Sample Size Calculator
Choose a method, enter your assumptions, and see the sample size they give, with every step of the working. Nothing here is universally correct: a different confidence level, margin of error, proportion or response rate gives a different number, so the calculator shows those alternatives together and explains each assumption.
Project details
Enter what you already know. Anything left blank is simply not compared.
One per line.
One per line.
One per line.
The measurement level from the Variables Builder, if you used it.
Your chosen design from the Research Design Builder, if you used it.
Your chosen technique from the Sampling Technique Builder, if you used it.
Method
Each method answers a slightly different question. Read what each assumes; the guide at the end explains them in full.
Your assumptions
Every input changes the result. Open “What this means” under each to see when it matters, typical values and its limitations.
What this means: unknown population
- What it means
- A population whose size isn't known, or is very large.
- When it matters
- When no reliable count exists; the calculation then assumes a very large population.
- Typical values
- All users of a public service, or a national population.
- Limitations
- Without a list of the population, random sampling is hard to achieve.
What this means: confidence level
- What it means
- How confident you want to be that the true value lies within your margin of error.
- When it matters
- Always. Higher confidence needs a larger sample.
- Typical values
- 95% is the most common; 90% and 99% are also used.
- Limitations
- It describes the sampling procedure, not certainty about any one result.
What this means: margin of error
- What it means
- How far above or below the sample's result the population's true value may be, such as ±5 percentage points.
- When it matters
- Always. Halving the margin roughly quadruples the sample.
- Typical values
- 5% is common for surveys; 3% for more precise estimates.
- Limitations
- It only covers random sampling error, not bias from non-response or poor measurement.
What this means: estimated proportion
- What it means
- Your best estimate of the percentage with the characteristic you are measuring, such as the share who agree.
- When it matters
- When you have a reliable prior estimate. Otherwise 50% is used because it gives the largest, most cautious sample.
- Typical values
- 50% if unknown; otherwise a figure from previous studies or a pilot.
- Limitations
- A wrong estimate gives a sample that is too small or too large for your margin of error.
Leave blank if you haven't estimated it yet.
What this means: expected response rate
- What it means
- The percentage of people you invite who you expect to take part.
- When it matters
- Whenever some people won't respond. It sets how many to invite.
- Typical values
- Varies widely by setting; base it on similar studies or a pilot.
- Limitations
- Inviting more people makes up for numbers, not for bias: non-responders may differ from responders.
What this means: design effect
- What it means
- How much larger the sample must be because of the sampling design, compared with simple random sampling.
- When it matters
- In cluster and multistage sampling, where people in the same group tend to be alike.
- Typical values
- 1 for simple random sampling; often 1.5 to 2 for cluster samples, depending on the clustering.
- Limitations
- Needs an estimate of how alike people in the same cluster are, which may come from previous studies.
Result
What your assumptions give. The number follows from them; it isn't correct in general.
With these assumptions, the sample size is 385.
- Required sample size
- 385
- After the design effect
- 385
- People to invite
- Add a response rate
- Expected responses
- Add a response rate
The working, step by step
1. Cochran's formula
- Formula
- n₀ = z² × p × (1 − p) ÷ e²
- Substitution
- n₀ = 1.959964² × 0.5 × (1 − 0.5) ÷ 0.05²
- z for 95% confidence
- 1.959964
- z²
- 3.841459
- p × (1 − p)
- 0.25
- e²
- 0.0025
- Result
- 384.1459, used as 385
With 95% confidence, a ±5% margin of error and an estimated proportion of 50%, a very large population needs 385 participants (384.1459 rounded up).
2. Design effect
- Formula
- n_adjusted = n × design effect
- Substitution
- n_adjusted = 385 × 1
- Result
- 385, used as 385
A design effect of 1 assumes simple random sampling, so the sample size is unchanged.
About your inputs
Response rate For you to review
No response rate is set, so the calculation doesn't say how many people to invite.
Assumptions behind this result
Each assumption, its value, and where it comes from in your project.
Population size: Unknown or very large
Source: Your entry
When no reliable count exists; the calculation then assumes a very large population.
Confidence level: 95% (z = 1.959964)
Source: Your choice
How confident you want to be that the true value lies within your margin of error.
