Confidence Interval Calculator - Mean and Proportion

A confidence interval calculator takes a point estimate - a sample mean x̄ or a sample proportion p̂ - and attaches a margin of error so you can report a range instead of a lonely number. The range is not a probability statement about the unknown parameter after you have computed it. The parameter is fixed; the interval would jitter if you drew a new sample. “95% confidence” describes the method’s long-run hit rate, not a 95% chance that this one printed interval contains μ or p.

This page builds two families of intervals. For a mean, the half-width is a critical value times s/√n (or σ/√n). For a proportion, it uses the Wald normal approximation p̂ ± z √(p̂(1−p̂)/n). The default mean case is x̄ = 50, s = 10, n = 40, 95%: with z* ≈ 1.96 the interval is about 46.9 to 53.1. Forcing t with df = 39 makes it a little wider.

Arithmetic stays in the browser. Nothing is uploaded. If you need a sample size that would hit a target margin, use the sample size calculator. If you only have a list of raw values, compute s first with a spread tool, then paste s and n here.

1. Choose mean or proportion, then 90%, 95%, or 99% confidence.
2. For a mean, enter x̄, a positive s or σ, and integer n ≥ 2. Pick auto / z / t. Auto uses z when n ≥ 30 and t otherwise; you can override.
3. For a proportion, enter successes x (0 through n) or p̂ in [0, 1], plus n. If n p̂ or n(1−p̂) is under 5 you still get a Wald interval, plus a warning - not a hard fail.
4. Read lower and upper bounds, standard error, critical value, and margin of error. Copy the block, or clear the fields to start over.

Mean intervals assume the sampling distribution of x̄ is approximately normal. Proportion intervals here are Wald, not Wilson, Agresti–Coull, or exact binomial.

The standard error of the mean is spread over √n. Doubling n does not cut the margin in half; you need four times the sample for that. z* comes from the standard normal: for 95% two-sided coverage, Φ(z*) = 0.975, so z* ≈ 1.96. This page finds that quantile by bisection on an erf-based CDF rather than hard-coding 1.96.

When σ is unknown, replace it with s and replace z* with t* at df = n − 1. t* is defined by P(|T| > t*) = 1 − confidence. The calculator locates t* by bisection on the Student-t CDF, which is implemented with a regularized incomplete beta (Lanczos log-gamma, continued-fraction betacf). For n = 40, df = 39, 95%, t* is a bit above 2.02, so the interval is slightly wider than the z version of the default example.

Known σ and a z interval belong together. Using t with a claimed known σ is conservative; using z with a tiny-sample s is optimistic. Auto mode follows the common n ≥ 30 split so homework matches a textbook’s “large sample, use z” line.

Wald’s interval is the first one most intro-stats courses write down. It can undershoot coverage when p̂ is near 0 or 1 or when n is small, and the bounds can fall outside [0, 1]. This calculator does not clip them and does not switch to Wilson. If you need a better-behaved binomial interval, say so in the write-up and use software that offers Wilson or Clopper–Pearson; this page will not pretend it did.

The np̂ ≥ 5 and n(1−p̂) ≥ 5 check is a teaching rule of thumb, not a theorem. Violating it produces a warning, not a refusal. You can still copy the Wald numbers and discuss why they may be untrustworthy.

p̂ must lie in [0, 1]. Entering 20 successes in 40 trials is the same as typing 0.5. Pair this with a z-score calculator if you want the critical z spelled out as a standard score, or with the p-value calculator if the next step is a test rather than an interval.

The margin of error is the quantity you add and subtract: critical value × standard error. Interval width is twice that. Raising confidence from 95% to 99% inflates the critical value (about 1.96 → 2.58 for z) and therefore the margin. That is the trade: fewer misses, a fatter range.

Does this 95% interval contain the true mean? You cannot know from one sample. In a simulation, about 95 of 100 correctly built 95% intervals would cover; the other five would miss entirely, and those five still look like ordinary numeric ranges. Coverage also collapses if the normal approximation is a lie - skewed data, a mean of a tiny n, or a Wald interval for 2 successes in 80 trials.

For a mixed toolbox of descriptive stats, see the statistics calculator.

It does not build a two-sample interval for μ₁ − μ₂, a paired difference interval, a variance interval, or a bootstrap percentile interval. It does not apply a finite-population correction: if you sampled 40 people from a village of 200, the usual s/√n overstates the SE, and this page will not shrink it by √((N−n)/(N−1)). It does not clip a Wald proportion interval into [0, 1], so a noisy p̂ near 0 can produce a negative lower bound - that is a known Wald pathology, left visible on purpose.

It also does not pick your confidence level for you. 95% is a convention, not a law of nature. Regulatory work sometimes wants 99%; an underpowered pilot might honestly report 90% and a huge margin rather than fake precision. Changing 95% to 99% on the default mean example moves z* from about 1.96 to about 2.58 and the half-width from about 3.1 to about 4.1, so the interval grows from roughly 46.9–53.1 to roughly 45.9–54.1.

Independence of observations is assumed and never checked. Cluster-sampled classrooms, overlapping time series, and “n = 40” that is really 8 people measured five times each all break the SE formula. The calculator cannot see your design; it only sees x̄, s, and n.

Margin of error scales with 1/√n. Keep s = 10 and 95% z. At n = 10, SE = 3.16 and the margin is about 6.2. At n = 40 (the default) the margin is about 3.1. At n = 160 it is about 1.55. Quadrupling n halves the margin; doubling n only cuts it by about 29%. If a protocol demands a margin of 1.0 with s ≈ 10 and z ≈ 1.96, you need n ≈ (1.96×10/1)² ≈ 384, which is a sample-size problem rather than an interval problem.

For proportions the same 1/√n story holds, but the SE also depends on p̂(1−p̂), which peaks at 0.5. A 95% Wald interval for p̂ = 0.5 and n = 40 has margin ≈ 0.155. For p̂ = 0.1 the formula gives a smaller number, yet that is exactly when the np̂ ≥ 5 warning may fire and the Wald coverage may sag. A smaller printed margin is not automatically a better interval.

t versus z also interacts with n. At n = 5, df = 4, 95% t* is near 2.78, far above 1.96, so forcing z on a tiny sample understates the margin. At n = 40 the gap is modest, which is why textbooks shrug and allow z. Auto mode encodes that shrug; Force t refuses to shrug.