AskANumber.com

Confidence Interval Calculator

The defaults — a sample mean of 50, a standard deviation of 10, and n = 100 — give a 95% interval of 48.04 to 51.96, a margin of error of ±1.96. A confidence interval turns a sample mean into a range that quantifies its uncertainty: mean ± z × (sd ÷ √n), where z is 1.645 for 90%, 1.96 for 95%, and 2.576 for 99% confidence.

95% confidence interval
48.04 – 51.96
50 ± 1.96
Margin of error
1.960
Standard error
1.000
Z
1.960
95% interval: 48.04 – 51.96
Mean ± 1.96; over many samples 95% of intervals built this way contain the true mean.
Inputs
Your sample
Sample mean
Standard deviation
Sample size
Confidence level
Confidence level
What "95% confident" means
Over many samples, 95% of the intervals built this way would contain the true mean — it is not a 95% probability that this one particular interval does.
Two ways to narrow the interval
The width scales with 1/√n, so quadrupling the sample size halves the margin. Lowering the confidence level also tightens the band — at the cost of catching the true mean less often.
Ask a follow-up
Uses your inputs above
48.04 – 51.96 95% confidence interval. Want to try a variation?

The math

Reviewed 2026
Formula
CI = mean ± z · (sd / √n)
Z-interval — for small samples (n < 30) a t-interval is more accurate

Related calculators

P-value
Statistics

Example: how confidence interval is calculated

Step-by-step with default inputs

Suppose you put the default values into Confidence Interval Calculator:

Sample mean
50
Standard deviation
10
Sample size
100
Confidence level
95%

Plug those into the formula CI = mean ± z · (sd / √n) and the result is:

95% confidence interval
48.04 – 51.96

How does the confidence interval calculator work?

Confidence Interval Calculator uses the formula shown in the math card and cites NIST/SEMATECH e-Handbook of Statistical Methods. Inputs are validated for sensible ranges; results are computed client-side for instant feedback and do not leave your browser.

References: NIST/SEMATECH e-Handbook of Statistical Methods.

Last reviewed July 2, 2026 · Editorial policy

Frequently asked questions

What does a 95% confidence interval actually mean?

If you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability the true mean falls in this particular interval — the true mean is fixed; the interval is what varies.

How do I make my confidence interval narrower?

Increase the sample size — the margin of error is z × sd ÷ √n, so quadrupling n halves the width. Lowering the confidence level also narrows it (90% uses z = 1.645 instead of 1.96), but that buys precision by accepting more risk of missing the true mean.

Why is a 99% interval wider than a 95% one?

Higher confidence demands a bigger multiplier: z rises from 1.96 to 2.576, stretching the margin by about 31%. To be more certain the interval captures the true mean, it has to cover more ground — width is the price of confidence.

What doesn't this account for?

Z-interval — for small samples (n < 30) a t-interval is more accurate For a more complete picture, combine with related calculators below.

How accurate is this confidence interval calculator?

The math is deterministic — the same inputs always produce the same output, and the formula is shown above. Accuracy of the answer for your situation depends on how well your inputs match reality and how well the formula models the question.

How do I share my result?

Hit Share at the top of the page. Every input you change is encoded in the URL, so a permalink reproduces exactly what you see. No account needed.