Margin of error: ±5%
Source: Your choice
How far above or below the sample's result the population's true value may be, such as ±5 percentage points.
Estimated proportion: 50%
Source: Your choice
50% gives the largest, most cautious sample when the true proportion isn't known.
Expected response rate: Not set
Source: Your entry
Inviting more people makes up for numbers, not for bias: non-responders may differ from responders.
Design effect: 1
Source: Your entry
How much larger the sample must be because of the sampling design, compared with simple random sampling.
Random sampling: Assumed
Source: The formulas' own assumption
The confidence level and margin of error only strictly apply to random samples.
How the calculation fits your project
Each check explains why, and names the parts of your project behind it.
Research design For you to review
Choose your research design to see whether a precision-based sample size suits it.
Sampling technique For you to review
Choose your sampling technique to see whether the formulas' assumption of random sampling holds.
Population For you to review
Describe your target population in the Sampling Technique Builder, so the population size can be traced to it.
Objectives For you to review
Add your objectives to see which the sample size serves.
Variables For you to review
Record your dependent variables and their measurement levels in the Variables Builder to see how they affect the calculation.
Hypotheses Looks aligned
Your project has no hypotheses, so a precision-based sample size is appropriate.
How the result changes with other assumptions
The same calculation under other common assumptions, everything else unchanged. None of these is recommended; they show what your choice of assumptions costs or saves.
| Assumption | Required sample size | After the design effect | People to invite |
|---|---|---|---|
| 95% confidence, ±5% marginYour current assumptions | 385 | 385 | Add a response rate |
| 95% confidence, ±3% margin | 1068 | 1068 | Add a response rate |
| 99% confidence, ±5% margin | 664 | 664 | Add a response rate |
| 99% confidence, ±3% margin | 1844 | 1844 | Add a response rate |
| Assumption | Required sample size | After the design effect | People to invite |
|---|---|---|---|
| 40% response rate | 385 | 385 | 963 |
| 60% response rate | 385 | 385 | 642 |
| 80% response rate | 385 | 385 | 482 |
| 100% response rate | 385 | 385 | 385 |
| Assumption | Required sample size | After the design effect | People to invite |
|---|---|---|---|
| 50% estimated proportionYour current assumptions | 385 | 385 | Add a response rate |
| 30% estimated proportion | 323 | 323 | Add a response rate |
| 10% estimated proportion | 139 | 139 | Add a response rate |
Your justification and report
Why these assumptions suit your study: where the population count, proportion and response rate come from.
Preview the report
Sample size Summary Using the Cochran formula, for a very large or unknown population, with a 95% confidence level, a ±5% margin of error and an estimated proportion of 50%, the required sample size is 385. This figure follows from the assumptions below; different assumptions give different figures. Assumptions - Population size: Unknown or very large (source: Your entry) - Confidence level: 95% (z = 1.959964) (source: Your choice) - Margin of error: ±5% (source: Your choice) - Estimated proportion: 50% (source: Your choice) - Expected response rate: Not set (source: Your entry) - Design effect: 1 (source: Your entry) - Random sampling: Assumed (source: The formulas' own assumption) Calculation steps 1. Cochran's formula Formula: n₀ = z² × p × (1 − p) ÷ e² Substitution: n₀ = 1.959964² × 0.5 × (1 − 0.5) ÷ 0.05² z for 95% confidence: 1.959964 z²: 3.841459 p × (1 − p): 0.25 e²: 0.0025 Result: 384.1459, used as 385 With 95% confidence, a ±5% margin of error and an estimated proportion of 50%, a very large population needs 385 participants (384.1459 rounded up). 2. Design effect Formula: n_adjusted = n × design effect Substitution: n_adjusted = 385 × 1 Result: 385, used as 385 A design effect of 1 assumes simple random sampling, so the sample size is unchanged. Method Cochran formula: Estimates the sample needed to measure a proportion with a chosen confidence level and margin of error, for a very large or unknown population. Formula: n₀ = z² × p × (1 − p) ÷ e² z: the z-score for the confidence level p: the estimated proportion e: the margin of error Limitations - Doesn't account for the population's size; use the finite population correction for smaller populations. - Not designed for testing hypotheses or comparing groups. - The margin of error covers random sampling error only, not bias from non-response or measurement.
