What margin of error means
When you survey a sample instead of an entire population, your result is an estimate, not an exact measurement. The margin of error tells you how far off that estimate is likely to be. If 60% of respondents say they'd recommend your product and your margin of error is ±4.9 percentage points, the true figure across everyone you could have asked is probably somewhere between 55.1% and 64.9%.
The "probably" is what the confidence level covers. At 95% confidence — the standard choice — you'd expect the true value to land inside that range in 95 out of 100 repeated surveys. It's not a guarantee for any single survey, but it's a well-calibrated bet.
Planning a survey and want to work the other direction — start from a target margin of error and find out how many responses you need? Use our sample size calculator, the companion to this tool.
The margin of error formula
The basic formula for a proportion is:
MoE = z × √(p(1−p) / n)
- z is the z-score for your confidence level: 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. Higher confidence means a wider interval.
- p is the response distribution — the proportion you expect to pick a given answer. If you don't know, use 50%, which produces the widest (most conservative) margin.
- n is your sample size — the number of completed responses.
If you know the total population size N and it isn't much larger than your sample, the margin shrinks a little. You apply the finite population correction by multiplying the result by √((N−n) / (N−1)). Surveying 200 people out of a 300-person company is far more precise than 200 people out of a million customers, and the correction captures exactly that.
A worked example
Say you collected 400 responses and want 95% confidence, with p at the conservative 50%:
MoE = 1.96 × √(0.5 × 0.5 / 400) = 1.96 × √0.000625 = 1.96 × 0.025 = 0.049
That's a margin of error of ±4.9 percentage points. If 62% of your 400 respondents chose "very satisfied", the true share among all your customers is likely between 57.1% and 66.9%. Whether that range is tight enough depends on the decision you're making — for a broad directional read it's plenty; for declaring a 3-point improvement over last quarter, it isn't.
What drives your margin of error
Three inputs move the number, and they don't move it equally:
- Sample size — with quadratic returns. Because n sits under a square root, halving your margin of error takes four times the responses. Going from ±9.8 to ±4.9 points means growing from 100 to 400 responses; getting to ±2.45 would take 1,600. Early responses buy a lot of precision, later ones progressively less.
- Confidence level. Moving from 90% to 99% confidence swaps a z-score of 1.645 for 2.576, widening the margin by more than half. You're trading a tighter interval for a higher chance the interval misses.
- Response distribution. The p(1−p) term peaks at p = 50% and shrinks toward the extremes. A question where answers split 90/10 has a noticeably smaller margin than one splitting 50/50, which is why 50% is the safe default when you can't predict the split.
Margin of error at common sample sizes
At the 95% confidence level with p = 50% and a large population, here's what different sample sizes buy you:
| Sample size | Margin of error |
|---|---|
| 100 | ±9.8 points |
| 200 | ±6.9 points |
| 400 | ±4.9 points |
| 1,000 | ±3.1 points |
| 2,000 | ±2.2 points |
Notice the flattening: the jump from 100 to 400 responses cuts the margin in half, but the jump from 1,000 to 2,000 shaves off less than a point. For most business surveys, somewhere between 400 and 1,000 responses is the sweet spot where precision is good and effort is still reasonable. If you're tracking a score like NPS, our NPS calculator pairs well with this one — a ±7-point margin on a 20-point NPS swing changes the story completely.
What margin of error doesn't cover
The margin of error only accounts for sampling error — the random noise that comes from asking a subset instead of everyone. It says nothing about the other ways a survey can mislead:
- Unrepresentative samples. If only your happiest customers reply, a thousand responses just measure the wrong group more precisely. Who answers matters as much as how many.
- Leading or confusing questions. "How much do you love our new design?" biases every response, and no sample size fixes that. Careful wording — see our guide on how to create a survey from scratch — matters more than the math.
- Comparing subgroups. Your overall margin applies to the full sample. Slice it into segments of 50 people each, and each segment carries a much larger margin of its own.
And if you're comparing two results — this quarter versus last, or variant A versus variant B — overlapping margins don't settle it. Use our statistical significance calculator to test whether the difference is real.