Skip to main content
    Sample Size CalculatorBeta

    Method document

    Sample size to estimate Bland–Altman limits of agreement

    How many participants, each measured by two methods, are needed to estimate both 95% limits of agreement within a chosen margin, from the standard deviation of the differences. Covers non-response.

    Engine version 1.0.0 · Reviewed 9 October 2026

    Open in calculator

    See also: Sample size to show that two methods agree within a tolerance

    https://statwisely.com/methods/precision-limits-of-agreement/

    When to use it

    • A method-comparison study, such as a new device against a reference one, reporting the limits of agreement with their confidence intervals.

    Not for

    • Showing that the limits fall within a clinical tolerance. Use the limits-of-agreement test under “Detect an effect”, Measurement.

    Notation

    nn
    Number of participants
    ss
    Standard deviation of the differences between methods
    tn−1t_{n-1}
    t quantile on n − 1 degrees of freedom for confidence 1 − α
    EE
    Margin of error for each limit, in the units of measurement
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    The smallest n whose limits are estimated within ±E

    n=min⁡{n:tn−1 s1n+1.9622(n−1)≤E}n = \min\left\{ n : t_{n-1}\, s \sqrt{\frac{1}{n} + \frac{1.96^{2}}{2(n-1)}} \le E \right\}

    Each limit d̄ ± 1.96s has a variance of about s²[1/n + 1.96²/(2(n − 1))], from the mean difference and the standard deviation [1]; the t quantile keeps small studies honest.

    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 error for each limitEE

    Values: Above 0, in the units of measurement

    Often a fraction of the clinical tolerance, so the interval of a limit can be told apart from it.

    SD of the differencesss

    Values: Above 0

    From a pilot comparison of the same two methods.

    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 differences between methods are normal, with a mean and spread that do not change across the range measured.
    • Non-response is unrelated to the outcome.

    Worked examples

    Each limit within ±1

    s = 2, E = 1: at n = 48, 2.012 × 2 × √(1/48 + 1.96²/94) = 0.999, so n = 48.

    Sample size: 48

    Open this example in the calculator

    Halving the margin

    E = 0.5: n = 183, nearly four times as many.

    Sample size: 183

    Open this example in the calculator

    Allowing for incomplete pairs

    48 complete pairs, with 10% incomplete: recruit 54.

    Needed for analysis: 48 · To approach: 54

    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.
    • Simulated method comparisons at the planned size had a mean half-width for each limit of 0.997 against a target of 1. An independent implementation in R recomputes these sizes and is compared with the engine’s.

    How to cite

    StatWisely. Sample size to estimate Bland–Altman limits of agreement. Method document, engine version 1.0.0, reviewed 2026-10-09. https://statwisely.com/methods/precision-limits-of-agreement/

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

    References

    1. 1.Bland JM, Altman DG. Measuring agreement in method comparison studies. Statistical Methods in Medical Research. 1999;8(2):135–160.