Your project draft
This tool writes only the Sample size section. Everything else is shown as you entered it.
- Sample size
- Cochran formula: 95% confidence, ±5% margin, unknown population.
Sample size methods explained
Every method the calculator covers. References for each will be added after academic review.
Cochran formula
Estimates the sample needed to measure a proportion with a chosen confidence level and margin of error, for a very large or unknown population.
Learn more about Cochran formula
Formula
n₀ = z² × p × (1 − p) ÷ e²
- z: the z-score for the confidence level
- p: the estimated proportion
- e: the margin of error
When it is commonly used
In surveys estimating a percentage, such as the share of people who agree with a statement, when the population is large.
Assumptions
- Participants are selected by simple random sampling.
- The outcome of interest is a proportion.
- The population is large enough that its size doesn't matter.
Strengths
- Makes every assumption explicit.
- Widely used and easy to report.
Limitations
- Doesn't account for the population's size; use the finite population correction for smaller populations.
- Not designed for testing hypotheses or comparing groups.
References
References awaiting academic review.
Cochran formula with finite population correction
Adjusts Cochran's sample size downwards when the population is small enough that the sample is a noticeable share of it.
Learn more about Cochran formula with finite population correction
Formula
n = n₀ ÷ (1 + (n₀ − 1) ÷ N)
- n₀: the sample size from Cochran's formula
- N: the population size
When it is commonly used
When the population is known and not very large, such as the staff of one organisation.
Assumptions
- The population size is known and accurate.
- Participants are selected by simple random sampling.
- The outcome of interest is a proportion.
Strengths
- Avoids collecting more data than a small population needs.
- Reduces to Cochran's formula for very large populations.
Limitations
- Depends on an accurate population size.
- Not designed for testing hypotheses or comparing groups.
References
References awaiting academic review.
Yamane formula
A simplified formula that gives a sample size from the population size and margin of error alone.
Learn more about Yamane formula
Formula
n = N ÷ (1 + N × e²)
- N: the population size
- e: the margin of error
When it is commonly used
In social science surveys where a quick estimate for a known population is wanted.
Assumptions
- A 95% confidence level: the formula has no term for confidence, so another level can't be chosen.
- An estimated proportion of 50%, the most cautious value.
- Simple random sampling from a known population.
Strengths
- Simple to calculate and explain.
- Needs only the population size and margin of error.
Limitations
- The confidence level and proportion are fixed, and not visible in the formula.
- Gives slightly larger samples than Cochran's formula with finite population correction for the same inputs.
References
References awaiting academic review.
Slovin formula
The same calculation as Yamane's formula. It is widely cited under this name, but its origin is unclear.
Learn more about Slovin formula
Formula
n = N ÷ (1 + N × e²)
- N: the population size
- e: the margin of error
When it is commonly used
Where a course or department asks for “Slovin's formula”; it gives the same result as Yamane's.
Assumptions
The same as Yamane's formula: 95% confidence, a proportion of 50%, and simple random sampling from a known population.
Strengths
- Simple to calculate.
- Needs only the population size and margin of error.
Limitations
- Its source is uncertain, which some examiners question.
- The same fixed assumptions as Yamane's formula.
References
References awaiting academic review.
Krejcie and Morgan
A formula, published with a widely used table, for the sample needed from a known population. It is algebraically the same as Cochran's formula with finite population correction.
Learn more about Krejcie and Morgan
Formula
s = X² × N × P × (1 − P) ÷ (d² × (N − 1) + X² × P × (1 − P))
- X²: the chi-square value for 1 degree of freedom at the confidence level (equal to z²)
- N: the population size
- P: the estimated proportion
- d: the margin of error
When it is commonly used
When a known population is sampled and a table-based figure is expected, as in many education and social science theses.
Assumptions
- The population size is known.
- Simple random sampling.
- The published table assumes 95% confidence, a proportion of 50% and a 5% margin of error.
Strengths
- Matches a table many readers recognise.
- Accounts for the population's size.
