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    Method document

    Sample size to estimate Cronbach’s alpha

    How many participants are needed to estimate a scale’s internal consistency, Cronbach’s alpha, within a chosen margin, from the expected alpha and the number of items. Covers non-response.

    Engine version 1.0.0 · Reviewed 9 October 2026

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

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

    When to use it

    • Reporting the internal consistency of a new or translated questionnaire with a confidence interval.

    Not for

    • Showing that alpha exceeds a benchmark such as 0.70. Use the test of Cronbach’s alpha under “Detect an effect”, Measurement.

    Notation

    nn
    Number of participants
    kk
    Number of items in the scale
    αc\alpha_c
    Expected Cronbach’s alpha
    EE
    Margin of error: the interval runs from α_c − E to α_c + E
    ee
    Ratio of the interval’s ends on the 1 − alpha scale, (1 − α_c + E)/(1 − α_c − E)
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Participants for an interval of width 2E

    n=⌈8kk−1(z1−α/2ln⁡e)2+2⌉n = \left\lceil \frac{8k}{k-1} \left(\frac{z_{1-\alpha/2}}{\ln e}\right)^{2} + 2 \right\rceil

    Feldt’s interval for alpha is symmetric for ln(1 − alpha), whose variance is about 2k/[(k − 1)(n − 2)]; requiring the interval to run from α_c − E to α_c + E on that scale gives Bonett’s formula [1, 2].

    Non-response

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

    The number to approach so that n respond. Applied last, after any clustering.

    Parameters

    Confidence level1−α1-\alpha

    Values: 90%, 95% or 99%

    95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

    Margin of errorEE

    Values: Above 0 and below 1

    Half the interval’s width on the coefficient’s scale: 0.05 gives an interval from 0.75 to 0.85 around an alpha of 0.80.

    Expected alphaαc\alpha_c

    Values: Between 0 and 1, with α_c ± E inside it

    From a pilot or an earlier study of the same scale in a similar population.

    Number of itemskk

    Values: At least 2

    More items make alpha more precise for the same number of participants, though the gain is small beyond about ten.

    Expected non-response rateqqOptional

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

    The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

    Assumptions

    • The items are essentially tau-equivalent, with normally distributed errors, and every participant answers all of them.
    • Non-response is unrelated to the outcome.

    Worked examples

    A ten-item scale

    Alpha 0.8 within ±0.05, 10 items: e = 0.25/0.15 = 1.667 and ln e = 0.511, so n = 8.889 × (1.96/0.511)² + 2 = 132.9, rounded up to 133.

    Sample size: 133

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    A four-item scale

    With 4 items the factor 8k/(k − 1) rises from 8.9 to 10.7: n = 160.

    Sample size: 160

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    Allowing for non-response

    133 completed questionnaires, with 10% not returned: approach 148.

    Needed for analysis: 133 · To approach: 148

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    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.
    • Feldt intervals from simulated ten-item scales at the planned size had a mean half-width of 0.052 against a target of 0.05: the formula plans for the expected width, so about half of studies fall a little short. An independent implementation in R recomputes these sizes and is compared with the engine’s.

    How to cite

    StatWisely. Sample size to estimate Cronbach’s alpha. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/precision-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