The plus-or-minushiding behind your result.
A survey result without its margin of error is half a sentence. Enter how many people you actually heard from and the calculator hands you the ± band around your number — the real range your '52%' lives in, the confidence interval you can put in a deck, and how much tighter it gets if you collect more.
Your numbers
How many people actually completed the survey — the responses you have in hand, not the invites you sent.
How many people are in the full group you're trying to measure. Leave it large if the population is effectively unlimited.
The percentage you got — say 52% chose option A. Unsure? Leave it at 50%, the worst case, which gives the widest (most conservative) margin.
The verdict
Margin of error
4.4%
The ± band from the standard formula, before any population correction.
With FPC applied
4.4%
The margin once the finite-population correction tightens it.
FPC factor
0.99×
The correction multiplier: closer to 1 means the population barely tightens the band.
Effective n
505
Infinite-population sample with the same precision.
Your 50% really means
45.6% – 54.4%
The confidence interval — the range your true value almost certainly sits in at 95% confidence.
How tight is your read?
Sensitivity · square-root law
What more responses actually buy you
| Responses (n) | Margin | Interval at 50% |
|---|---|---|
| 200 | ±6.9% | 43.1–56.9% |
| 500 | ±4.4% | 45.6–54.4% |
| 1,000 | ±3.1% | 46.9–53.1% |
| 2,000 | ±2.1% | 47.9–52.1% |
Same 50% result, same 95% confidence — only the sample size changes. Notice the diminishing returns: the margin shrinks with the square root of n, so quadrupling responses only halves the band. Past about 1,000 responses you're paying a lot to gain very little.
Where the band lands
Your result, with its honest range
You measured 50%, but a sample of 500 only pins it to 45.6% – 54.4% at 95% confidence. If a rival number falls inside that band, you can't honestly claim a difference — it's a statistical tie. This is the inverse of the sample size calculator: there you pick a target margin and solve for n; here you have n and read off the margin.
Your move
You've got the band. I'll make sure the survey was worth running.
50% result on 500 responses → ±4.4% margin (45.6% – 54.4%) at 95% confidence.
A tight margin on the wrong question still tells you nothing. Send me what you're trying to learn and I'll help you ask it cleanly, hit a sample that holds up, and turn the read into a decision — not a slide nobody trusts. Free 30-minute call, yours whether we work together or not.
Plain English
Margin of error is the honesty tax on a sample.
Margin of error is the ± range around a survey result that accounts for the fact you asked a sample, not everyone. When a poll says 52% with a ±4% margin, the honest reading is 'somewhere between 48% and 56%.' Drop the ± and you're quoting a level of precision your sample never bought you.
Three things set the width: how many people you heard from, how confident you want to be, and how the answers split. Sample size is the big one — but its grip is loose. Because the margin shrinks with the square root of n, quadrupling your sample only halves the margin. Going from 500 to 1,000 responses barely moves the needle; going from 100 to 1,000 transforms it.
This is the mirror image of the sample-size question. There you pick a target margin and solve for how many responses you need; here you already have the responses and want to know how precise they actually made you. Plug in your numbers and you'll see the margin, the confidence interval, and — for a small known population — the finite-population correction that quietly tightens the band.
The formula
MoE = z × √( p(1 − p) / n ) × FPC, where FPC = √( (N − n) / (N − 1) )
At 95% confidence z = 1.96. With 500 responses (n) and a 50% result (p = 0.5): MoE = 1.96 × √(0.25 / 500) = 1.96 × 0.02236 = ±4.38%. So a reported 50% really means 45.6% to 54.4%. Against a 50,000-person population the finite-population correction barely budges it (±4.36%); against a 3,000-person list it tightens to about ±4.20%.
