Statistics · Survey Research
Sample Size
Calculator
Find the required sample size, margin of error, or confidence level for surveys and research. Includes finite population correction, step-by-step solutions, and trade-off chart.
—
Sample Size
—
Margin of Error
—
Confidence
Sample Size Calculator
%
%
e.g. 5 means ±5%
%
50% = most conservative
N
For finite population correction (FPC)
Sample Size Needed
—
respondents
With FPC
—
Z-score
—
Confidence
—
Margin of Error
—
Proportion
—
Population
—
📋 Step-by-Step Solution
📊 Sample Size Comparison Table
📈 Sample Size vs Margin of Error Trade-off
Smaller margin of error requires exponentially more respondents
💼 Typical Sample Sizes by Use Case
How to Calculate Sample Size
Sample size determines how many observations you need to get statistically reliable results. The key trade-off: larger samples give more precision (smaller margin of error) but cost more time and resources.
Sample Size Formulas
Infinite population:
n = Z² × p(1−p) / E²
Finite population (FPC):
n_adj = n / (1 + (n−1)/N)
Where:
Z = Z-score for confidence level (90%→1.645, 95%→1.96, 99%→2.576)
p = population proportion (0.5 = most conservative)
E = margin of error as decimal (5% → 0.05)
N = population size
Why is 50% the most conservative proportion?
The sample size formula includes p(1−p), which is maximized when p=0.5, giving 0.5×0.5=0.25. Any other value gives a smaller product (e.g., 0.3×0.7=0.21). So using p=50% gives the largest — most conservative — sample size estimate. If you know the proportion will be far from 50% (e.g., a rare condition affecting only 5%), you can use that proportion to reduce the required sample size.
What is the finite population correction (FPC)?
When your sample is a significant fraction of the total population (generally more than 5%), you can apply the FPC to reduce the required sample size. The formula is: n_adj = n / (1 + (n−1)/N). For example, if the infinite-population formula gives n=385 but your population is only 1,000: n_adj = 385/(1+384/1000) = 385/1.384 ≈ 278. You need 278 instead of 385 — a meaningful saving.
How does doubling precision affect sample size?
Sample size is inversely proportional to the square of the margin of error. Halving the margin of error (e.g., from ±5% to ±2.5%) quadruples the required sample size. Going from ±5% to ±1% increases sample size by 25x. This quadratic relationship explains why very precise surveys are expensive — each incremental improvement in precision costs disproportionately more.
What confidence level should I choose?
95% is the scientific and industry standard. It means if you repeated the survey 100 times, 95 confidence intervals would contain the true value. Use 99% when the cost of being wrong is high (medical, safety, legal decisions). Use 90% for preliminary or exploratory research where budget is tight. Political polls typically use 95% with ±3% margin of error, requiring about 1,067 respondents.