Cronbach’s Alpha – Advanced

Property 1 (of Cronbach’s Alpha): Let xj = tj + ej where each ej is independent of tj and all the ej are independent of each other. Also let x0 = \sum\nolimits_{j=1}^k x_k and t0 = \sum\nolimits_{j=1}^k t_k. Then the reliability of x0 ≥ α where α is Cronbach’s alpha.

Here we view the xj as the measured values, the tj as the true values and the ej as the measurement error values.

Proof: By Property 5 of Basic Concepts of Correlationvar(ti–tj) = var(ti) + var(tj)  2 cov(titj) , and since var(ti–tj) ≥ 0, it follows that var(ti) + var(tj) ≥ 2 cov(titj). Since for each i there are k – 1 j for which j ≠ i, it follows that

image2626

and so by symmetry,
image2627

It now follows by Property 5 of Basic Concepts of Correlation

image2628 image2629

But  for  i ≠ j
image2630

Also
image2631

Thus,
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5 Responses to Cronbach’s Alpha – Advanced

  1. Bekhal says:

    Hello ,
    How do I know that my questionnaire is reliable after I use Cronbach’s alpha for that purpose ? I mean the result of the process should be around what number?

    Bekhal

  2. Bekhal says:

    Hello ,
    I used a questionnaire of five Likert scale and the questionnaire itself has many categories and each category has many subsets .is it possible to use Cronbach’s alpha to assure the reliability of the questionnaire and how can i use it ?if you could clarify that for me, I would be grateful
    thanks a lot
    Bekhal /Iraq

    • Charles says:

      Bekhal,
      As long as all the questions in the questionnaire are measuring the same concept you can calculate Cronbach’s Alpha. If each of the categories is measuring a different concept then you need to calculate a separate Cronbach’s Alpha for each concept/category.
      Charles

  3. SEIDJEMAL says:

    IT is v.nice

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