Interactive guide
95% confident isn't about this one interval
The most common misreading of a confidence interval is "there's a 95% chance the true mean is in this range." That's not it. The true mean is a fixed number, not a random one, it either sits inside a given interval or it doesn't. What's actually true 95% of the time is the procedure: draw a sample, build an interval this same way, and about 95% of the intervals you'd ever build will contain it. Below, build a lot of them and watch that play out for real, on a population from the Central Limit Theorem page.
One sample, one interval, repeated
Every interval here is built the same way: draw n values, take their mean x̄, then go out z × σ/√n in each direction. σ/√n is the standard error, exactly the spread the Central Limit Theorem page derived for the sampling distribution of the mean; z just sets how many standard errors wide the net needs to be to catch a chosen share of that distribution.
Because x̄ itself is random, jumping around from sample to sample, the interval built around it moves too, sometimes landing on the true mean, sometimes just missing it. A 95% confidence level means the net is wide enough that, over many repeats, roughly 95% of those moving intervals land on target. It says nothing about whether any one interval, the one you happen to be holding, is one of the hits.
Building one interval
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1
Draw a sample of size n
Independently, from the chosen population, exactly like the CLT page.
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2
Compute the sample mean x̄
The center of the interval, and the one number that varies from draw to draw.
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3
Go out z × standard error
z is bigger for a higher confidence level, a wider net catches more, at the cost of being less precise.
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4
Check it against the true mean, then repeat
In real life you never get this check. Here, because the population is synthetic, we do.
Each row is one sample's interval. Green touches the true mean, red misses it.
- True mean μ
- —
- Hit rate so far
- —
Population
Confidence level
Sample size
—
- Trials so far
- 0
- Nominal confidence
- 95%
Hit rate, converging to the nominal line
Try the bimodal population at n=2: the empirical hit rate for a "95%" interval isn't close to 95% at all here, small samples from a two-humped population badly break the normal-shaped assumption the whole z × SE formula leans on. Push n up toward 30 and the same population's hit rate settles back in near the nominal line, exactly the Central Limit Theorem's own guarantee arriving late but on schedule.
The jargons
Every term here describes some part of the same repeated procedure: sample, estimate, bound, repeat.
The long-run share of intervals, built this same way, that actually contain the true parameter. Not the probability for any single interval.
How far the interval extends on each side of the sample mean. Wider for higher confidence, narrower for larger n.
The spread of the sampling distribution of the mean, the exact quantity the CLT page derives and this page's margin of error is built from.
The actual, empirical share of intervals that contain the true mean. Should match the nominal confidence level, but can drift when the sampling distribution isn't yet normal-shaped.
How many standard errors wide the net needs to be. 1.96 for 95%, because 95% of a normal distribution sits within ±1.96 standard deviations of its mean.
"95% chance the true mean is in this interval" treats a fixed number as random. The randomness is in which interval you happened to draw, not in where the true mean sits.