What a given sample size buys you
At 95% confidence and a 50/50 split (the widest, most conservative case), here's the ± margin each sample size delivers for a large population. Note how the gains flatten: the jump from 100 to 400 buys you a lot, the jump from 1,000 to 2,000 buys you almost nothing. These are rules of thumb from the standard formula, not a single published dataset.
| Sample size (n) | Margin of error (±) |
|---|---|
| 100 | ±9.8% |
| 200 | ±6.9% |
| 400 | ±4.9% |
| 500 | ±4.4% |
| 1,000 | ±3.1% |
| 1,500 | ±2.5% |
| 2,000 | ±2.2% |
Rule-of-thumb figures from the standard formula at 95% confidence and a 50/50 split for a large population — not a published dataset. A real split away from 50% or a small known population tightens these further.
Your margin came back wide. Now what?
Margin is wider than the gap you're measuring.
If A leads B by 3 points but your margin is ±5, you can't call it. The result is a statistical tie. Either collect more responses or report it honestly as 'too close to call.'
You need the margin roughly halved.
You'll need about 4× the responses, not 2×. The margin shrinks with the square root of n, so precision gets expensive fast. Decide whether the extra certainty is worth the extra collection.
Your result is far from 50%.
Good news: lopsided results carry a smaller margin. A 90/10 split has a noticeably tighter band than 50/50 at the same sample, because p(1 − p) is smaller. Don't quote the worst-case 50% margin if your real result is extreme.
You surveyed most of a small, known list.
Use the finite-population correction. If you heard from 800 of a 1,200-person team, your real margin is far tighter than the textbook 'infinite population' figure — you've measured most of the group.
How to report a result you can stand behind
01Always quote the ±
A number without its margin is a guess wearing a percentage sign. State the result, the margin, and the confidence level together, every time.
02Don't over-collect for false precision
Past ~1,000 responses the margin barely moves for a large population. If ±3% is fine, stop collecting — extra responses are diminishing returns.
03Use the real split, not 50%
50% is the conservative default, but if your actual result is 80/20 the margin is genuinely smaller. Reporting the worst case undersells your precision.
04Mind the overlap before calling a winner
Two results are only meaningfully different when their confidence intervals don't overlap. Eyeball the bands before you declare a lead.
05Segment with care
The margin applies to the whole sample. Slice it into four segments and each sub-result carries a much wider margin — roughly double — because each n is a quarter the size.
06Correct for small populations
When you've surveyed a big share of a known, finite group, the finite-population correction legitimately tightens your margin. Use it instead of pretending the population is infinite.
07Match confidence to the stakes
95% is the default, but a board-level call might warrant 99% (wider margin, more certainty) while a quick internal pulse is fine at 90% (tighter, less sure).
08Remember it only covers sampling error
Margin of error says nothing about a biased sample, a leading question, or non-response. A tiny margin on a skewed sample is still wrong — clean data first.
The vocabulary
- Margin of error (MoE)
- The ± range around a result that accounts for sampling. ±4% on a 50% result means the true value is likely 46% to 54%.
- Confidence interval
- The actual range the result spans: the observed value minus and plus the margin of error (e.g. 46%–54%).
- Confidence level
- How often the true value falls inside the interval if you repeated the survey. 95% is standard; it sets the z-score.
- Z-score
- The standard-normal multiplier tied to confidence: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
- Sample size (n)
- The number of completed responses. The margin shrinks with the square root of n — quadruple n to halve the margin.
- Finite-population correction
- An adjustment that tightens the margin when you've sampled a large share of a small, known population.
Margin of error questions, straight answers
Margin of error = z × √(p(1 − p) / n), where z is the z-score for your confidence level (1.96 for 95%), p is your observed proportion as a decimal, and n is your sample size. For a 50% result on 500 responses at 95% confidence: 1.96 × √(0.25 / 500) = ±4.38%. If you're sampling a small known population, multiply by the finite-population correction to tighten it.
Keep going
A calculator tells you what. A call tells you what to do about it.
Send me the account behind these numbers. I'll tell you straight where the money's leaking and what I'd fix first — free, and you keep it whether you hire me or not.