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How to Work Out a Survey's Margin of Error
Manesh Jayawardhana
CIO & Co-founder
A survey of 1,000 people finds 52% support for something. Reported as “52% support”, it sounds decisive. Reported honestly it’s “somewhere between about 49% and 55%”, which includes the possibility that it’s a minority position.
Same data, two very different headlines, and the difference is one number nobody quotes.
The calculation
For a proportion:
MOE = z × √( p(1−p) ÷ n )
Where p is the observed proportion, n the sample size, and z the value for your confidence level — 1.96 at 95%.
At n = 1,000 and p = 0.5: 1.96 × √(0.25/1000) ≈ 0.031, or ±3.1 percentage points.
Two properties worth internalising.
It shrinks with the square root of n. Halving the margin requires quadrupling the sample. Going from ±3.1 to ±1.5 points means moving from 1,000 responses to about 4,000 — which is why large surveys cost so much for relatively modest precision gains.
It’s largest at 50%. The p(1−p) term peaks there, so a result near a split has the widest interval and a lopsided result a narrower one. Using p = 0.5 gives the conservative worst case, which is why sample size calculators default to it.
| Sample size | Margin at 95% |
|---|---|
| 100 | ±9.8 points |
| 400 | ±4.9 points |
| 1,000 | ±3.1 points |
| 4,000 | ±1.5 points |
Population size barely matters
A common surprise: polling 1,000 people gives about the same margin whether the population is 100,000 or 100 million.
The finite population correction only becomes significant when the sample is a substantial fraction of the population — surveying 300 people in a company of 500, for instance. For any large population, sample size alone determines the margin.
Which means national polls and city polls need similar sample sizes for similar precision, and the intuition that a bigger country needs a bigger sample is simply wrong.
What a confidence level actually says
95% confidence does not mean a 95% probability the true value is in this interval.
It means that if you repeated the same survey many times, about 95% of the intervals produced would contain the true value. It’s a statement about the long-run behaviour of the procedure, not about this particular result.
The distinction sounds pedantic and matters when interpreting a single survey: this interval either contains the true value or it doesn’t, and you can’t know which.
The error the formula doesn’t measure
This is the important caveat, and it’s usually the larger problem.
Margin of error covers sampling error only — the variation from surveying a subset rather than everyone. It assumes a genuinely random sample from the population you care about.
It says nothing about:
Non-response bias. If 8% of people contacted respond, and those 8% differ systematically from the 92% who didn’t, the sample is skewed in a way no sample size fixes.
Question wording. Two differently-worded questions on the same topic reliably produce different results, sometimes by more than the margin.
Sampling frame. A survey of people who answer landlines, or who are in a panel, or who follow your account, doesn’t sample the general population at all.
Order effects, interviewer effects, and social desirability bias.
Each can exceed the sampling error, and none appears in the calculation. A survey reporting ±3% with a 6% response rate is quoting precision it doesn’t have.
Common mistakes to avoid
- Reporting a difference between two results without checking whether it exceeds the margin.
- Treating the margin as covering all survey error.
- Quoting a margin for a subgroup, which has a much smaller n and a much wider margin than the headline.
- Interpreting 95% confidence as a 95% probability about this result.
- Increasing sample size to fix a bias problem, which it cannot.
How to do it with Margin of Error Calculator
The Margin of Error Calculator computes the margin and the sample needed for a target.
- Enter the sample size and confidence level.
- Read the margin and apply it to your result rather than quoting the result alone.
- To hit a target margin, use the reverse calculation — and note the square-root relationship.
- Report the response rate alongside, since it speaks to the errors the margin doesn’t cover.
Other statistics tools are in the tools directory.
Frequently asked questions
Why does population size barely matter?
Because for populations much larger than the sample, the margin depends almost entirely on sample size. Polling 1,000 people gives a similar margin whether the population is a million or a hundred million.
What does a 95% confidence level mean?
That if the same survey were repeated many times, about 95% of the intervals produced would contain the true value. It’s a statement about the procedure, not a 95% probability about this result.
Does margin of error cover all survey error?
No. It covers sampling error only. Non-response bias, question wording and an unrepresentative sampling frame are usually larger sources and appear nowhere in the calculation.
Final thought
Quote the interval, not the point estimate. “52%, ±3” is the honest version, and it prevents a statistical tie being reported as a lead.