QM Bench

Sample size for quality engineers: theory, guide and examples

How many parts do you need for a capability study, to estimate a mean or a defect rate, or to demonstrate reliability with a zero-failure test, and why too small samples give misleading results.

Ľubomír Blažek
Automotive quality manager · updated 2026-10-25

Try it while you read

Open the free Sample Size Calculator tool and click Load sample: it reproduces the case study below.

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Theory: precision costs parts

Every estimate from a sample has uncertainty, and the uncertainty shrinks only with the square root of the sample size: four times the parts give twice the precision. Planning the sample size before a study prevents both wasted effort and results that cannot be trusted.

Capability study

The lower confidence bound of Cpk (Bissell’s approximation) must still meet the requirement:

Cpk − z · √(1/(9n) + Cpk² / (2(n − 1))) ≥ Cpk_required

Estimating a mean or a proportion

n = (z · σ / E)²    n = z² · p (1 − p) / E²

E is the margin of error you can accept, σ the expected standard deviation, p the expected proportion.

Zero-failure (success-run) test

n = ln(1 − C) / ln(R)

Key point

The required n depends strongly on how close the expected value is to the requirement. A process expected at Cpk 2.0 needs few parts to prove 1.33; one expected at 1.45 needs more than 200.

Step by step

  1. Choose the question in the tool.
  2. Enter the expected value, from a pilot run or a similar process.
  3. Enter the requirement or the margin and the confidence.
  4. Read n and look at the table to see the trade-off.

Example: planning a capability study

A pilot run suggests Cpk ≈ 1.67; the customer requires 1.33 at 95 % confidence (one-sided). The tool returns at least 37 parts: with 37 values the lower 95 % bound of a true Cpk 1.67 is just at 1.33.

The engineer plans 50 parts (10 subgroups of 5) for a margin. If the pilot had shown only Cpk 1.45, the required number would rise to more than 200, a clear signal to improve the process before the formal study. For the study itself, use the capability tool.

Common mistakes

  • Using a fixed “30 parts” rule regardless of the requirement.
  • Planning with an optimistic expected value.
  • Ignoring that subgroups must be rational, not just numerous.

Frequently asked questions

Why is 30 parts not enough for Cpk?

With 30 values the 95 % confidence interval of Cpk is about ±0.4 around the estimate. A measured Cpk of 1.33 could be a true 1.0. Use at least 100 values, or calculate how many you need so the lower bound still meets the requirement.

What is the success-run theorem?

For a zero-failure test, n = ln(1 − C) / ln(R): to show 90 % reliability with 90 % confidence, 22 parts must pass without a failure.