Cronbach’s Alpha Proof

Proof

Property 1 (of Cronbach’s Alpha Basic Concepts): 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 Correlation, var(ti–tj) = var(ti) + var(tj) – 2 cov(ti, tj) , and since var(ti–tj) ≥ 0, it follows that var(ti) + var(tj) ≥ 2 cov(ti, tj). 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,
image2632Reference

Astivia, O. L. O. (2024) Why is Cronbach’s alpha a lower bound to reliability?
https://psychometroscar.com/why-is-cronbachs-alpha-a-lower-bound-to-reliability/

Bendermacher, N. (2010) Beyond alpha: lower bounds for the reliability of tests
https://core.ac.uk/download/pdf/56682969.pdf

10 thoughts on “Cronbach’s Alpha Proof”

  2. Hello Sir,
    I have collected the data of 128 respondents by survey. Each questionnaire contains 37 questions in YES and NO format. Now, how i can calculate Cronbach’s alpha to check the reliability of questionnaire? Is 37 items (37 questions) are sufficient for Cronbach’s alpha?

    Reply

    • Pankaj,
      The Real Statistics website explains how to perform these calculations. Alternatively you can use the Real Statistics software to do the calculations for you.
      Cronbach’s alpha can certinly be calculated with 37 items.
      Charles

      Reply

  3. 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

    Reply

  4. 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

    Reply

    • 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

      Reply

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