Standard error of difference

Standard error of difference (SEdiff) 為2次測驗分數差異的標準差 (X:第一次測量；Y: 第二次測量)

Var (Dbar) = (SEdiff)²

         =Var(X - Y)

         =Var(X)+Var(Y)–2*Cov(X, Y)

         =(SE_X)²+(SE_Y)²–2*SE_X*SE_Y

        

Cov(X, Y) =Σ(X-Xbar)*(Y-Ybar)/n-1

Σ(X-Xbar)*(Y-Ybar) =Σ(X-Xbar)*(Y-Ybar)

                  =Σ(T_X,i+ε_X,i-T_X,i)*(T_Y,i+ε_Y,i-T_Y,i)

                  =Σ(ε_X,i*ε_Y,i)

                  =0

èCov(X, Y)=0

èSEdiff=√[(SE_X)²+(SE_Y)²]

bar: mean (e.g., Xbar is mean of X)

T: true score

ε: error

i:第i個人

在CTT，個別受測者之第一次測量和第二次測量的測量誤差(standard error of measurement, SEM)會一樣èSEdiff=√2(SEM)²

在IRT，個別受測者之第一次測量和第二次測量的測量誤差(standard error)不一樣èSEdiff=√[(SE_X)²+(SE_Y)²]