation model is comparable to Lord's (1952) integral equation model in that the former aims to analyze directly the frequency distribution of test scores and the latter to estimate true-score or latent trait distribution.