Calculate a confidence interval for a mean or proportion. Enter your sample data and confidence level to get the interval and margin of error.
Pick a scenario, then enter your own sample data.
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A confidence interval is a range of values, built from your sample data, that's likely to contain the true population value — a mean or a proportion. Instead of reporting a single estimate (like an average of 72), a confidence interval reports a range (like 68 to 76) along with how confident you are, usually 95%. This calculator builds the interval for you: choose whether you're estimating a mean or a proportion, enter your sample data and confidence level, and it returns the interval, the margin of error, the standard error, and the critical value (z or t) used. It handles both the z-interval (large samples or a known population standard deviation) and the t-interval (small samples with an unknown standard deviation), so your result is statistically correct either way.
A confidence interval is the point estimate plus or minus a margin of error, where the margin equals a critical value times the standard error. For a mean when the population standard deviation is known, use the z-interval: x̄ ± z·(σ/√n), with z = 1.645 for 90%, 1.96 for 95%, and 2.576 for 99%. When the standard deviation is unknown — the usual case — use the t-interval: x̄ ± t·(s/√n), where t comes from the t-distribution with df = n − 1 degrees of freedom and is slightly larger than z for small samples. For a proportion, use p̂ ± z·√(p̂(1−p̂)/n), where p̂ = x/n. For example, with x̄ = 72, σ = 12, and n = 36 at 95% confidence: the standard error is 12/√36 = 2, the margin of error is 1.96 × 2 = 3.92, and the interval is 68.08 to 75.92. A larger sample or a lower confidence level produces a narrower, more precise interval.
Confidence Interval Formulas
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Build a correct t-interval when your sample is under 30 and σ is unknown.
Put a confidence interval around a conversion rate to judge a result.
Estimate a process mean or defect rate within a confidence range.
Verify a stats problem with the full step-by-step working.
Build an interval for an average or a percentage from your sample data.
Uses the t-distribution for small samples automatically, so the interval is statistically valid.
Returns the interval, margin of error, standard error, and the exact critical value used.
A sensitivity table shows how the interval widens at 90%, 95%, and 99% confidence.
Shows the formula and each step, so it's easy to learn or check homework.
No signup — the standard statistical method, done for you.
A confidence interval is a range of plausible values for an unknown population parameter (a mean or proportion), calculated from sample data. It's reported with a confidence level — usually 95% — that describes how often such intervals capture the true value over many samples.
Take your point estimate and add and subtract a margin of error equal to the critical value times the standard error. For a mean with known σ: x̄ ± 1.96·(σ/√n). For an unknown σ, use the t-value for df = n−1 instead of 1.96. For a proportion: p̂ ± 1.96·√(p̂(1−p̂)/n).
Use z when you're estimating a proportion, or estimating a mean and the population standard deviation is known (or your sample is large). Use the t-distribution when you're estimating a mean and the population standard deviation is unknown — which is the typical case — especially with small samples. This calculator picks the right one based on your inputs.
The margin of error is the ± part of the interval — the critical value multiplied by the standard error. It sets the interval's width: a 60% result with a ±5% margin gives an interval of 55% to 65%. Larger samples shrink the margin of error.
Divide successes by sample size to get p̂, then use p̂ ± z·√(p̂(1−p̂)/n). For example, 520 of 1,000 gives p̂ = 0.52, a standard error of about 0.0158, and a 95% interval of roughly 48.9% to 55.1%.
Larger samples produce narrower (more precise) intervals, because the standard error shrinks with the square root of n. To halve the margin of error you need about four times the sample size. Smaller samples give wider intervals and also use larger t-values.
It means that if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true value. It does not mean there's a 95% probability that this specific interval contains it — the true value is fixed, the interval is what varies.
The confidence level (e.g. 95%) is the long-run success rate of the method; the confidence interval is the actual range you get for your data (e.g. 68 to 76). A higher confidence level produces a wider interval for the same sample.