Sample Size Calculator (Live Survey & Research n-Size Tool)
Calculate the exact sample size ($n$) required for surveys, clinical research, and academic studies. Supports Cochran's formula across 90%, 95%, and 99% confidence levels, Finite Population Correction (FPC), and response-rate distribution buffers.
To calculate sample size using Cochran's formula, square the critical value ($Z^*$), multiply by the estimated variance ($p \times (1 - p)$), and divide by the square of your acceptable margin of error ($e$): n = (Z² × p × (1 − p)) / e². For unknown populations, assume $p = 0.5$ (50%) to ensure a conservative, statistically valid sample size.
The Mathematics of Cochran's Sample Size Formula
Standardized by statistician William G. Cochran in his landmark 1963 text Sampling Techniques, Cochran's formula determines the minimum sample size required to estimate a population proportion with a specified level of precision:
Where $Z$ represents the critical value corresponding to your chosen confidence level ($1.96$ for 95% confidence), $p$ is the estimated proportion of an attribute present in the population ($0.5$ for maximum variance), and $e$ represents the desired margin of error ($0.05$ for ±5%).
Why Sample Size Flattens Out (The "384 Rule")
A frequent surprise for researchers is discovering that surveying an entire national electorate (335 million citizens) requires virtually the exact same sample size as surveying a mid-sized metropolitan area (100,000 citizens):
- At a 95% confidence level with a ±5% margin of error, Cochran's formula yields an unadjusted baseline of 384.16 (∼385) respondents.
- Whether your population ($N$) is 100,000 or 1,000,000,000, the required sample size remains ~384 to 385 people.
- Total population size only reduces the required sample size when surveying small, closed groups ($N < 5,000$), where the Finite Population Correction (FPC) narrows sampling variability.
Population Size vs. Required Sample Size Reference Matrix
Standardized reference values derived from the Krejcie & Morgan (1970) and Cochran formulas across varying population sizes:
| Population Size ($N$) | ±5% Margin of Error (95%) | ±3% Margin of Error (95%) | ±1% Margin of Error (99%) |
|---|---|---|---|
| 100 | 80 | 92 | 99 |
| 500 | 217 | 341 | 476 |
| 1,000 | 278 | 516 | 906 |
| 10,000 | 370 | 964 | 4,899 |
| 100,000+ (Large) | 384 | 1,056 | 14,228 |
Related Statistical & Research Calculators:
- Margin of Error Calculator — Calculate survey sampling error and sample size precision with finite population correction.
- Confidence Interval Calculator — Calculate confidence intervals and critical values using Z and t-distributions.
- Standard Error (SEM) Calculator — Calculate sample mean precision and standard error of the mean.
- Z-Score Calculator — Calculate standard scores, percentiles, and normal distribution p-values.
- Percentage Calculator Hub — Master tool for general percentages, ratios, and variance.
Frequently Asked Questions
How many survey respondents do I need?
For most surveys, a sample size of 385 respondents is recommended. This provides a 95% confidence level with a ±5% margin of error, which represents the standard academic and market research benchmark for populations over 50,000.
Why is 50% used as the default population proportion?
Setting p = 0.5 (50%) maximizes the mathematical product p(1 − p) = 0.25. This yields the largest possible sample size, guaranteeing that your sample will be sufficiently large regardless of the actual survey results.
When should I enter a population size?
Only enter a population size if your target audience is a small, finite group (e.g., surveying all 500 employees at a company). If your audience exceeds 50,000 people (such as all internet users or national voters), leave it blank.
What is the difference between sample size and response rate?
Sample size is the number of completed survey responses you need. If you need 385 completed surveys and anticipate a 20% response rate, you must distribute your survey to 1,925 total candidates (385 / 0.20 = 1,925).
Is my research data saved or logged?
No. All calculations run 100% locally inside your browser's memory using JavaScript. No population metrics, sample targets, or research parameters are transmitted over an HTTP network or saved in an external database.