Research Guide

Sample Size and Margin of Error: A Practical Guide

How sample size controls margin of error, the formulas behind both, the finite-population correction, and the practical caveats researchers should not skip.

Sample size and margin of error are two sides of the same coin. For a given population and confidence level, a larger sample gives a narrower margin of error. Deciding how many respondents you need is really a decision about how much error you can tolerate.

This guide covers the standard formulas, when the finite-population correction matters, and the assumptions behind the numbers — because the arithmetic is only as good as the sampling design behind it.

The margin of error for a proportion

For a proportion (a percentage-type result), the margin of error at confidence level α is E = z·√(p̂(1−p̂)/n), where z is the z-score for the confidence level (1.96 for 95%, 1.645 for 90%, 2.576 for 99%), p̂ is the observed proportion, and n is the sample size. The formula comes from the standard error of a proportion.

Because p̂(1−p̂) is largest when p̂ = 0.5, the margin of error is widest near 50%. Many sample-size calculators use p̂ = 0.5 to get the conservative (largest) sample size, which is why 50% is the default in the PanelRoster sample size calculator.

Sample size for a proportion

Rearranging the margin-of-error formula gives the required sample size for a proportion: n = z²·p(1−p)/E², where p is the expected proportion, E is the desired margin of error, and z is the confidence-level z-score. With p = 0.5, E = 0.05 (5%) and 95% confidence, this gives n = 1.96²·0.25/0.0025 ≈ 384.

For a mean, the comparable formula is n = (z·σ/E)², which requires an estimate of the population standard deviation σ.

The finite population correction

When the sample is a large fraction of a small population, the margin of error shrinks slightly. The finite population correction multiplies the standard error by √((N−n)/(N−1)), where N is the population size. In practice the correction is negligible when n is below about 5–10% of N, which is why national surveys use the same sample size whether the population is 30 million or 300 million.

The PanelRoster sample size calculator applies the correction when you enter a finite population size.

The assumptions behind the numbers

These formulas assume a simple random sample: every member of the population has a known, equal chance of selection. Real surveys use stratification, clustering and weighting, which change the effective sample size (the design effect). The formulas also assume responses are measured without systematic (non-sampling) error.

A low response rate can introduce nonresponse bias that no margin of error can fix. Margin of error describes sampling variability only — it says nothing about bias.

Key takeaways

Larger samples give narrower margins of error; the relationship is a square root, so quadrupling n halves the error.
Use p = 0.5 for a conservative proportion sample size when you have no better estimate.
The finite population correction only matters when the sample exceeds roughly 5–10% of the population.
Random sampling assumptions matter more than the arithmetic; bias is not covered by margin of error.

Tools used in this guide

References

References are provided for further reading; PanelRoster is not affiliated with the linked resources.