Sample Size Calculator

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Sample Size

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Table of Contents

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What is Sample Size?

This sample-size calculator estimates how many independent observations are needed to estimate a population proportion with a chosen two-sided confidence level and margin of error. It uses a normal approximation and can apply a finite-population correction when the eligible population size is known.

The result addresses random sampling precision under the formula’s assumptions. It does not fix selection bias, low response, clustering, weighting, measurement error, or a poorly defined target population. Complex studies often need design effects, expected response rates, subgroup targets, and professional statistical planning.

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How to Use This Calculator

  1. Enter a confidence level above 50% and below 100%, such as 95.
  2. Enter the desired margin of error as a percentage, such as 5.
  3. Enter an expected proportion as a percentage. Use 50 when no defensible prior estimate is available and a conservative maximum is desired.
  4. Optionally enter a positive whole-number population for sampling without replacement.
  5. Select Calculate. The result is always rounded upward to a whole observation.

All three percentage fields now consistently expect percentages. Do not enter 0.05 for 5% or 0.5 for 50%.

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Formula Used

n₀ = z²p(1 − p) ÷ e²

Here z is the two-sided normal critical value, p is the expected proportion in decimal form, and e is the absolute margin in decimal form. With finite population N:

n = n₀ ÷ [1 + (n₀ − 1)/N]

The page calculates z from the entered confidence percentage and applies the correction only when N is supplied.

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Variables Explained

Entry or resultMeaning
Confidence levelDesired two-sided normal confidence percentage.
z valueNormal critical value corresponding to that confidence level.
Margin of error eMaximum planned absolute percentage-point error under the approximation.
Proportion pExpected population share, entered as a percentage.
Population NOptional count of eligible units when sampling without replacement.
Sample sizeCalculated requirement rounded upward to the next whole observation.
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Worked Example

Enter 95% confidence, 5% margin of error, and a 50% proportion, leaving population blank. The normal critical value is about 1.959964. Then n₀ = 1.959964² × 0.5 × 0.5 ÷ 0.05² = about 384.146.

Round upward, so the required sample is 385. If the population were only 1,000, the finite correction would reduce the result to about 278 completed independent responses.

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Why Use This Calculator?

The result is the number of usable observations, not necessarily the number of people to contact. If only 60% are expected to respond, divide the required completed count by 0.60 and adjust for ineligible or incomplete responses.

Using p = 50% produces the largest p(1 − p) and therefore a conservative size when p is unknown. A prior estimate can reduce the number, but it should come from credible evidence. Cluster sampling and unequal weights usually increase variance; apply an appropriate design effect. Very rare outcomes, hypothesis tests, comparisons between groups, and continuous outcomes require different planning methods.

Precision changes nonlinearly. Halving the margin of error roughly quadruples the uncorrected sample size because e is squared in the denominator. Raising confidence also increases the critical value and sample requirement. These tradeoffs help explain why a very tight target can become expensive quickly.

The finite correction matters most when the planned sample is a meaningful share of the population. It assumes a simple random sample without replacement from a known list. Do not enter the number of invitations or an estimated audience as N unless it truly represents the eligible finite population.

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Common Mistakes

    • Entering decimal proportions even though the form expects percentages.
    • Rounding the unrounded requirement down.
    • Treating contacts invited as completed usable responses.
    • Applying finite correction to a vague or changing population.
    • Ignoring clustering, stratification, weights, nonresponse, or subgroup goals.
    • Believing a large convenience sample removes selection bias.
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Frequently Asked Questions

Why is 50% used when the proportion is unknown?

The product p times one minus p is largest at 0.5, so it gives the largest and most conservative size under this formula.

Should I round the result?

Yes, always upward. A fractional observation cannot be collected, and rounding down would fall below the calculated requirement.

When should I enter population size?

Use it for a known finite population sampled without replacement. Leave it blank for a very large or undefined population.

Does this number include nonresponses?

No. It is a target for usable observations. Inflate invitations based on expected eligibility, response, and completion rates.

Can this plan a comparison between two groups?

Not by itself. Group comparisons require effect size, allocation, power, significance level, and the planned analysis.

Does a larger sample prevent bias?

No. A large unrepresentative sample can remain biased. Sampling frame, recruitment, measurement, and response need separate attention.

Why is 95% confidence not a guarantee?

The confidence statement belongs to repeated use of the method under its assumptions. It is not a probability that one fixed interval contains the parameter after collection.

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Sources & References

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