Confidence Interval Calculator

What "95% confident" actually claims

It is a statement about the method, not about this one interval. If you repeated the whole sampling process many times and built an interval each time, about 95% of those intervals would contain the true value. The interval you happen to have either contains it or it does not; you simply cannot know which.

This is why "there is a 95% chance the true value is in this range" is technically wrong, and why the correct phrasing sounds so much more awkward. In practice the interval is still the right thing to report: it shows both the estimate and how much the data can bear.

Why the margin of error shrinks so slowly

The margin of error is proportional to 1 / √n. To halve it you need four times the data; to cut it to a tenth you need a hundred times. This is the single most useful fact in study design and the reason national polls settle around 1,000 respondents — roughly ±3 points. Going to ±1.5 points would need 4,000 people, and the extra cost rarely changes any decision.

t or z, and when the difference bites

If the standard deviation came from your own sample — which it almost always does — the correct critical value comes from the t-distribution with n − 1 degrees of freedom. At n = 10 the 95% critical value is 2.26 rather than 1.96, making the interval 15% wider. At n = 100 it is 1.98 and the distinction stops mattering. Using z on a small sample produces an interval that is too narrow, which is to say too flattering.

Proportions and the finite population correction

For proportions the standard error is √(p(1 − p) / n), which is largest at p = 0.5 — a 50/50 split is the hardest thing to pin down. If your sample is a large fraction of a small population, the finite population correction shrinks the interval: surveying 400 of 500 employees is far more informative than surveying 400 out of a million.

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