Calculate confidence intervals for a mean or a proportion — choose your confidence level (90%, 95%, 99% or custom), and see the margin of error and lower and upper bounds.
Enter your values on the left, then press Calculate.
25 patient wait times, mean 18.4 min, SD 4.2 min. Enter the lab and build the 95% confidence interval for the true average.
Use the calculator above to compute confidence intervals for a sample mean or proportion at 90%, 95%, 99% or any custom confidence level. The interval quantifies the precision of a sample estimate — and turns a single number into an honest range that reflects sampling uncertainty.
A confidence interval (CI) is a range of values that, given the sampling method, is expected to contain the true population parameter at a stated level of confidence. A 95% CI means that if you repeated the sampling many times, 95% of those intervals would contain the true value — not that there is a 95% chance the true value is inside this particular one.
For a mean: CI = x̄ ± z × (s ÷ √n), where x̄ is the sample mean, s the standard deviation, n the sample size, and z the critical value from the normal distribution at the chosen confidence level (1.96 for 95%). For a proportion: CI = p̂ ± z × √(p̂(1−p̂)/n).
A survey of 400 customers gives a satisfaction score with mean 7.8 and standard deviation 1.5. At 95% confidence: CI = 7.8 ± 1.96 × (1.5 ÷ √400) = 7.8 ± 0.147. The true mean satisfaction is between 7.65 and 7.95 with 95% confidence.
If you doubled the sample to 800, the interval shrinks to 7.8 ± 0.104 — a 30% improvement in precision for double the sampling cost. CIs make the precision-vs-cost trade-off explicit before committing to a sample size.
Confidence intervals replace point estimates with honest ranges. They protect against over-claiming precision and force decisions to be made against a range of plausible outcomes rather than a single number that may be coincidence.
Use CIs every time you report a sample mean, sample proportion or process capability index. Also use them to test whether a process change has produced a statistically meaningful improvement.
CIs sit alongside hypothesis testing and DOE in the inferential statistics toolkit. CIs quantify precision; hypothesis tests answer yes/no questions; DOE optimises across multiple inputs.
Use the CI width to decide whether your sample size is adequate. If the interval is wider than the difference you need to detect, increase the sample. Re-run after any process change to see if the new CI overlaps the old one.
Pair CIs with hypothesis testing and sample size calculation to make the inferential workflow watertight.
A range of values that, given the sampling method, is expected to contain the true population parameter at the chosen confidence level (commonly 95%).
If you repeated the same sampling process many times, 95% of the intervals you produce would contain the true value. It is not a probability statement about this particular interval.
95% is the default. 90% gives narrower intervals (less rigour) and 99% gives wider intervals (more rigour). The right choice depends on the cost of being wrong.
Increase the sample size — width shrinks proportional to 1/√n. Reducing variability or accepting lower confidence also narrows the interval.
When the sample size is small (under 30) or the population standard deviation is unknown — which is most real cases. The t-distribution has heavier tails, giving slightly wider, more honest intervals.
Want to understand how confidence intervals inform decision-making in improvement projects? The Green Belt covers this in full.
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