Free tool Margin of error calculator and formula
“52% of customers prefer the new plan, ± 3 points.” Where does the ± come from? Enter how many people answered and the percentage you got. The calculator gives the margin of error, the range it implies, and a table of how the margin shrinks as the sample grows.
- 90%, 95%, 99% confidence
- Finite population correction
- Runs locally
Enter your sample
Use completed responses for the sample size. Leave the proportion at 50% for the widest, most cautious margin.
Completed responses or observations. Thousands separators are fine.
95% is the convention for surveys and polls.
The percentage you got, for example 52%. Leave at 50% for the largest margin, which holds for every question.
Fill in only if the sample is a noticeable share of a small population, such as 300 of 1,200 employees.
Margin of error (±)
3.10%
Percentage points either side of the proportion.
Lower bound
46.90%
Upper bound
53.10%
z value
1.960
For the chosen confidence level.
Margin of error by sample size
Same proportion and population; your sample size is highlighted.
| Sample size | 90% confidence | 95% confidence | 99% confidence |
|---|---|---|---|
| 100 | ±8.22% | ±9.80% | ±12.88% |
| 200 | ±5.82% | ±6.93% | ±9.11% |
| 300 | ±4.75% | ±5.66% | ±7.44% |
| 400 | ±4.11% | ±4.90% | ±6.44% |
| 500 | ±3.68% | ±4.38% | ±5.76% |
| 600 | ±3.36% | ±4.00% | ±5.26% |
| 800 | ±2.91% | ±3.46% | ±4.55% |
| 1,000 | ±2.60% | ±3.10% | ±4.07% |
| 1,500 | ±2.12% | ±2.53% | ±3.33% |
| 2,000 | ±1.84% | ±2.19% | ±2.88% |
| 3,000 | ±1.50% | ±1.79% | ±2.35% |
| 5,000 | ±1.16% | ±1.39% | ±1.82% |
| 10,000 | ±0.82% | ±0.98% | ±1.29% |
How it was calculated
| Step | Calculation |
|---|---|
| z for 95% confidence | z = 1.9600 |
| Standard error of the proportion | √(p(1 − p) ÷ n) = √(0.5 × 0.5 ÷ 1000) = 0.0158114 |
| Margin of error | e = z × √(p(1 − p) ÷ n) = 1.9600 × 0.0158114 = 0.0309898 |
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https://tapcub.com/tools/margin-of-error-calculator
Version v1.0Updated Update history
What ± 3 points means
In the example, 1,000 people answered and half said yes. The margin of error is ±3.10 percentage points at 95% confidence.
Margin of error
If the survey were repeated many times with fresh random samples, about 95% of the resulting ranges would contain the true share. It is measured in percentage points, not percent of the result: 50% ± 3.1 means 46.9% to 53.1%.
Lower and upper bound
The range the true value plausibly falls in. If you are comparing against a threshold or a competitor’s number and it sits inside this range, the survey cannot tell them apart.
Proportion
The margin is widest at 50% and narrows towards 0% or 100%: with 1,000 responses it is ±3.10 at 50%, ±2.84 at 30% and ±1.86 at 10%. Reporting the 50% figure covers every question in the survey.
Sample size table
Precision grows with the square root of n. Four times the responses halve the margin: 400 responses give ±4.90, 1,600 give ±2.45. Beyond a few thousand, gains are small for the extra cost.
The margin of error only covers random sampling. Leading questions, self-selected respondents and non-response bias are not in it, and they are often larger.
Two numbers, one choice
The result updates as you type.
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1
Enter the sample size
Count completed responses to the question you are reporting, not invitations sent.
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2
Enter the proportion
Use the result you are reporting, or keep 50% to state one margin for the whole survey.
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3
Add the population if it is small
If you surveyed more than about 5% of a known group, entering its size narrows the margin. Then read the table to see what a bigger sample would buy you.
z × standard error, corrected for small populations
The margin of error is the critical z value multiplied by the standard error of the proportion. When the sample is a sizeable share of a finite population, the finite population correction shrinks it.
