Math Tools
Determine the minimum sample size needed for a statistically valid survey or study using confidence levels and margin of error.
Recommended Sample Size
370responses
Using Cochran formula with z=1.96, p=50%, e=5%
Tip: A 95% confidence level with 5% margin of error is standard for most research. Adjust values to see how precision requirements affect sample size.
Sample size is the number of respondents required for a survey to produce reliable results that represent the broader population. It depends on three factors: population size, confidence level (how certain you want to be), and margin of error (how close your results should be to the true population value). A larger sample size increases precision but requires more resources to collect.
The calculator uses the Cochran formula: n0 = z² × p(1−p) / e². Here, z is the z-score for your confidence level (90%, 95%, or 99%), p is the response distribution (typically 50% for unknown distributions), and e is your margin of error expressed as a decimal. If you specify a finite population, the calculator applies a finite population correction to reduce the sample size slightly. The result is always rounded up to ensure statistical validity.
n = n0 / (1 + (n0 − 1) / N) where n0 = z² × p(1 − p) / e²n0 is the raw sample size from Cochran's formula. The correction factor reduces n0 when your population (N) is finite. z is your z-score (1.645 for 90%, 1.96 for 95%, 2.576 for 99% confidence). p is response distribution (0.5 for 50%, the most conservative). e is margin of error as a decimal (0.05 for 5%). The final n is rounded up.
| Input | Result | Notes |
|---|---|---|
| Population: 10,000 | Confidence: 95% | Margin of Error: 5% | Distribution: 50% | 370 respondents | Standard research survey: z=1.96, p=0.5, e=0.05 → n0≈384, then finite correction applied to get 370 |
| Population: 50,000 | Confidence: 95% | Margin of Error: 3% | Distribution: 50% | 1,067 respondents | Stricter precision requirement: smaller margin (3%) requires larger sample; z=1.96, e=0.03 |
| Population: unlimited | Confidence: 90% | Margin of Error: 10% | Distribution: 50% | 68 respondents | Less strict requirements (90% confidence, 10% error) yield a smaller sample; useful for quick pilot studies |
Confidence level (90%, 95%, 99%) is how sure you are your results are accurate. Margin of error (e.g., 5%) is how far your results can deviate from the true population value. Higher confidence and smaller margins of error require larger samples.
95% is standard for most research and business decisions. Use 90% for quick, exploratory studies or when resources are very limited. Use 99% when precision is critical (e.g., medical trials, regulatory compliance).
It's the expected proportion answering 'yes' (or the primary category). If you expect 60% will say 'yes,' use 60%. If unknown, use 50%—it's the most conservative and yields the largest sample size.
No. The sample size is the number of completed responses you need. If you expect a 70% completion rate, plan to reach (sample size ÷ 0.7) total people to account for drop-outs.
Leave population size blank. Large populations barely affect the calculation—the finite correction becomes negligible. Most national surveys use 1,000–2,000 respondents regardless of country population.
Yes. The formula is the same. The only difference is response rate—online typically has lower completion rates, so plan accordingly by increasing your outreach target.