Confidence Interval Calculator
The margin of error made honest: x̄ ± z*·σ/√n with z* derived at runtime by bisection of the normal curve — no tabulated constant anywhere — and the t-based boundary printed where the z method ends.
Confidence Interval Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Z-based intervals only — known σ required; t-intervals refused with the boundary stated
- Normal-model coverage; tiny skewed samples violate it
In short: A sample of 16 gives a mean of 52 with a known population σ of 5. For a 95% interval the page bisects the normal curve for the z* that leaves 2.5% in each tail — z* = 1.959964 — and the margin is 1.959964 × 5/√16 = 2.449955, giving the interval (49.550045, 54.449955). The quantile is not a tabulated constant: the page computes the error function from its Taylor series and bisects it live, so the interval inherits the precision of the arithmetic rather than the memory of a table.
Formula
CI = x̄ ± z* · σ/√n
z* solves P(Z < z*) = 1 − (1 − level)/2, found by bisection
the margin is the critical value times the standard error — and every symbol is either typed by you or computed in front of you.
Worked Example
- Standard error. σ/√n — the sample mean’s own uncertainty, shrinking by the square root of n.
- Critical value. the page bisects the erf-series curve for the z* matching your level: 1.644854 at 90%, 1.959964 at 95%, 2.575829 at 99% — computed, not recited.
- The interval. x̄ ± the margin, with the interpretation said carefully: the METHOD captures the true mean 95% of the time, not this interval a probability.
x̄ = 52, σ = 5, n = 16, 95%: margin 2.449955, interval (49.550045, 54.449955). At n = 100 the same data gives margin 0.979982. At 99% the margin widens to 3.219787 — confidence buys width.
Strengths & Limits Of This Model
Where this engine is strong
- z* computed live by bisection — six honest decimals
- The interpretation card says what 95% actually means
Where it stops
- No t-based or bootstrap intervals
- No proportion (p̂) intervals
Practical Use Cases
Survey reporting
the ±2.1% beside every poll headline
Lab reporting
measurement intervals at stated coverage
Coursework
z* shown to six decimals, derived live
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Confidence Interval Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What does “95% confident” actually mean?
That the METHOD — draw a sample, build this interval — captures the true mean in 95% of samples. The particular interval on the page either contains the true mean or does not; the probability lives in the procedure, not the endpoint. Saying “there is a 95% chance the mean is in THIS interval” is the most common statistics sentence that is subtly wrong.
Why is z* computed instead of taken from a table?
Because it can be. The page computes the normal curve from the erf Taylor series — machine precision — and bisects it for the quantile your level needs. Six decimals of z* fall out of the arithmetic; a tabulated 1.96 would put a rounding wedge under every margin built on it.
Why does the interval need the population σ?
Because z intervals assume the standardiser is exact. When σ is estimated from the sample, the honest critical value comes from the t distribution — and this page refuses that case rather than pretending z covers it. The refusal is the honesty: the t quantile machinery is not implemented here, and an approximate coverage claim is worse than a stated boundary.
Why does the margin shrink by √n, not n?
Because the standard error of a mean is σ/√n: averaging cancels noise only at the square-root rate. Four times the data buys half the margin — the diminishing returns every survey budget knows.
What does the confidence level trade?
Width. Higher confidence pulls a wider net from the same data: at 99% the default margin grows from 2.45 to 3.22. Precision and confidence are a budget — the only way to raise one without spending the other is more data.
Does this interval assume anything about the data?
Yes — that the sample mean is approximately normal, which the central limit theorem supplies for large n and gentle populations. Very small samples from skewed populations violate it, and the page’s n = 4 example exists to show the shape of the risk, not to endorse it.
What if I only have the sample standard deviation?
Then the z interval is the wrong tool and the page refuses it. The t-based interval needs the t quantile machinery this site does not implement — carry x̄, s and n to your statistics package, or quote the t statistic page’s exact figure alongside a printed table.
Why does the page print z* to six decimals?
Because it can compute them honestly — and because the six-decimal z* is the guard against the 1.96 shortcut compounding through every downstream margin.