Sample Size and Margin of Error Calculator

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finite population sample size correction

Sample size for a finite population

In the Size: proportion tab, turn on the "Finite population" option and type N. The result shows the infinite-population n₀, the corrected n and the formula with the values filled in, ready to cite in your paper.

When and how to correct for a finite population

The basic formula assumes a population so large that drawing one element does not change the odds of the others. When the total N is known and the sample would be a relevant part of it, you gain precision from having already observed part of the universe, and Cochran's correction shrinks the size: n = n₀ / (1 + (n₀ − 1) / N), where n₀ is the infinite-population result (384.15 for 95%, 5% and p = 50%). In practice it is worth correcting when the sample exceeds about 5% of the population.
The effect is large for small populations: with N = 100, 80 respondents; with N = 300, 169; with N = 800, 260; with N = 2,000, 323. It grows slowly afterwards: 357 for N = 5,000 and 382 for N = 50,000, almost the 385 of the infinite population. A thesis example: a school with 300 students, 95% confidence and a 5% margin needs 169 students; with a 10% margin, 73; at 90% confidence and a 10% margin, only 56.
A few cautions. The correction assumes random selection without replacement from a complete list of the population; without the list, N is only an estimate. It also does not replace the non-response adjustment: if only half of the 169 students answer, invite 338. And if the corrected n gets close to N, the best option is a census. The tool warns when the size asked for, after the design effect, exceeds the total population.

Checked examples

Population of 100

Input: N = 100, 95%, 5%, p = 50%
Expected output: n = 80

Versus 385 for an infinite population.

School with 300 students

Input: N = 300, 95%, 5%, p = 50%
Expected output: n = 169

384.15 / (1 + 383.15 / 300).

N = 1,000

Input: N = 1.000, 95%, 5%, p = 50%
Expected output: n = 278

The most quoted case.

N = 2,000

Input: N = 2.000, 95%, 5%, p = 50%
Expected output: n = 323

About 16% of the population.

A 10% margin

Input: N = 300, 90%, 10%, p = 50%
Expected output: n = 56

At 95%, 10% and N = 300 it would be 73.

Full tool FAQ

How do I calculate the sample size for a survey?

For a proportion, use n₀ = z² · p · (1 − p) / E². At 95% confidence (z = 1.96), a 5% margin and p = 50%, n₀ = 384.15, so 385 people. If the population is finite, apply n = n₀ / (1 + (n₀ − 1) / N). Then multiply by the design effect, if any, and divide by the expected response rate to learn how many to invite.

Why does everyone use 385 and p = 50%?

When should I use the finite population correction?

Which confidence level and margin of error should I choose?

How do I interpret the margin of error?

How do I calculate the sample size for a mean?

What are the design effect and the response rate?

Wald, Wilson or Clopper-Pearson: which interval method should I use?

How do I calculate the sample size to compare two groups?

Do these formulas work for convenience samples?

Frequently asked questions

Do I need to correct if the population is 1 million?

It makes no difference: the result is 384 against 385 without correction. In practice, ignore the correction when the sample is under 5% of the population.

Does a finite population change the margin of error?

Yes, in the sense that the same margin is reached with fewer people. The Margin of error tab also accepts N and multiplies the variance by (N − n) / (N − 1); with n = N (a census), the margin is zero.

Do I need a sample per stratum, for example per class?

If you want reliable results per class, compute each class size as its own population. If the survey only reports the total, a sample proportional to class sizes usually suffices and can use a deff below 1.

Next steps

Respondents needed with the proportion and population entered
Margin	90%	95%	99%
±1%	6,764	9,604	16,588
±2%	1,691	2,401	4,147
±3%	752	1,068	1,844
±4%	423	601	1,037
±5%	271	385	664
±7%	139	196	339
±10%	68	97	166

Respondents needed with the proportion and population entered

Margin

90%

95%

99%

±1%

6,764

9,604

16,588

±2%

1,691

2,401

4,147

±3%

752

1,068

1,844

±4%

423

601

1,037

±5%

271

385

664

±7%

139

196

339

±10%

166

When and how to correct for a finite population

The basic formula assumes a population so large that drawing one element does not change the odds of the others. When the total N is known and the sample would be a relevant part of it, you gain precision from having already observed part of the universe, and Cochran's correction shrinks the size: n = n₀ / (1 + (n₀ − 1) / N), where n₀ is the infinite-population result (384.15 for 95%, 5% and p = 50%). In practice it is worth correcting when the sample exceeds about 5% of the population.

The effect is large for small populations: with N = 100, 80 respondents; with N = 300, 169; with N = 800, 260; with N = 2,000, 323. It grows slowly afterwards: 357 for N = 5,000 and 382 for N = 50,000, almost the 385 of the infinite population. A thesis example: a school with 300 students, 95% confidence and a 5% margin needs 169 students; with a 10% margin, 73; at 90% confidence and a 10% margin, only 56.

A few cautions. The correction assumes random selection without replacement from a complete list of the population; without the list, N is only an estimate. It also does not replace the non-response adjustment: if only half of the 169 students answer, invite 338. And if the corrected n gets close to N, the best option is a census. The tool warns when the size asked for, after the design effect, exceeds the total population.

Checked examples

Population of 100

Input: N = 100, 95%, 5%, p = 50%
Expected output: n = 80

Versus 385 for an infinite population.

School with 300 students

Input: N = 300, 95%, 5%, p = 50%
Expected output: n = 169

384.15 / (1 + 383.15 / 300).

N = 1,000

Input: N = 1.000, 95%, 5%, p = 50%
Expected output: n = 278

The most quoted case.

N = 2,000

Input: N = 2.000, 95%, 5%, p = 50%
Expected output: n = 323

About 16% of the population.

A 10% margin

Input: N = 300, 90%, 10%, p = 50%
Expected output: n = 56

At 95%, 10% and N = 300 it would be 73.

Full tool FAQ

How do I calculate the sample size for a survey?

Why does everyone use 385 and p = 50%?

When should I use the finite population correction?

Which confidence level and margin of error should I choose?

How do I interpret the margin of error?

How do I calculate the sample size for a mean?

What are the design effect and the response rate?

Wald, Wilson or Clopper-Pearson: which interval method should I use?

How do I calculate the sample size to compare two groups?

Do these formulas work for convenience samples?

Frequently asked questions

Do I need to correct if the population is 1 million?

It makes no difference: the result is 384 against 385 without correction. In practice, ignore the correction when the sample is under 5% of the population.