Plan your study or check your results. Work out the sample size you need, or test whether a difference between two groups is statistically significant.
Required sample size
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Whether you’re designing a study or writing up results, this covers sample-size planning and significance testing in one place.
Find the number of participants you need to estimate a proportion or mean, or to compare two groups with enough statistical power.
Compare two proportions (a z-test) or two means (Welch's t-test) and get a two-sided p-value, confidence interval, and effect size.
Set your confidence level, margin of error, and statistical power, and the calculator handles the maths behind them.
To estimate a single percentage — such as "what proportion of students agree?" — the standard formula is:
You want to estimate a proportion to within ±5% at 95% confidence, and you're unsure of p, so you use 50%.
n = 1.96² × 0.5 × 0.5 / 0.05² = 3.8416 × 0.25 / 0.0025 = 385 participants.
If you're estimating an average (a mean) rather than a percentage, the formula uses the standard deviation (σ) instead of p(1−p):
A ready reckoner for the most common question — how many people should I survey? These use a 50% expected proportion (the safest assumption) and a large or unknown population.
| Margin of error | Sample size (95% confidence) | What it means |
|---|---|---|
| ±10% | 97 | Rough, quick pulse-check |
| ±7% | 196 | Small survey, broad findings |
| ±5% | 385 | The usual standard for surveys |
| ±3% | 1,068 | Higher precision (e.g. polling) |
| ±2% | 2,401 | Very precise estimates |
| ±1% | 9,604 | National-scale precision |
Your confidence level sets the z-value in the formula. Higher confidence needs a larger sample:
| Confidence level | z-value | Common use |
|---|---|---|
| 80% | 1.28 | Exploratory work |
| 90% | 1.645 | Lower-stakes estimates |
| 95% | 1.96 | The academic standard |
| 99% | 2.576 | High-stakes / clinical |
When you're comparing two groups — a treatment vs a control, for example — sample size also depends on statistical power: the chance your study detects a real effect if one exists. The convention is 80% power, with 90–95% used for higher-stakes studies. Bigger differences are easier to detect and need fewer participants; small differences need many more.
To detect a difference between 30% and 50% at 95% confidence with 80% power, you need about 93 participants per group (≈186 in total). Halve the gap you're trying to detect and the required sample roughly quadruples — which is why realistic effect sizes matter so much.
A p-value is the probability of seeing a difference at least as large as yours if there were really no difference between the groups. A small p-value means your result would be unlikely by chance alone, so you have evidence of a real effect.
Compare the p-value to your significance level (alpha, α — usually 0.05):
| p-value | Verdict (at α = 0.05) | Interpretation |
|---|---|---|
| p < 0.01 | Statistically significant | Strong evidence of a real difference |
| p < 0.05 | Statistically significant | Evidence of a real difference |
| p ≥ 0.05 | Not significant | Not enough evidence to conclude a difference |
Group 1: 45 of 100 (45%). Group 2: 30 of 100 (30%). A two-proportion z-test gives p = 0.0285. Because 0.0285 is below 0.05, the difference is statistically significant at the 95% level — with a 15-percentage-point difference and a 95% confidence interval of roughly 1.7 to 28.3 points.
| Cohen's d | Interpretation |
|---|---|
| 0.2 | Small effect |
| 0.5 | Medium effect |
| 0.8 | Large effect |
Confusing "not significant" with "no effect." A non-significant result often just means your sample was too small to detect a real difference.
A margin-of-error sample size isn't enough to compare two groups — you also need to plan for statistical power.
Always report the effect size and a confidence interval too, so readers know how big the difference is.
Recruit more than the minimum so that missing data and withdrawals don't leave you underpowered.
This calculator handles the quick numbers. If you'd like a statistician to run your full analysis, or an editor to make sure the results read clearly in your manuscript, our team can help — no pressure, just here if you need it.
The number of participants or observations in your study.
How sure you want to be that your interval contains the true value (e.g. 95%).
How far your estimate may reasonably be from the true value (e.g. ±5%).
The probability of your result (or a bigger one) if there were no real difference.
Your cut-off for calling a result significant — usually 0.05.
The chance of detecting a real effect if one exists — usually set at 80%.
How large the difference is (e.g. Cohen's d), independent of sample size.
A range that likely contains the true difference or value.
Pick your study design, then enter your confidence level, margin of error (or the effect you want to detect), and expected values. For estimating one proportion the formula is n = z² · p(1 − p) / e². The calculator does this for you and returns the number of participants you need.
At 95% confidence with a ±5% margin of error and a large or unknown population, you need about 385 respondents. Tighter margins need more: ±3% needs about 1,068 and ±1% needs about 9,604. Use the “Estimate one proportion” design to adjust for your exact confidence, margin, and population size.
Run a significance test and compare the p-value to your significance level (alpha, usually 0.05). If the p-value is below alpha, the result is statistically significant; if it’s equal to or above alpha, it isn’t. The calculator returns the p-value, the verdict, a confidence interval, and an effect size.
A p-value is the probability of seeing a difference at least as large as yours if there were really no difference between the groups. A small p-value (typically below 0.05) suggests the difference is unlikely to be due to chance alone.
Power is the chance your study detects a real effect if one exists. 80% is the common minimum — an 80% chance of finding a true effect. Higher power (90–95%) needs a larger sample. The calculator lets you set power when comparing two groups.
Use a z-test to compare two proportions (counts), and a t-test to compare two means. This calculator uses Welch’s t-test for means, which doesn’t assume the two groups have equal variances — the safer, more robust choice.
Yes. It’s completely free with no sign-up, and it runs entirely in your browser — nothing you enter is uploaded or stored anywhere.
Yes. This tool gives quick estimates, but for a full analysis or an independent check of your results and reporting, our data-analysis team can run or verify your statistics, and our scientific editors can make sure they read clearly in your manuscript.
Use the calculator as often as you like. When you’re ready to run, verify, or write up your analysis, ManuscriptLab’s statisticians and editors are here to help.
Tell us a bit about your project and we’ll get back to you within 24 hours.