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Power and sample size

Before comparing supplier A with B, a new method with the old one, or before with after, find out how many parts to measure so a difference that matters is not missed, and what it costs.

Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.

CheckRevised 11/03/2026
What to calculate
Difference and variation
Estimate the standard deviation from data (paste from Excel)

Paste historical or pilot-study measurements, one per line. The sample standard deviation and its confidence interval appear below; use the upper limit to plan with a margin.

See what was read
Cost (optional)

Zero hides the cost. Helps compare scenarios: raising power from 50% to 90% can cost three times as many parts.

Your inputs stay only on this browser and are not sent anywhere. Untick to delete them.

—Sample size
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Method and results

Same layout as the power and sample size report: hypotheses, assumed values and the results table.

Power curve

Chance of detecting each size of difference, for the calculated sample size and for half and twice it. The dot is your choice.

Tap the chart to see the values.

Power vs sample size

How power grows with the sample for the chosen difference: beyond a point, each extra part adds little.

Tap the chart to see the values.

Sample size vs difference

Sample needed for each difference at 80%, 90% and 95% power (log scale). Half the difference needs about four times the sample.

Tap the chart to see the values.

Why the power is what it is

Distribution of the test statistic if there is no difference (H0) and if the chosen difference exists (H1). Red: false alarm α; green: power; amber: β, the chance of missing it.

Tap the chart to see the values.

Scenario table

Next step

Take this page’s results to another calculator with the fields already filled in.

Run the test on the collected data →ANOVA on the collected data →Estimate the standard deviation (graphical summary) →

The calculation report gathers this page's inputs, results, equations, assumptions and references, with fields for project, engineer and revision. The full report also includes the charts and interpretations, in color. In the print dialog, choose Save as PDF.

Method, assumptions and references

Equations used

1-sample, paired and 2-sample t: δ = d/SE, SE = σ/√N (1-sample, paired) or σ·√(2/N) (2-sample, N per group); DF = N − 1 or 2N − 2; two-sided power = 1 − T′(t₁₋α/₂; DF, δ) + T′(−t₁₋α/₂; DF, δ), T′ = noncentral t
Proportions and rates (normal approximation): power = 1 − Φ[(z·SE₀ − Δ)/SE₁] + Φ[(−z·SE₀ − Δ)/SE₁]; 1 proportion SE₀ = √[p₀(1 − p₀)/N], SE₁ = √[p(1 − p)/N]; 2 proportions SE₀ = √[2p̄(1 − p̄)/N], SE₁ = √{[p₁(1 − p₁) + p₂(1 − p₂)]/N}; Poisson SE = √[λ/(N·T)]
Variances (normal): 1 variance P[ρ²·χ²(N − 1) outside χ²α/₂ and χ²₁₋α/₂], ρ = σ/σ₀; 2 variances P[ρ²·F(N − 1, N − 1) outside Fα/₂ and F₁₋α/₂], ρ = σA/σB
One-way ANOVA: λ = N·D²/(2σ²), noncentral F with (k − 1, k(N − 1)) DF; 2-level factorial: λ = r·R·E²/(4σ²), error DF = r·R + c·b − (1 + terms + [c > 0] + b − 1), r = corner points per replicate, R = replicates, c = center points per block, b = blocks
Equivalence (TOST): power = T′(−t₁₋α; DF, (d − UL)/SE) − T′(t₁₋α; DF, (d − LL)/SE); estimation: margin for the mean = t₁₋α/₂(N − 1)·σ/√N (or z); for the proportion, Clopper-Pearson with x = N·p

Assumptions and limits

  • Random, independent samples; for mean and variance tests, approximately normal data. The standard deviation, proportions and rates entered are planning values: if the real ones are worse, the real power will be lower.
  • The 2-sample t assumes equal variances and the same N in both groups; Welch’s test with unequal variances needs a slightly larger sample. Proportions and rates use the normal approximation; with few expected occurrences, check with the exact test.
  • In ANOVA, the maximum difference D assumes the other means lie in the middle (worst case for power). In the factorial, the effect is the difference between the mean responses at the high and low levels of a factor; aliased terms do not count.

Standards and technical documents

  • Montgomery, D. C. Design and Analysis of Experiments, 9th ed. Wiley, 2017 (caps. 2, 3 e 6)
  • Montgomery, D. C.; Runger, G. C. Applied Statistics and Probability for Engineers, 7th ed. Wiley, 2018 (cap. 9)
  • Fleiss, J. L.; Levin, B.; Paik, M. C. Statistical Methods for Rates and Proportions, 3rd ed. Wiley, 2003
  • Chow, S.-C.; Shao, J.; Wang, H. Sample Size Calculations in Clinical Research, 2nd ed. Chapman & Hall, 2008
  • Schuirmann, D. J. A comparison of the two one-sided tests procedure and the power approach for assessing equivalence. J. Pharmacokinet. Biopharm. 15:657–680, 1987
  • Clopper, C. J.; Pearson, E. S. The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26:404–413, 1934
  • Cohen, J. Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Erlbaum, 1988
  • NIST/SEMATECH e-Handbook of Statistical Methods, seções 7.2.2 e 7.2.4

U.S. regulations

  • ISO 9001:2015, 9.1.3: analysis and evaluation
  • ISO 3534-1: statistics, vocabulary and symbols
  • 21 CFR 820.250 (statistical techniques and sampling plans based on a valid statistical rationale)

Frequently asked questions

How do I calculate the sample size to compare two suppliers?

Choose 2-sample t, enter the smallest mean difference that matters (for example 0.5 mm), the process standard deviation (0.4 mm) and the desired power (80%). The page gives the number of parts per supplier: in the example, 12 parts from each, with an actual power of 83%.

What is the power of a test?

It is the chance that the test flags a difference that really exists. With 50% power, a well-run study misses half of the real differences; with 80% or 90%, you rarely miss a difference of the chosen size.

Which standard deviation should I use if I have no data yet?

Use historical data from the same process, the within σ from a control chart, or a pilot study of 10 to 30 parts (paste them into "Estimate the standard deviation from data"). To plan with a margin, use the upper limit of the confidence interval.

Why does half the difference need four times the sample?

The precision of a mean improves with the square root of the number of parts. To see a difference half as large, the error of the mean must halve, and that takes four times as much data.

When should I use a one-sided test?

Only when, before collecting, you decided that just one direction matters (for example, the improvement is only worth it if it reduces defects). It needs fewer parts but cannot see changes in the opposite direction.

What is the difference between a standard test and an equivalence test?

A standard test tries to prove there is a difference; failing to find one does not prove the products are the same. An equivalence test tries to prove the difference lies within acceptable limits, which is the right approach to validate a change of supplier or method.

How many replicates should a factorial experiment have?

Choose 2-level factorial, enter the factors, model terms, effect to detect and standard deviation. The page calculates the replicates and total runs; center points help test curvature and add degrees of freedom to the error.

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