Sample Size Calculator
Determine how many survey respondents or study participants you need. Enter your desired confidence level, margin of error, and (optionally) population size to calculate the minimum sample size required for statistically valid results.
n = Z² × p(1-p) / e²
where Z = z-score, p = proportion, e = margin of error
Z = 1.960 — standard for most research
Use 50% if unknown — gives the largest (most conservative) sample size.
Sample Size Quick Reference
Required sample size at 95% confidence, p = 50%:
| Margin of Error | Infinite Pop. | Pop. 10,000 | Pop. 1,000 |
|---|---|---|---|
| ±1% | 9,604 | 4,900 | 906 |
| ±2% | 2,401 | 1,937 | 707 |
| ±3% | 1,068 | 965 | 517 |
| ±5% | 385 | 370 | 278 |
| ±10% | 97 | 96 | 88 |
* Smaller margin of error requires a larger sample. Use finite population correction when your population is small.
Frequently Asked Questions
How do you calculate sample size for a survey?
Use the formula n = Z² × p × (1-p) / e², where Z is the z-score for your confidence level (1.96 for 95%), p is the expected proportion (use 0.5 if unknown), and e is your desired margin of error as a decimal. For example, at 95% confidence with ±5% margin and p = 0.5: n = (1.96)² × 0.25 / (0.05)² = 384.16 → 385 respondents.
What is margin of error in a survey?
Margin of error (also called confidence interval half-width) is the maximum expected difference between your sample result and the true population value. A ±5% margin of error means your survey result is within 5 percentage points of the true answer. Smaller margins of error require larger sample sizes.
What confidence level should I use for my survey?
The standard is 95% confidence for most business and social science research. Use 99% for high-stakes decisions (medical trials, financial studies, legal matters). Use 90% for exploratory or pilot studies where speed and cost matter more than precision. Increasing confidence level significantly increases the required sample size.
What proportion should I use if I don't know it?
Use p = 0.5 (50%) when the true proportion is unknown. This gives the most conservative (largest) sample size estimate, ensuring your study is adequately powered regardless of the actual proportion. If past data or pilot studies suggest a different value, using that estimate will reduce the required sample size.
What is finite population correction (FPC)?
Finite population correction reduces the required sample size when your population is small relative to the estimated sample. The formula is: n_adjusted = n₀ / (1 + (n₀ - 1) / N), where N is the total population size. FPC is most impactful when the sample is more than 5% of the population. For a population of 1,000, the standard 385-person sample reduces to about 278 after FPC.
How does sample size affect accuracy?
Larger samples give smaller margins of error and more precise estimates. However, the relationship is not linear — accuracy improves with the square root of sample size. Doubling your sample size reduces margin of error by about 30%, not 50%. The biggest gains come from increasing small samples; going from 100 to 400 respondents cuts margin of error in half.
What is the minimum sample size for a survey?
There is no universal minimum, but common guidelines suggest at least 30 respondents for basic analysis, 100 for simple descriptive statistics, and 385+ for nationally representative surveys at ±5% margin with 95% confidence. For subgroup analysis (age groups, regions, etc.), each subgroup should ideally have at least 100 respondents.
Does a larger population require a larger sample size?
Mostly no — for large populations (100,000+), the required sample size is nearly the same regardless of total population size. A survey of a city of 100,000 and a country of 300 million need roughly the same sample (~385 at ±5%, 95% CI). Population size only significantly reduces required sample size for small populations under about 10,000 through the finite population correction.
Sample Size Formula
The standard sample size formula for estimating a proportion in a population is derived from the normal distribution. For an infinite (or very large) population:
Infinite Population Formula:
n = Z² × p × (1 - p) / e²
Where:
- n = required sample size
- Z = z-score for the desired confidence level (e.g., 1.960 for 95%, 1.645 for 90%, 2.576 for 99%)
- p = expected proportion (use 0.5 if unknown — maximizes sample size)
- e = margin of error as a decimal (e.g., 0.05 for ±5%)
| Confidence Level | Z-Score | Common Use |
|---|---|---|
| 80% | 1.282 | Exploratory research, quick estimates |
| 85% | 1.440 | Preliminary studies |
| 90% | 1.645 | Market research, less critical decisions |
| 95% | 1.960 | Standard social science and business research |
| 99% | 2.576 | Medical studies, high-stakes decisions |
Finite Population Correction
When your population is not very large relative to the required sample, the standard formula overestimates the needed sample size. The Finite Population Correction (FPC) adjusts for this:
Finite Population Formula:
nₐ = n₀ / (1 + (n₀ - 1) / N)
where n₀ = infinite-population estimate, N = total population size
The FPC becomes significant when the sample represents more than about 5% of the population. For example, surveying 385 people from a population of 1,000 (38.5% sample fraction) yields a corrected sample size of around 278 — a 28% reduction.
n₀ = 385 (95% CI, ±5%, p = 0.5, infinite pop.)
N = 1,000
nₐ = 385 / (1 + (385 - 1) / 1,000)
nₐ ≈ 278 respondents needed
Leave the population field blank if you don't know your population size, or if it is very large (e.g., a national survey). The standard formula will be used in those cases.
Choosing Margin of Error
The margin of error (MoE), also called the confidence interval half-width, defines how close your sample estimate will be to the true population value. A ±5% margin means your results are within 5 percentage points of the actual answer.
Guidelines by Research Type
| Research Type | Recommended MoE | Rationale |
|---|---|---|
| National polls, political surveys | ±2–3% | Close races require precision |
| Customer satisfaction surveys | ±5% | Industry standard for business decisions |
| Market research, internal surveys | ±5–10% | Trend detection, directional insights |
| Exploratory / pilot studies | ±10% | Quick, cheap screening before full study |
| Academic / clinical research | ±1–3% | Rigorous evidence requirements |
Sample size grows quadratically as margin of error decreases — cutting your MoE in half requires four times as many participants. Choose the largest MoE your research objectives can tolerate.
Understanding Confidence Levels
A 95% confidence level means that if you repeated the survey 100 times with different random samples, 95 of the resulting intervals would contain the true population value. It does notmean there is a 95% probability that the true value lies in your specific interval — once you collect data, the true value either is or isn't in the interval.
Higher confidence levels require larger samples:
- Going from 90% to 95% confidence increases sample size by approximately 41%.
- Going from 95% to 99% confidence increases sample size by approximately 73%.
For most surveys and business research, 95% confidence is the standard. Use 99% for high-stakes decisions (medical, legal, financial). Use 90% to reduce cost in exploratory or screening studies.
Sample Size Calculation Examples
Example 1: National Survey (Unknown Proportion)
A polling organization wants to estimate voting intent with ±3% margin of error at 95% confidence. Since the true proportion is unknown, they use p = 0.5 (worst case).
Z = 1.960, e = 0.03, p = 0.5
n = (1.960)² × 0.5 × 0.5 / (0.03)²
n = 3.8416 × 0.25 / 0.0009
n = 1,068 respondents needed
Example 2: Customer Survey (Known Proportion)
A company surveys customers about satisfaction. From past data, ~70% are satisfied. They want ±5% margin of error at 95% confidence.
Z = 1.960, e = 0.05, p = 0.70
n = (1.960)² × 0.70 × 0.30 / (0.05)²
n = 3.8416 × 0.21 / 0.0025
n = 323 respondents needed (vs. 385 with p = 0.5)
Example 3: Small Population (Finite Correction)
An HR team surveys 500 employees. They want ±5% margin at 95% confidence. Without correction: 385 needed. With FPC:
n₀ = 385, N = 500
nₐ = 385 / (1 + 384 / 500)
nₐ ≈ 218 employees needed