- z: 1.645 for 90%, 1.960 for 95%, 2.576 for 99% confidence
- Uses the normal approximation (Wald), the convention for survey margins
- For small samples or proportions near 0% or 100%, the Wilson interval is more accurate
- Halving the margin requires four times the sample
Margin of error
e = z × √( p(1 − p) ÷ n )
n = 1,000, p = 50%, 95%: 1.96 × √(0.25 ÷ 1,000) = 1.96 × 0.0158 = ±3.10%
Finite population correction
e = z × √( p(1 − p) ÷ n ) × √( (N − n) ÷ (N − 1) )
Same sample from N = 5,000: × 0.8945 → ±2.77%
Rule of thumb at 95% and p = 50%
e ≈ 1 ÷ √n
n = 1,000: 1 ÷ 31.6 ≈ ±3.2%
Margin of error for common sample sizes
95% confidence, large population. The 50% column is the figure pollsters quote; the other columns show how the margin narrows for less even results.
| Sample size | p = 50% | p = 30% or 70% | p = 10% or 90% |
|---|---|---|---|
| 100 | ±9.80% | ±8.98% | ±5.88% |
| 400 | ±4.90% | ±4.49% | ±2.94% |
| 1,000 | ±3.10% | ±2.84% | ±1.86% |
| 1,600 | ±2.45% | ±2.25% | ±1.47% |
| 2,500 | ±1.96% | ±1.80% | ±1.18% |
| 10,000 | ±0.98% | ±0.90% | ±0.59% |
e = 1.95996 × √(p(1 − p) ÷ n). For n = 100 and p = 10% only 10 people are expected in the smaller group, which is where the normal approximation starts to strain. Example values.
How margins of error get misread
The formula is standard; the errors are in how the number is applied.
Applying the overall margin to subgroups
A ±3.1% margin for 1,000 people becomes about ±9.8% for a subgroup of 100. Every breakdown needs its own margin.
Comparing two results with one margin
The margin on a difference between two groups is larger than either group’s margin, roughly √2 times for equal groups. 52% vs 48% with ±3.1% each is not a clear lead.
Confusing points with percent
±3 means percentage points: 50% becomes 47%–53%, not 48.5%–51.5%.
Treating it as total error
An opt-in web poll with 10,000 responses has a tiny margin of error and can still be badly biased. The margin assumes a random sample.
Precise numbers start with clean ones
Most of the percentages you report come from analytics, not surveys. TapCub keeps the denominators honest so the margin is the only uncertainty left.
Funnels
Conversion between every step with fixed time windows, split by channel, device or segment.
Learn moreSegments
Live, rule-based groups whose size updates as data arrives: the population behind a percentage.
Learn moreBots counted separately
Crawlers inflate the sample without adding information. They are kept out of reports by default.
Learn moreMargin of error questions
Still have a question?
Chat with the team behind TapCub — we usually reply within a few hours on business days.
What is the margin of error formula, and how do you calculate it?
Multiply the z value for your confidence level by the standard error of the proportion: e = z × √(p(1 − p) ÷ n). For 1,000 responses, p = 50% and 95% confidence: 1.96 × √(0.25 ÷ 1,000) = 0.031, or ±3.1 percentage points.
What is a good margin of error?
It depends on the decision. Published national polls usually aim for ±3% (about 1,000 respondents); internal surveys often accept ±5% (about 385). If the options you compare could be closer than your margin, you need a larger sample.
Why is the margin largest at 50%?
The variance of a proportion, p(1 − p), peaks at p = 0.5. Results near 50% are the most uncertain; results near 0% or 100% have narrower margins for the same sample size.
Does population size affect the margin of error?
Only when the sample is a noticeable share of the population. 1,000 responses from 5,000 people give ±2.77% instead of ±3.10%; from a million people, ±3.10%. For large populations it can be ignored.
What sample size do I need for a ±5% margin?
About 385 at 95% confidence for a large population, fewer for a small one. The survey sample size calculator works it out for any margin, population and response rate.
Is margin of error the same as a confidence interval?
The margin of error is half the width of the confidence interval. This page uses the normal (Wald) formula pollsters use. For small samples or extreme proportions, the confidence interval calculator uses the more accurate Wilson interval.
How do I reduce the margin of error?
Collect more responses, or accept a lower confidence level. Doubling precision needs four times the sample. Better sampling does not shrink the margin, but it reduces bias, which the margin never shows.
Update history
New features, fixes and data refreshes for this tool, newest first.
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· v1.0
LaunchedFirst release.
Missing something or found an error? Tell us — requests that come up often go into the next update. All tool updates
Plan the next survey with the margin in mind
If the margin you got is too wide, work out how many responses the next round needs, or check how precise a mean is rather than a percentage.