Optimal Sample Size Calculator

Calculator inputs

The form uses a single-column page with a responsive input grid.

Large screens: 3 columns · Smaller screens: 2 columns · Mobile: 1 column

Study type

Confidence level (%)

Power (%)

Margin of error (%)

Population size (optional)

Design effect

Expected nonresponse (%)

Estimated proportion p (%)

Standard deviation (σ)

Minimum detectable difference (Δ)

Expected proportion in group A (%)

Expected proportion in group B (%)

Allocation ratio (B/A)

Plotly graph

The chart updates after calculation and helps compare the sensitivity of sample size to margin of error or effect size.

Example data table

Scenario	Confidence	Power	Main assumption	Error or effect	Estimated result
Survey proportion	95%	80%	p = 50%	±5%	385
Customer satisfaction mean	95%	80%	σ = 12	±3 units	62
Two means trial	95%	90%	σ = 10	Δ = 4	132 per group
Two proportions A/B test	95%	80%	50% vs 60%	10 point gap	387 per group

Formula used

Single proportion: n₀ = Z² × p × (1 − p) / e²

Single mean: n₀ = Z² × σ² / e²

Two means: n per group = 2 × (Zα/2 + Zβ)² × σ² / Δ²

Two proportions: n per group = [Zα/2√(2p̄(1−p̄)) + Zβ√(p₁(1−p₁)+p₂(1−p₂))]² / (p₁−p₂)²

Finite population correction: n = n₀ / [1 + (n₀ − 1) / N]

Nonresponse adjustment: Final n = Adjusted n / (1 − nonresponse rate)

These formulas estimate a defensible sample size, then adjust it for real-world collection constraints such as complex design and incomplete response.

How to use this calculator

Choose the study type that matches your project.
Enter the confidence level and, when relevant, the statistical power.
Supply the key assumption such as proportion, standard deviation, or effect difference.
Add population size if the study targets a limited universe.
Include design effect for clustered or complex sampling plans.
Enter expected nonresponse so the final recommendation stays practical.
Submit the form to see the result above the form.
Use the chart and downloads to compare and document scenarios.

FAQs

1. What is an optimal sample size?

It is the smallest sample that still meets your statistical target. A good estimate balances precision, power, budget, fieldwork effort, and expected missing responses.

2. Why does confidence level change the answer?

Higher confidence uses a larger Z value. That makes the formula demand more observations, because you want a tighter guarantee that your interval or conclusion is reliable.

3. When should I use 50% for a proportion?

Use 50% when you do not know the likely proportion. It gives the maximum variance and usually produces the most conservative sample size for surveys.

4. What does statistical power mean here?

Power is the chance of detecting a real effect when it exists. Comparative studies often use 80% or 90% power to reduce missed findings.

5. Why add a design effect?

Design effect inflates the sample when sampling is clustered, stratified, weighted, or otherwise more complex than a simple random design. It protects precision estimates.

6. What is finite population correction?

When the target population is limited, the correction reduces the needed sample. It matters most when your planned sample is a noticeable share of the population.

7. Why adjust for nonresponse?

Some people will not respond or will provide unusable data. Inflating the sample beforehand helps preserve the effective completed sample you actually need.

8. Can one calculator fit every study?

No. These formulas are strong planning tools, but complex trials, multilevel models, rare outcomes, and regulatory studies may need specialist design assumptions.