Sample Size Formula
Formula for calculating the minimum sample size needed for surveys and research studies at a given confidence level and margin of error.
The Formula (Infinite Population)
n = (z² × p × (1 - p)) / e²
With Finite Population Correction
n_adj = n / (1 + (n - 1) / N)
Variables
| Symbol | Meaning |
|---|---|
| n | Required sample size |
| z | Z-score for desired confidence level |
| p | Expected proportion (use 0.5 if unknown) |
| e | Margin of error (as a decimal) |
| N | Total population size |
| n_adj | Adjusted sample size for finite population |
Z-Values for Common Confidence Levels
| Confidence Level | Z-Value |
|---|---|
| 80% | 1.282 |
| 85% | 1.440 |
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Quick Reference
| Margin of Error | 95% Confidence | 99% Confidence |
|---|---|---|
| ±1% | 9,604 | 16,590 |
| ±3% | 1,068 | 1,844 |
| ±5% | 385 | 664 |
| ±10% | 97 | 166 |
Example
How many people should you survey for ±5% margin at 95% confidence?
z = 1.960, e = 0.05, p = 0.5 (unknown proportion)
n = (1.96² × 0.5 × 0.5) / 0.05²
n = (3.8416 × 0.25) / 0.0025
n = 0.9604 / 0.0025 = 385 responses needed