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    Sample size to estimate a regression slope

    How many participants are needed to estimate the slope of a simple linear regression, the change in a numerical outcome per unit of a numerical predictor, within a chosen margin. Planned from the residual spread of the outcome and the spread of the predictor. Covers non-response.

    Engine version 1.0.0 · Reviewed 4 October 2026

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    https://statwisely.com/methods/precision-slope/

    When to use it

    • Estimating how much an outcome changes with a measured predictor, such as blood pressure per year of age, when the effect itself rather than a test is the aim.

    Not for

    • Showing that the slope differs from zero. Use linear regression under “Detect an effect”.
    • A slope adjusted for other predictors, whose precision also depends on how they relate to this one.
    • A strength of association on a unit-free scale. Use the correlation.

    Notation

    nn
    Number of participants
    EE
    Margin of error, in outcome units per predictor unit
    σe\sigma_e
    Residual standard deviation of the outcome around the line
    σx\sigma_x
    Standard deviation of the predictor
    t1−α/2, n−2t_{1-\alpha/2,\,n-2}
    t quantile on n − 2 degrees of freedom
    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 interval is narrow enough

    n=min⁡{n≥3:t1−α/2, n−2 σeσxn−1≤E}n = \min\left\{ n \ge 3 : t_{1-\alpha/2,\,n-2}\,\frac{\sigma_e}{\sigma_x\sqrt{n-1}} \le E \right\}

    The standard error of the least-squares slope is σₑ divided by the root of the predictor’s sum of squares, (n − 1)σₓ² [1]. Because the t quantile depends on n, n is found by search rather than in closed form. If only the outcome’s SD σ_y and the correlation ρ are known, σₑ = σ_y√(1 − ρ²).

    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, in outcome units per predictor unit

    Half the width of the interval wanted around the slope.

    Residual standard deviationσe\sigma_e

    Values: Above 0

    From a previous regression of the same outcome, or σ_y√(1 − ρ²).

    Standard deviation of the predictorσx\sigma_x

    Values: Above 0

    How widely the predictor will vary in the sample. Recruiting across a wider range of the predictor makes the slope more precise.

    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 relationship is linear, and the residuals are independent with constant variance, roughly normal for small samples.
    • The predictor’s sample SD equals σₓ. With a randomly sampled predictor this holds on average, and simulation shows the typical interval within about 1% of the target.
    • Non-response is unrelated to the outcome.

    Worked examples

    Residual SD 10, predictor SD 5

    A margin of 0.5: at n = 65, t₀.₉₇₅,₆₃ × 10 / (5√64) = 1.998 × 0.25 = 0.4996, within the margin, while n = 64 gives 0.504; so n = 65.

    Sample size: 65

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    Doubling the margin

    A margin of 1 needs n = 19: roughly a quarter, plus the small-sample cost of t.

    Sample size: 19

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    Allowing for non-response

    With 10% expected not to respond: 65 / 0.9 = 72.2, so 73 are approached.

    Needed for analysis: 65 · To approach: 73

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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.
    • The same search in R, on qt(0.975, n − 2) × σₑ / (σₓ√(n − 1)), returns 65, 19, 250 and 388 for four test cases, matching the engine. Fitting lm() to 5,000 simulated samples at each size gave median interval half-widths of 0.499, 0.990, 0.250 and 0.0998 against targets of 0.5, 1, 0.25 and 0.1.

    How to cite

    StatWisely. Sample size to estimate a regression slope. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/precision-slope/

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

    References

    1. 1.Draper NR, Smith H. Applied Regression Analysis. 3rd ed. New York: John Wiley & Sons; 1998.