Sample Size and Margin of Error Calculator

Continue in the tool

More pages for this tool

confidence interval for a proportion Wilson

Confidence interval for a proportion: Wald, Wilson and exact

In the Confidence interval tab, choose Proportion, type the successes and the total and see the four methods in the same table. Wilson is highlighted as the recommended default, and the interpretation text comes ready to copy.

Which confidence interval method to use

The Wald interval, p ± z · √(p(1 − p)/n), is the one taught in textbooks because it is simple, but it has two problems: it covers less than nominal with small n or p near 0 and 100%, and it collapses to zero width when there are 0 or n successes. With 0 out of 15, Wald gives [0%; 0%], a certainty the sample does not support. The Wilson interval starts from the test equation itself and fixes this: 0 out of 15 becomes 0% to 20.4%.
The four methods side by side for 3 successes out of 10 show the difference: Wald runs from 1.6% to 58.4%, Wilson from 10.8% to 60.3%, Agresti-Coull from 10.3% to 60.8% and the exact Clopper-Pearson from 6.7% to 65.2%. With 250 out of 1,000 the four almost coincide (22.3% to 27.7% for Wald, 22.4% to 27.8% for Wilson). Rule of thumb: with large n and p far from the extremes any will do; with small n, use Wilson, or Clopper-Pearson if you want the conservative interval that never covers less than nominal.
The confidence level describes the method: at 95%, if we repeated the collection many times, about 95% of the intervals built this way would contain the true proportion. It is not the chance that this specific interval contains the value. Raising it to 99% widens the interval (250 out of 1,000 goes to 21.6% to 28.7% for Wilson). Brown, Cai and DasGupta (2001) compare these methods and recommend Wilson or Agresti-Coull over Wald. For the interval of a mean, the tool uses Student's t, in the same tab.

Checked examples

12 out of 50, Wilson

Input: x = 12, n = 50, 95%
Expected output: 14,3% a 37,4%

Wald: 12.2% to 35.8%.

3 out of 10, four methods

Input: x = 3, n = 10, 95%
Expected output: Exato: 6,7% a 65,2%

Wald: 1.6% to 58.4%.

Zero successes

Input: x = 0, n = 15, 95%
Expected output: Wilson: 0% a 20,4%

Exact: 0% to 21.8%; Wald: 0%.

Large sample

Input: x = 250, n = 1.000, 95%
Expected output: Wilson: 22,4% a 27,8%

The four methods almost equal.

More confidence

Input: x = 250, n = 1.000, 99%
Expected output: Wilson: 21,6% a 28,7%

z = 2.5758 instead of 1.96.

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

My Wald interval came out negative. What do I do?

It is the classic sign that Wald does not suit your case. The tool clips at 0% and 100% and warns you. Use Wilson or the exact Clopper-Pearson, which always stay between 0 and 100%.

Can I enter the proportion as a percentage instead of successes?

You can: choose "Percentage and total". The tool converts to the nearest whole number of successes (30% of 100 becomes 30) and shows the count used. The exact method needs a whole count.

What is the smallest n for the normal approximation?

A common rule is at least 5 successes and 5 failures (n·p ≥ 5 and n·(1 − p) ≥ 5), and some authors ask for 10. The tool shows a warning when Wald is outside that safe zone.

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

Which confidence interval method to use

The Wald interval, p ± z · √(p(1 − p)/n), is the one taught in textbooks because it is simple, but it has two problems: it covers less than nominal with small n or p near 0 and 100%, and it collapses to zero width when there are 0 or n successes. With 0 out of 15, Wald gives [0%; 0%], a certainty the sample does not support. The Wilson interval starts from the test equation itself and fixes this: 0 out of 15 becomes 0% to 20.4%.

The four methods side by side for 3 successes out of 10 show the difference: Wald runs from 1.6% to 58.4%, Wilson from 10.8% to 60.3%, Agresti-Coull from 10.3% to 60.8% and the exact Clopper-Pearson from 6.7% to 65.2%. With 250 out of 1,000 the four almost coincide (22.3% to 27.7% for Wald, 22.4% to 27.8% for Wilson). Rule of thumb: with large n and p far from the extremes any will do; with small n, use Wilson, or Clopper-Pearson if you want the conservative interval that never covers less than nominal.

The confidence level describes the method: at 95%, if we repeated the collection many times, about 95% of the intervals built this way would contain the true proportion. It is not the chance that this specific interval contains the value. Raising it to 99% widens the interval (250 out of 1,000 goes to 21.6% to 28.7% for Wilson). Brown, Cai and DasGupta (2001) compare these methods and recommend Wilson or Agresti-Coull over Wald. For the interval of a mean, the tool uses Student's t, in the same tab.

Checked examples

12 out of 50, Wilson

Input: x = 12, n = 50, 95%
Expected output: 14,3% a 37,4%

Wald: 12.2% to 35.8%.

3 out of 10, four methods

Input: x = 3, n = 10, 95%
Expected output: Exato: 6,7% a 65,2%

Wald: 1.6% to 58.4%.

Zero successes

Input: x = 0, n = 15, 95%
Expected output: Wilson: 0% a 20,4%

Exact: 0% to 21.8%; Wald: 0%.

Large sample

Input: x = 250, n = 1.000, 95%
Expected output: Wilson: 22,4% a 27,8%

The four methods almost equal.

More confidence

Input: x = 250, n = 1.000, 99%
Expected output: Wilson: 21,6% a 28,7%

z = 2.5758 instead of 1.96.

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

My Wald interval came out negative. What do I do?

It is the classic sign that Wald does not suit your case. The tool clips at 0% and 100% and warns you. Use Wilson or the exact Clopper-Pearson, which always stay between 0 and 100%.

Can I enter the proportion as a percentage instead of successes?

You can: choose "Percentage and total". The tool converts to the nearest whole number of successes (30% of 100 becomes 30) and shows the count used. The exact method needs a whole count.

What is the smallest n for the normal approximation?

A common rule is at least 5 successes and 5 failures (n·p ≥ 5 and n·(1 − p) ≥ 5), and some authors ask for 10. The tool shows a warning when Wald is outside that safe zone.