Margin of Error Calculator (Mean, z-Interval)
The margin of error for a sample mean from the standard deviation, sample size and confidence level, working shown.
Inputs
- Working
- 1.96 × 10 ÷ √100 = 1.96 × 1 = 1.96
- Standard error
- 1
- Z critical
- 1.96
- Sample size
- 100
- To halve the margin
- n = 400
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What each Margin of Error input means
- Std dev
- The standard deviation of the measurement — the population figure if known, or the sample's for a reasonably large sample.
- Sample size
- The sample size. The margin shrinks with its square root.
- Confidence
- The confidence level: 90%, 95% or 99%, with z = 1.645, 1.96 or 2.576. 95% is what nearly every report means.
Margin of Error formula
margin of error = z × σ ÷ √n · z = 1.645 (90%), 1.96 (95%), 2.576 (99%) · the confidence interval is the sample mean ± this
The margin of error is how far a sample mean can plausibly be from the population mean at a chosen confidence level. It is the standard error, σ ÷ √n, scaled by a z value: 1.96 standard errors for 95%, 1.645 for 90%, 2.576 for 99%. The standard deviation sets the scale of the noise, the sample size shrinks it — by the square root, so four times the data halves the margin — and the confidence level sets how many standard errors count as 'plausibly'. OpenStax calls the same quantity the error bound. Reports that say 'accurate to within ±3 points, 19 times in 20' are quoting a 95% margin of error; the page also shows the sample size that would halve whatever margin it finds.
Margin of Error worked example
A standard deviation of 10 and a sample of 100 at 95% confidence: the standard error is 10 ÷ √100 = 1, and the margin is 1.96 × 1 = ±1.96. A sample mean of 50 would therefore be reported as 50 ± 1.96, or (48.04, 51.96). At 99% the margin is 2.576; with n = 400 it drops to ±0.98; with n = 25 it rises to ±3.92.
More Margin of Error examples
A standard deviation of 12 and 36 observations at 95%: 1.96 × 12 ÷ 6 = ±3.92.
Thirty observations with a standard deviation of 15 at 90%: 1.645 × 15 ÷ √30 = 1.645 × 2.7386 = ±4.505, and 120 observations would halve it.
Where the Margin of Error calculator can give the wrong answer
- This is the margin for a mean with a known or well-estimated standard deviation. A poll percentage uses the proportion formula z × √(p(1 − p) ÷ n), which is where the familiar ±3% for n ≈ 1,000 comes from.
- With a small sample and a standard deviation estimated from that sample, the t multiplier applies and the true margin is larger than the z figure here — about 15% larger at n = 10.
- The margin covers sampling error only. A biased sample, a badly worded question or measurement error can put the truth outside the margin no matter how large n is.
- Halving the margin needs four times the sample, because √n is in the denominator; the page shows that n.
- Only 90%, 95% and 99% are offered — the three z values every text tabulates — and 95% is what almost every published margin means.
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Margin of Error reference table
| Sample size | Mean, standard deviation 10 | Percentage near 50% |
|---|---|---|
| 25 | ±3.92 | ±19.6 points |
| 50 | ±2.77 | ±13.9 points |
| 100 | ±1.96 | ±9.8 points |
| 200 | ±1.39 | ±6.9 points |
| 400 | ±0.98 | ±4.9 points |
| 1,000 | ±0.62 | ±3.1 points |
| 1,067 | ±0.60 | ±3.0 points |
Margin of Error FAQ
- Multiply the z value for your confidence level by the standard deviation and divide by the square root of the sample size. For SD 12, n = 36 and 95%: 1.96 × 12 ÷ 6 = 3.92. Enter the three values and the page shows the multiplication.
- That the sample estimate is expected to be within 3 units of the population value at the stated confidence — usually 95%, meaning the method lands within ±3 in 95% of samples. It does not mean the estimate cannot be further off; one sample in twenty is.
- Collect more data — four times as much for half the margin — or accept a lower confidence level (90% instead of 95% cuts it by 16%). Reducing the variability of the measurement itself also helps and is often cheaper than more samples.
- The standard error is σ ÷ √n, the typical distance between a sample mean and the true mean. The margin of error is that multiplied by z for the confidence level, so at 95% the margin is 1.96 standard errors. The standard-error calculator gives the first on its own.
- Because they ask about 1,000 people: at 95% confidence and a result near 50%, 1.96 × √(0.25 ÷ 1,000) = 3.1 points. Asking 4,000 would only halve it.