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    Method document

    Sample size to show that Cronbach’s alpha exceeds a benchmark

    How many participants are needed to show that a scale’s internal consistency exceeds a benchmark such as 0.70, by Feldt’s exact F test, from the expected alpha and the number of items. Covers dropout.

    Engine version 1.0.0 · Reviewed 9 October 2026

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    See also: Sample size to estimate Cronbach’s alpha

    https://statwisely.com/methods/measurement-cronbach-alpha/

    When to use it

    • Validating a questionnaire whose internal consistency must meet a stated standard.

    Not for

    • Reporting alpha with a confidence interval, with no benchmark to meet. Use Cronbach’s alpha under “Estimate a value”.

    Notation

    nn
    Number of participants
    kk
    Number of items
    α0, α1\alpha_0,\ \alpha_1
    Benchmark alpha, and the alpha expected
    Fa,bF_{a,b}
    F distribution on a and b degrees of freedom
    α\alpha
    Significance level: the chance of a false positive
    1−β1-\beta
    Power: the chance of detecting the effect if it is real
    τ\tau
    Number of tails: 2 for a two-sided test, 1 for one-sided
    qq
    Expected dropout

    Formulas

    Feldt’s exact F test

    1−β=P ⁣(Fn−1, (n−1)(k−1)>1−α11−α0 Fn−1, (n−1)(k−1); 1−α)1-\beta = P\!\left(F_{n-1,\,(n-1)(k-1)} > \frac{1-\alpha_1}{1-\alpha_0}\, F_{n-1,\,(n-1)(k-1);\,1-\alpha}\right)

    (1 − α₀)/(1 − α̂) follows an F distribution when alpha is α₀; when it is α₁ the same statistic is that F scaled by (1 − α₀)/(1 − α₁). The smallest n reaching the power is found by search [1, 2].

    Dropout

    nrecruit=⌈n1−q⌉n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

    Applied last, to the participants actually recruited.

    Parameters

    Expected alphaα1\alpha_1

    Values: Between 0 and 1

    From a pilot or an earlier study of the scale.

    Benchmark alphaα0\alpha_0

    Values: 0 or more, below 1

    Often 0.70 for group comparisons, 0.90 for decisions about individuals.

    Number of itemskk

    Values: At least 2

    Every participant answers all of them.

    Significance levelα\alpha

    Values: Between 0 and 1; usually 0.05

    The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

    Power1−β1-\beta

    Values: Between 0 and 1; usually 0.80 or 0.90

    The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

    One- or two-sidedτ\tau

    Values: 2 (two-sided) or 1 (one-sided)

    Two-sided is the default. A one-sided test is justified only when an effect in the other direction would be treated exactly as no effect, and should be stated in advance.

    Expected dropoutqqOptional

    Values: 0% up to, but not including, 100%

    The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

    Assumptions

    • The items are essentially tau-equivalent with normal errors.
    • Benchmark tests are one-sided by default: the property must be shown to exceed the benchmark, not merely to differ from it.
    • Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

    Worked examples

    Alpha 0.80 against 0.70

    A 10-item scale with alpha 0.8 against 0.7, one-sided: n = 83, with power 0.8026042293383746.

    Sample size: 83

    Protocol statement

    To show that alpha exceeds 0.7 when it is 0.8 with 80% power at a one-sided significance level of 0.05, using Feldt’s F test of Cronbach’s alpha for a 10-item scale, the required sample size is 83 participants. Calculated with StatWisely (statwisely.com) using: N: Feldt's F test, (1-α₀)/(1-α̂) ~ F(n-1, (n-1)(k-1)). These are planning estimates; confirm the final design with a qualified statistician.

    Open this example in the calculator

    A four-item scale

    With 4 items: n = 100.

    Sample size: 100

    Open this example in the calculator

    A two-sided test

    Testing in both directions at α = 0.05: n = 104.

    Sample size: 104

    Open this example in the calculator

    Validation

    • Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
    • An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
    • Simulated ten-item scales at the planned size, tested with Feldt’s F test, gave power of 0.804 against a planned 0.803.

    How to cite

    StatWisely. Sample size to show that Cronbach’s alpha exceeds a benchmark. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/measurement-cronbach-alpha/

    Cite the original references for the formulas your design uses as well.

    References

    1. 1.Feldt LS. The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika. 1965;30(3):357–370.
    2. 2.Bonett DG. Sample size requirements for testing and estimating coefficient alpha. Journal of Educational and Behavioral Statistics. 2002;27(4):335–340. doi:10.3102/10769986027004335