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Confidence interval calculator

Confidence interval for a mean (Student’s t or normal) and for a proportion (Wilson, with Wald for comparison).

Parameters

%95 by convention; 90 for a narrower interval, 99 for more certainty.
Note

Wilson interval; Wald’s for comparison.

Result

Confidence interval

[0.17205; 0.5159]

Calculation
Wilson: centre = (p̂ + z²/2n)/(1 + z²/n), half-width = z·√(p̂(1−p̂)/n + z²/4n²)/(1 + z²/n)
Margin of error
± 0.17192
Critical value
1.96
Observed proportion p̂
0.32
Wald interval (for comparison)
[0.13714; 0.50286]
Observed counts
8 successes, 17 failures
0.20.4proportionWilsonWald

Notes (2)

The true proportion is estimated to lie in [0.17205; 0.5159] (Wilson interval).

Over many samples, 95% of the intervals built this way would contain the true value.

Calculation, assumptions and sources

OverviewConfidence interval for a mean, with the population standard deviation unknown (Student’s t) or known (normal), and for a proportion.

Confidence interval for a mean, with the population standard deviation unknown (Student’s t) or known (normal), and for a proportion. For the proportion, the Wilson interval is used. The Wald interval, shown for comparison, can leave [0; 1] or collapse to a point when no success is observed; this case is flagged.

Method & formulasmean, σ unknown: x̄ ± t(1−α/2, n−1) · s/√n
mean, σ unknown: x̄ ± t(1−α/2, n−1) · s/√nmean, σ known: x̄ ± z(1−α/2) · σ/√nproportion (Wilson): (p̂ + z²/2n)/(1 + z²/n) ± z·√(p̂(1−p̂)/n + z²/4n²)/(1 + z²/n)proportion (Wald, for comparison): p̂ ± z·√(p̂(1−p̂)/n)

The interval is the estimate plus or minus a critical value times the standard error; only the distribution providing the critical value changes between cases. Estimating the standard deviation from the sample adds uncertainty, which Student’s t accounts for: at n = 5 its critical value is 2.78 against 1.96 for the normal, a 42% wider interval. The centre of the Wilson interval is shifted from p̂ toward 1/2, which keeps it inside [0; 1].

Confidence level
the proportion of intervals that would contain the true value if sampling were repeated many times.
Margin of error
half the width of the interval; it shrinks as 1/√n.
Wilson interval
an interval for a proportion (Wilson, 1927) whose actual coverage stays close to the stated level even for p near 0 or 1, unlike Wald’s.
Validity domainThe interval on a mean assumes random sampling and, for small samples, an approximately normal population; normality is not tested.

The interval on a mean assumes random sampling and, for small samples, an approximately normal population; normality is not tested. On a large sample the central limit theorem makes the assumption mild, but a sampling bias remains whatever the size. No finite-population correction is applied (it matters beyond ~5% of the population).

Interpreting the confidence levelThe true value is fixed; it is the interval that varies from sample to sample.

The true value is fixed; it is the interval that varies from sample to sample. Reading it as “a 95% chance that the true value lies in the interval” is wrong in frequentist statistics. That statement belongs to the Bayesian credible interval, which is not computed here.