# 卡隆巴系數

${\displaystyle \rho _{T}={k^{2}{\overline {\sigma _{ij}}} \over \sigma _{X}^{2}}}$，當中
${\displaystyle {\overline {\sigma _{ij}}}}$ 係指每對題目之間嘅協方差（covariance）嘅平均值
${\displaystyle \sigma _{X}^{2}}$ 指「啲題目嘅變異數（variance）嘅總和」加埋「題目之間嘅協方差總和」；即係話
${\displaystyle \sigma _{X}^{2}=\sum _{i=1}^{k}\sum _{j=1}^{k}\sigma _{ij}=\sum _{i=1}^{k}\sigma _{i}^{2}+\sum _{i=1}^{k}\sum _{j\neq {i}}^{k}\sigma _{ij}}$（有關呢啲數學符號嘅意思，可以睇吓加總）；

## 例子

${\displaystyle k=4,{\overline {\sigma _{ij}}}=6,}$
${\displaystyle \sigma _{X}^{2}=\sum _{i=1}^{k}\sigma _{i}^{2}+\sum _{i=1}^{k}\sum _{j\neq {i}}^{k}\sigma _{ij}=(10+11+12+13)+4*(4-1)*6=118,}$
${\displaystyle \rho _{T}={4^{2}*6 \over 118}=.8135}$

${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{3}}$ ${\displaystyle X_{4}}$
${\displaystyle X_{1}}$ ${\displaystyle 10}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 6}$
${\displaystyle X_{2}}$ ${\displaystyle 6}$ ${\displaystyle 11}$ ${\displaystyle 6}$ ${\displaystyle 6}$
${\displaystyle X_{3}}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 12}$ ${\displaystyle 6}$
${\displaystyle X_{4}}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 6}$ ${\displaystyle 13}$

## 攷

1. Cho, E. (2016). Making reliability reliable: A systematic approach to reliability coefficients. Organizational Research Methods, 19(4), 651–682.
2. Green, S. B., & Yang, Y. (2009). Commentary on coefficient alpha: A cautionary tale. Psychometrika, 74(1), 121–135.