Sample Size Calculation: Formula and Worked Example

Sample size: what it is and when to calculate it

The sample size (n) is the number of parts or measurements needed to estimate a mean or a proportion with a defined margin of error and confidence level. Calculating it before collecting data avoids two problems: measuring too much and wasting time, or measuring too little and drawing weak conclusions.

The goal here is to estimate a process characteristic. To accept or reject a lot, the route is different: AQL sampling per ISO 2859-1. To watch the process over time, use SPC control charts.

Visual map of sample size

The map shows both formulas (mean and proportion), the finite population correction and the worked examples.

Formulas: mean and proportion

To estimate a mean with margin of error E, knowing (or estimating) the standard deviation σ:

n = (z × σ ÷ E)²

To estimate a proportion p (for example, the fraction defective):

n = z² × p × (1 − p) ÷ E²

The z value comes from the confidence level: 1.645 for 90%, 1.96 for 95% and 2.576 for 99%. Always round the result up. If p is unknown, use p = 0.5, which gives the largest possible n and is therefore the safe case. For more on σ and the process curve, see normal distribution and standard deviation.

Example 1: sachet weight (mean)

Illustrative example. A line fills sachets and history shows σ = 2.0 g. You want to estimate the mean weight within ±0.5 g at 95% confidence.

n = (1.96 × 2.0 ÷ 0.5)² = 7.84² = 61.47 → 62 sachets

σ changes the result, and so does the margin. See how n reacts to each decision:

Doubling the margin of error cuts n to a quarter. Raising confidence from 95% to 99% nearly doubles the collection effort.

Example 2: fraction defective (proportion)

Illustrative example. You want to estimate a process fraction defective within ±3 percentage points at 95% confidence.

• With p estimated at 5% (from history): n = 1.96² × 0.05 × 0.95 ÷ 0.03² = 0.1825 ÷ 0.0009 = 202.75, so 203 parts.
• With no information (p = 0.5) and a ±5 point margin: n = 1.96² × 0.5 × 0.5 ÷ 0.05² = 384.16, so 385 parts.

When the process has few defects, using p = 0.5 greatly overstates the size. Even rough history saves collection effort.

Finite population correction

If the lot or population (N) is small and the sample would exceed about 5% of it, the formula overstates n. Apply:

n_adjusted = n₀ ÷ [1 + (n₀ − 1) ÷ N]

With N = 1,000 and n₀ = 384.16 (proportion, ±5 points, 95%): n = 384.16 ÷ (1 + 383.16 ÷ 1,000) = 384.16 ÷ 1.38316 = 277.7, so 278 parts instead of 385. When N is very large relative to n, the correction is negligible and the result is unchanged.

Step by step to set the sample

1. Define what to estimate: a mean (variable data) or a proportion (attribute data).
2. Set the margin of error E from the decision to be made, not from what fits the routine.
3. Choose the confidence level (95% is usual) and get z.
4. Get σ or p: from history, from a pilot of 15 to 30 parts or, failing that, use p = 0.5.
5. Calculate n, round up and correct for a finite population if needed.
6. Sample randomly or stratified by shift, machine and lot, and record on a check sheet.

Sample size, AQL and control charts

Common mistakes

• Rounding down. The calculated n is a minimum; 61.47 becomes 62.
• Guessing σ or p without a basis. A small pilot or history gives a better estimate and makes the calculation defensible.
• Unrepresentative sample. Collecting only at shift start or from one machine skews the result, whatever n is.
• Ignoring the measurement system. More parts do not fix a poor gauge. See MSA and Gauge R&R.
• Using the proportion formula with very few defects without checking that the sample will contain enough occurrences for the estimate to make sense.
• Forgetting the finite population in small lots and inspecting more parts than needed.

Frequently asked questions

How do you calculate sample size?

For a mean, n = (z × σ ÷ E)². For a proportion, n = z² × p × (1 − p) ÷ E². Always round up.

Which z value should I use?

1.645 for 90% confidence, 1.96 for 95% and 2.576 for 99%.

What if I do not know the standard deviation or the proportion?

Use historical data or a pilot of 15 to 30 parts to estimate σ. For a proportion with no information, use p = 0.5, which gives the largest n.

What is the difference between sample size and an AQL plan?

Sample size estimates a characteristic with a margin of error. An AQL plan defines how many parts to inspect and how many defectives to accept in order to approve a lot.

When do I need the finite population correction?

When the sample exceeds about 5% of the lot size. With N = 1,000, n drops from 385 to 278 in the example.

How many parts are enough to calculate Cpk?

As a practical reference, 30 to 50 measurements under stable conditions, but ideally set it by the precision the study requires.

About the author

Vagner Soares

Lean Manufacturing & Behavioral Management Specialist

Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.

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