Statistics
Confidence Interval Calculator
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How to use it
Using the confidence interval calculator
- 01
Enter your sample statistics
The mean, standard deviation and size from your collected data.
- 02
Pick a confidence level
95% is the scientific default; 99% widens the net, 90% narrows it.
- 03
Read the range
The primary output brackets where the true population mean plausibly lies; the margin-of-error row gives the half-width.
Good to know
What “95% confident” really means
It describes the procedure, not any single interval: if you repeated the sampling many times, about 95% of the intervals built this way would capture the true mean. Your particular interval either contains it or does not; the confidence level quantifies long-run reliability.
The levers you control
- Quadrupling n halves the margin of error.
- Raising confidence from 95% to 99% widens the interval by ~31%.
- Lower data variability (smaller s) narrows the interval directly.
How it's calculated
The math behind this calculator
CI = x̄ ± z·(s / √n)The interval centers on your sample mean and extends z standard errors to each side, where the standard error is the sample SD divided by √n. Wider intervals mean less certainty; larger samples shrink the margin proportionally to √n.
z multipliers are 1.6449 (90%), 1.9600 (95%) and 2.5758 (99%). The normal approximation is appropriate when n ≥ 30 or the population SD is genuinely known; smaller samples should use a t-distribution instead.
Assumptions & limitations
- Large-sample normal approximation (n ≥ 30 or known σ).
- Observations are independent and randomly sampled.
- For small samples, a t-based interval would be wider.
Worked example
A sample of 900 measurements with mean 100 and SD 15 gives a 95% CI of [99.02, 100.98]; a margin of error of ±0.98 around the mean.
FAQ
Frequently asked questions
- Should I use z or t here?
- This tool uses z, appropriate for n ≥ 30 or known population σ. For small samples with estimated SD, the analogous t-interval will be slightly wider.
- Why did my margin of error come out small?
- Margins shrink with √n; big surveys produce tight intervals. Check that n reflects completed responses, not invitations sent.
- Can the interval contain impossible values?
- Yes, mechanically; e.g. a negative lower bound for a weight. That signals the model (often normality or sample size) fits poorly rather than a calculator bug.
- Does higher confidence mean a better estimate?
- Not inherently. You trade width for confidence: 99% intervals are safer but vaguer. Report the level you commit to before seeing results.
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