Sample Size Calculator

How to Calculate Required Sample Size

The standard sample size formula for estimating a proportion balances three things: how confident you want to be, how much error you'll tolerate, and how variable the outcome is expected to be (captured by the expected proportion).

$$n_0 = \frac{Z^2 \cdot p(1-p)}{E^2}$$

For example, at 95% confidence (Z = 1.96), a 5% margin of error, and an expected proportion of 50%, the formula gives $n_0 = \frac{1.96^2 \times 0.5 \times 0.5}{0.05^2} \approx 384.16$, rounded up to 385 respondents.

Why 50% Is the "Safe" Default

The term p(1-p) is maximized when p = 0.5, which is why using 50% as your expected proportion — when you genuinely don't have a better estimate — produces the largest, most conservative required sample size. If you have solid prior data suggesting the true proportion is closer to, say, 20% or 80%, using that value will give you a smaller, more efficient required sample size.

Finite Population Correction

The base formula assumes an effectively infinite population. If you're sampling from a small, fully known population — like all 800 employees at a company — you can shrink the required sample size using the finite population correction, entered above as "Population Size." This correction only matters meaningfully when your sample would represent a substantial share of the total population; for large populations, the correction is negligible.

Trade-offs in Sample Size Planning

Every choice here has a cost: a smaller margin of error requires a larger sample, a higher confidence level requires a larger sample, and a proportion closer to 50% requires a larger sample than one closer to 0% or 100%. Sample size planning is fundamentally about deciding how much precision and confidence you actually need, then finding the minimum sample that delivers it.

A Brief History of Sample Size Planning

Formal sample size and survey sampling theory developed largely in the early-to-mid 20th century, driven substantially by the emergence of political polling and the U.S. Census Bureau's efforts to make reliable inferences from surveying only a fraction of the population rather than everyone. A pivotal, cautionary moment came in the 1936 U.S. presidential election, when the Literary Digest magazine predicted a landslide loss for Franklin D. Roosevelt based on a huge but badly biased sample of over 2 million respondents (drawn largely from car and telephone owners, skewing wealthier), while pollster George Gallup correctly predicted Roosevelt's win using a much smaller but carefully, randomly selected sample of a few thousand people. The episode became a foundational lesson in statistics: sample size alone doesn't guarantee accuracy — how the sample is selected matters just as much, if not more, than how large it is.

Common Sample Size Mistakes

Assuming a bigger sample is always better without considering diminishing returns is a common misconception — beyond a certain point, doubling your sample size only marginally tightens your margin of error, since the required sample size scales with the square of the desired precision. Using a convenience sample (whoever is easiest to reach) and treating it as if it were randomly selected from the full population is a much more serious error, echoing the Literary Digest's mistake above — no sample size calculation can fix a fundamentally biased sampling method. Forgetting to apply finite population correction when sampling from a small, known population can also lead to recommending an unnecessarily large sample size.

Sample Size Terms You Should Know

Margin of Error — the range above and below a survey result within which the true population value is expected to fall, at a given confidence level.

Confidence Level — the long-run percentage of times a properly conducted survey's margin of error would capture the true population value.

Expected Proportion — an estimate of the outcome you're measuring, used in the sample size formula; 50% is used as a conservative default when this is unknown, since it requires the largest sample.

Finite Population Correction — an adjustment that reduces the required sample size when sampling from a small, known total population rather than an effectively unlimited one.