Sample Size and Margin of Error Calculator

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how to calculate the margin of error of a survey

How to calculate the margin of error of a survey

In the Margin of error tab, type the number of respondents, the observed proportion (50% for the worst case) and the confidence level. The result comes in percentage points, with the range of plausible values and the formula.

What the margin of error of a survey means

The margin of error of a proportion is E = z · √(p · (1 − p) / n), where n is the number of respondents. At 95% confidence and p = 50%, it depends only on n: 400 respondents give ±4.90 percentage points, 600 give ±4.00, 1,000 give ±3.10, 2,000 give ±2.19 and 2,500 give ±1.96. Because n sits under a square root, quadrupling the sample only halves the margin, which is why national polls stop at 1,000 to 3,000 interviews.
How to read the number: if a survey of 1,000 interviewees shows a candidate at 40% and the margin is 3.1 points, the real value is very likely between 36.9% and 43.1%. For proportions far from 50% the margin is smaller: with p = 20%, the same sample gives ±2.5 points. Two candidates at 40% and 38% are in a statistical tie when the intervals overlap, but the correct calculation for the gap between them is different, and the gap is more uncertain than each isolated result.
The margin covers only the error of drawing a sample instead of asking everyone. It does not include refusals, biased questions, incomplete coverage or post-stratification weights. The design effect corrects part of that: with deff = 1.5 and 1,000 interviews, the margin rises from ±3.10 to ±3.80. That is why margins published by serious polling firms are often a bit larger than the simple formula.

Checked examples

400 respondents

Input: n = 400, p = 50%, 95%
Expected output: ±4,90 pontos

Almost the 5% of a sample of 385.

1,000 respondents

Input: n = 1.000, p = 50%, 95%
Expected output: ±3,10 pontos

The standard of many polls.

2,000 respondents

Input: n = 2.000, p = 50%, 95%
Expected output: ±2,19 pontos

Double the sample, only 30% less margin.

A proportion of 20%

Input: n = 1.000, p = 20%, 95%
Expected output: ±2,48 pontos

p(1 − p) = 0.16.

With a design effect

Input: n = 1.000, deff = 1,5
Expected output: ±3,80 pontos

3.10 · √1.5.

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

Does the margin of error depend on the population size?

Hardly at all, if the population is large. A survey of 1,000 people has a similar margin in a city of 500,000 and in the whole country. It only matters when the sample is a relevant part of the population.

Does a 3-point margin of error mean a 3% chance of error?

No. The margin is the width of the interval, ±3 points, and the confidence (95%) is the chance the method captures the real value. The remaining 5% is the chance the interval misses the real value.

How does the margin change for subgroups?

For a subgroup with smaller n, the margin is larger. In a survey of 1,000 people, the 200 young people aged 18 to 24 have a margin of ±6.9 points. Compute the margin with the subgroup's n.

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

What the margin of error of a survey means

The margin of error of a proportion is E = z · √(p · (1 − p) / n), where n is the number of respondents. At 95% confidence and p = 50%, it depends only on n: 400 respondents give ±4.90 percentage points, 600 give ±4.00, 1,000 give ±3.10, 2,000 give ±2.19 and 2,500 give ±1.96. Because n sits under a square root, quadrupling the sample only halves the margin, which is why national polls stop at 1,000 to 3,000 interviews.

How to read the number: if a survey of 1,000 interviewees shows a candidate at 40% and the margin is 3.1 points, the real value is very likely between 36.9% and 43.1%. For proportions far from 50% the margin is smaller: with p = 20%, the same sample gives ±2.5 points. Two candidates at 40% and 38% are in a statistical tie when the intervals overlap, but the correct calculation for the gap between them is different, and the gap is more uncertain than each isolated result.

The margin covers only the error of drawing a sample instead of asking everyone. It does not include refusals, biased questions, incomplete coverage or post-stratification weights. The design effect corrects part of that: with deff = 1.5 and 1,000 interviews, the margin rises from ±3.10 to ±3.80. That is why margins published by serious polling firms are often a bit larger than the simple formula.

Checked examples

400 respondents

Input: n = 400, p = 50%, 95%
Expected output: ±4,90 pontos

Almost the 5% of a sample of 385.

1,000 respondents

Input: n = 1.000, p = 50%, 95%
Expected output: ±3,10 pontos

The standard of many polls.

2,000 respondents

Input: n = 2.000, p = 50%, 95%
Expected output: ±2,19 pontos

Double the sample, only 30% less margin.

A proportion of 20%

Input: n = 1.000, p = 20%, 95%
Expected output: ±2,48 pontos

p(1 − p) = 0.16.

With a design effect

Input: n = 1.000, deff = 1,5
Expected output: ±3,80 pontos

3.10 · √1.5.

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

Does the margin of error depend on the population size?

Does a 3-point margin of error mean a 3% chance of error?

No. The margin is the width of the interval, ±3 points, and the confidence (95%) is the chance the method captures the real value. The remaining 5% is the chance the interval misses the real value.

How does the margin change for subgroups?

For a subgroup with smaller n, the margin is larger. In a survey of 1,000 people, the 200 young people aged 18 to 24 have a margin of ±6.9 points. Compute the margin with the subgroup's n.