Limitations
- The published table rounds to the nearest whole number, so it can be one below the rounded-up figure.
- The table only covers its standard assumptions; other values need the formula.
References
References awaiting academic review.
Power analysis
Works out the sample needed to detect an effect of a given size with a chosen probability (power), for a specific statistical test.
Learn more about Power analysis
Formula
Depends on the statistical test, the effect size, the significance level and the power.
- Effect size: how large an effect you expect or need to detect
- α: the significance level
- 1 − β: the power
When it is commonly used
When a study tests hypotheses or compares groups, such as in experiments and correlational studies.
Assumptions
- The planned statistical test is known.
- A realistic effect size can be justified from previous research or a pilot.
Strengths
- Links the sample size directly to the hypothesis test.
- The expected approach in most experimental research.
Limitations
- This calculator can't compute it yet; a dedicated tool is planned.
- Results are very sensitive to the effect size assumed.
References
References awaiting academic review.
Unknown population guidance
When the population's size isn't known, Cochran's formula for a very large population gives a cautious sample size, because a finite population would only need fewer.
Learn more about Unknown population guidance
Formula
n₀ = z² × p × (1 − p) ÷ e²
- z: the z-score for the confidence level
- p: the estimated proportion
- e: the margin of error
When it is commonly used
When the population can't be counted, such as all users of a public service.
Assumptions
- The population is at least large enough that its size wouldn't reduce the sample much.
- Simple random sampling, or a close approximation.
Strengths
- Cautious: never smaller than needed for a finite population.
- Doesn't need a population count.
Limitations
- May collect more data than necessary if the population is actually small.
- Random sampling is hard to achieve without a list of the population.
References
References awaiting academic review.
Next: Questionnaire
Sections and questions, each linked to the variable and indicator it measures.
What should I do next?
Next in the research journey: data.
Questionnaire Builder
Assembles a structured questionnaire from your variables and indicators, explains where every question comes from, and exports it to Word, PDF and Markdown.
SPSS Research Lab
Connects a research question and selected analysis to IBM SPSS Statistics procedures, output tables, assumptions and reporting tools.
- New
Research Table Builder
Builds publication-ready research tables in APA, IEEE, Harvard, Chicago or MLA style, from your data, your results or your project, ready to paste into Word.
SPSS: From data preparation to reporting
A connected guide to preparing research data, choosing a supported analysis, running its IBM SPSS Statistics procedure, checking assumptions, reading output and reporting results.
How the calculator works
- The calculator reads your project draft: research question, objectives, variables, hypotheses, research design and sampling plan. Enter them in the first step; later versions will bring them across from the other research tools.
- Every result shows its formula, the numbers substituted, the intermediate values and how it was rounded, so you can check it by hand.
- The sensitivity tables show how the result changes under other common assumptions. They describe the differences; they don't recommend any of them.
- Checks compare your inputs with your project and explain what to look at. They never score or rank your choices.
- The calculator writes only the sample size section of your project draft: the method and assumptions, not the numbers, so later tools recalculate rather than copy.
What this tool can't do
- No sample size is universally correct. Each result follows from the assumptions shown; change them and the result changes.
- The formulas estimate a proportion. They don't size studies that test hypotheses or compare groups; power analysis is needed for those, and isn't available here yet.
- The confidence level and margin of error strictly apply only to random samples, and cover sampling error, not bias.
- Estimating a mean needs its standard deviation, which these formulas don't use.
- Qualitative studies usually decide sample size by saturation or depth, not by formula.
- Nothing is saved unless you start a project in your research workspace, which keeps it in this browser only. Otherwise your plan is lost when you leave the page.
Awaiting review
- Statistical review: the methods, their assumptions and limitations, the checks, and the choice to round each step up.
- Formula verification: each formula and z-score has been checked against an independent calculation and the published Krejcie and Morgan table, and needs sign-off by a statistician.
- References: each method needs its original source and a methods textbook. None have been added yet.
- Examples: worked examples for each method, for the guide.
Privacy
Everything stays in your browser. Nothing is sent anywhere or saved, and it is cleared when you leave the page. If you start a project in your research workspace, your work is also saved in this browser, and only this browser, until you delete the project.