Enter your sample statistics and confidence level to get the interval and margin of error.
Alongside the mean-based interval above, this page also handles proportions — a poll, a conversion rate, a defect rate — with its own default: 60% measured across a sample of 30, at 95% confidence. The formula swaps in proportion-specific variance, p(1 − p), in place of a standard deviation: margin = z × √(p(1 − p) ÷ n). That comes to a margin of ±17.53 percentage points, an interval of 42.47% to 77.53%.
This page's own mean-based default (mean 100, standard deviation 15, n = 30) produces a tight ±5.37 margin — about 5% of the mean. The proportion default, from the identical n = 30, produces a margin more than three times as wide in relative terms. Small samples are simply worse at pinning down a proportion than a mean tends to suggest, which is exactly why a "60% of 30 people surveyed said yes" headline carries far less certainty than the clean 60% figure implies on its own.
Proportion variance, p(1 − p), peaks at p = 0.5 and shrinks toward either extreme — a proportion near 0% or 100% produces a narrower interval than one near 50%, for the exact same sample size. That is why opinion polls near an even split are the hardest to call precisely, and why a lopsided result (5% or 95%) can be stated with more confidence from a smaller sample than a genuine toss-up ever can. For the raw sample statistics — mean, standard deviation, variance — behind the mean-based mode above, see the Statistics Calculator, and for the underlying probability arithmetic a survey response rate is itself an estimate of, the Probability Calculator.
A confidence interval gives a range that's likely to contain the true population parameter at the stated confidence level. A common misconception is that a 95% confidence interval means "there's a 95% chance the true value is in this specific interval" — technically, the 95% describes the long-run reliability of the method (if you repeated the sampling many times, about 95% of the resulting intervals would contain the true value), not a probability statement about this one result.
Larger samples produce narrower, more precise intervals. The margin of error shrinks in proportion to the square root of the sample size, so quadrupling your sample size only halves the margin of error.
Higher confidence requires a wider interval, all else equal, because capturing the true value more reliably requires casting a wider net — a 99% confidence interval is always wider than a 95% one built from the same data.
The Sample Size Calculator works backward from a target margin of error to tell you how large a sample you need. This tool works forward from a sample you've already collected to compute the resulting confidence interval.
Worked example: a sample mean of 100, standard deviation of 15, n=30, at 95% confidence (z=1.96): margin of error = 1.96×(15/√30) ≈ 5.37, giving a confidence interval of roughly 94.63 – 105.37.
It does not mean there is a 95% probability the true value sits in your interval. The true value is fixed; your interval is what varies. The correct reading is procedural: if you repeated this sampling process many times, about 95% of the intervals you produced would contain the true population value.
In practice this distinction matters most when an interval is wide. A wide interval is not a weaker claim about the same estimate — it is a statement that your data barely constrains the answer.
With a sample of 100, a sample mean of 50 and a standard deviation of 10, the standard error is 10/√100 = 1. At 95% confidence the critical z value is 1.96, so the margin of error is 1.96 and the interval is [48.04, 51.96].
Note what drives that width: the standard error divides by the square root of the sample size. Quadrupling your sample halves the margin of error — it does not quarter it. Precision gets expensive quickly, which is the single most useful thing to know when planning a study.
Higher confidence always means a wider interval. There is no setting that gives you both precision and certainty; you are choosing where to sit on that trade.
The z-based interval assumes a roughly normal sampling distribution — safe for large samples by the central limit theorem, questionable below about 30, where the t-distribution is the correct choice. It also assumes genuine random sampling. No confidence interval fixes a biased sample; it will report a precise-looking range around the wrong number.
To work out the sample size needed for a target margin of error, use the Sample Size Calculator.