📋Sample Size Calculator
Calculate the minimum sample size needed for a survey or find the margin of error given your sample. Supports all confidence levels (80%–99.9%), any population proportion, and finite population correction for known population sizes.
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385
Sample Size by Confidence Level
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Sample Size Calculator: How Many Responses Do You Need?
The sample size is the number of survey responses needed to achieve a desired margin of error at a given confidence level. The Cochran formula n = z²×p×(1−p)/E² gives the minimum n for large populations. For a 95% confidence level (z*=1.96), 5% margin of error, and 50% proportion: n = 1.96² × 0.25 / 0.05² = 384.16 → 385 responses.
Formula: n = z² × p(1−p) / E² — with finite correction: n_adj = n / (1 + (n−1)/N)
| Confidence | MoE = 5% | MoE = 3% |
|---|---|---|
| 90% | 271 | 752 |
| 95% | 385 | 1,068 |
| 99% | 664 | 1,844 |
Understanding sample size is essential for survey design, clinical trials, quality control, and any study requiring statistical inference. Too small a sample produces unreliable estimates with wide confidence intervals; too large a sample wastes resources without improving accuracy beyond a point of diminishing returns.
Why Use 50% as the Population Proportion?
The term p×(1−p) in the formula is maximized when p = 0.5 (giving 0.25). Using 50% when you don't know the true proportion gives the most conservative (largest) required sample size — guaranteeing your survey will have sufficient power regardless of the actual proportion. If you have prior knowledge (e.g., from a pilot study) that the proportion is around 20%, using p=0.2 gives p×(1−p) = 0.16, requiring a smaller sample: n = 1.96²×0.16/0.05² ≈ 246 instead of 385.
When to Use Finite Population Correction
The basic Cochran formula assumes an infinite (or very large) population. When your sample is a substantial fraction of the population, the finite population correction (FPC) reduces the required sample size. The FPC factor is √((N−n)/(N−1)), and the adjusted formula is n_adj = n₀/(1 + (n₀−1)/N). As a rule of thumb: apply FPC when the sample fraction n/N exceeds 5%. For a town with N=500 residents needing n₀=385: n_adj = 385/(1+384/500) = 385/1.768 ≈ 218 — a 43% reduction!
The Diminishing Returns of Larger Samples
Sample sizes follow a square root relationship with margin of error: to halve the margin of error, you must quadruple the sample. Going from 5% MoE to 2.5% MoE at 95% confidence requires going from n=385 to n=1,537. From 5% to 1%: n=9,604. This is why national polls with ±3% margins typically use n≈1,000 respondents — going to ±2% would require ~2,400 respondents without a proportionally greater gain in insight.
Frequently Asked Questions
How many survey responses do I need?
The standard formula is n = z²×p×(1−p)/E², where z is the critical value for your confidence level, p is the estimated proportion (use 0.5 if unknown), and E is the margin of error as a decimal. For the most common case (95% CI, 5% MoE, unknown proportion): n = 1.96² × 0.25 / 0.05² = 3.8416 × 0.25 / 0.0025 = 384.16 → 385 responses. For a tighter ±3% margin: n = 1.96² × 0.25 / 0.03² = 1,068. For ±1%: n = 9,604. Always round up to ensure the desired precision.
What is margin of error and how is it calculated?
Margin of error (MoE) is the maximum expected difference between the survey result and the true population value, at the specified confidence level. For a proportion estimate: MoE = z × √(p×(1−p)/n), where z depends on confidence level (1.96 for 95%) and n is sample size. A poll result of 48% ± 3% (at 95% CI) means: if the survey were repeated many times, 95% of results would fall within 3 percentage points of the true value. With n=1,000 and p≈0.5: MoE = 1.96 × √(0.25/1000) = 1.96 × 0.0158 ≈ 3.1%.
Does my population size matter for sample size?
For large populations (thousands or more), population size barely matters — the key factors are confidence level, margin of error, and proportion. The finite population correction (FPC) only makes a meaningful difference when your sample would be more than 5% of the population. Example: surveying a company's 200 employees (N=200) needing n₀=132 (from the formula): n_adj = 132/(1+131/200) ≈ 79 — you only need 79 responses. But for a city of 1 million: the FPC reduces n=385 by less than 0.04% — negligible.
What confidence level should I use for my survey?
95% confidence is the most common standard in academic research, market research, and polling — it means you accept a 5% chance that the true value falls outside your interval. 99% confidence is used for high-stakes decisions (medical research, safety testing), requiring roughly 73% more responses than 95%. 90% confidence requires fewer responses (about 70% of the 95% sample) and is acceptable for preliminary research or informal surveys. In general: use 95% for published research, 90% for quick business decisions, 99% for critical medical or safety decisions.
Why does sample size not scale proportionally with population size?
This is the fundamental and often counterintuitive insight of statistics: a poll of 1,000 people is nearly as accurate for a city of 100,000 as for a country of 300 million. The reason is that sample accuracy depends primarily on the absolute sample size, not the sampling fraction. The standard error SE = √(p×(1−p)/n) depends only on n, not on N. Larger populations don't have more variable proportions — they just have more individuals sharing the same underlying probability. This is why national polls of 1,000 respondents can estimate opinions of 330 million people to within ±3%.