How to Calculate Sample Size
Determining the right sample size is one of the most important steps in research and statistics. Too small a sample gives unreliable results; too large wastes resources. The ideal sample size depends on your desired confidence level, margin of error, and population size.
The Formula (Unknown Population Size)
n = (Z² × p × (1 − p)) / e²
Where:
- n = required sample size
- Z = Z-score for your confidence level
- p = estimated proportion (use 0.5 if unknown, for maximum sample size)
- e = margin of error (as a decimal)
Common Z-Scores
| Confidence Level | Z-Score |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Step-by-Step Example
You want 95% confidence with a ±5% margin of error, and don't know the expected proportion.
- Z = 1.96 (for 95% confidence)
- p = 0.5 (conservative estimate)
- e = 0.05
n = (1.96² × 0.5 × 0.5) / 0.05² n = (3.8416 × 0.25) / 0.0025 n = 0.9604 / 0.0025 n = 384.16 → round up to 385
Adjusting for Finite Population
If your population is small (N < 10,000), use the finite population correction:
n_adjusted = n / (1 + (n − 1)/N)
Practical Considerations
- A 95% confidence level with ±5% margin of error is the standard for most surveys
- Increasing confidence to 99% or reducing error to ±3% significantly increases sample size
- Increase sample by 10–20% to account for non-responses or dropouts
Use our sample size calculator to determine the right n for your study.