Every confidence interval for a mean has the same three-part shape: x̄ ± (multiplier × standard error). The only real question is which multiplier you have earned. If σ is genuinely known, the standard error involves no guessing and the multiplier is z*. If σ is unknown — which is nearly always — you must estimate it with s from the very same sample, and that extra guess has to be paid for with a wider multiplier t* on n − 1 degrees of freedom. Draw real samples below, build both intervals side by side, then run thousands of studies and see which one actually keeps its promise.
This sample → this interval
the n values you drewx̄ — point estimateμ — the true valuezoom window
Where the multiplier is read off
The multiplier is never a style choice — it is the reach, in standard errors, that encloses the middle C% of a curve. Which curve depends on what you had to estimate. Knowing σ buys you the normal curve; estimating it with s costs you the t curve, whose heavier tails push the same middle percentage further out.
How fast t* settles down toward z*
The t penalty is not a fixed surcharge — it is a small-sample tax that fades as your estimate of σ gets trustworthy. Every extra observation sharpens s, so the curve falls toward the z* floor and never quite touches it.
Resting heart ratepress “Draw a new sample” to build an interval
What do you actually know about the spread?
What you are measuring
12
95%
Estimate x̄—
Spread used s—
Std. error s/√n—
Multiplier t*—df = 11
The interval, built from this sample
—
resulting interval—
Which multiplier keeps its promise?
Talk is cheap, so let the machine settle it. Below, the whole study is repeated 2,000 times at your current settings, and three different intervals are built from each and every sample. All that is counted is how often each one actually contained μ — and how wide it had to be to manage that.
The recipe, and the fine print
x̄ ± (multiplier × standard error) — then answer one question to fill it in. Is σ known? If yes, the standard error is σ/√n and the multiplier is z*. If no — the usual answer — the standard error is s/√n and the multiplier is t* with df = n − 1. That is the entire difference between the two cases. Everything else about the arithmetic is identical, which is why it is worth being precise about the one thing that changes: the √n underneath is the same either way, so a wider t interval is never punishment for a noisier sample — it is the price of having estimated the spread instead of knowing it.
Six things to keep straight.(1) The confidence level describes the procedure, not this interval. Once you have computed two actual numbers, μ is either between them or it is not; no chance remains. The 95% is the long-run hit rate of the recipe, which is precisely what the 2,000-study panel above is measuring. (2) It is an interval for μ, not for individuals. Compare the two panels in the first chart: the interval is far narrower than the data it came from. It is not the range where 95% of values fall. (3) df = n − 1, not n. One degree of freedom is spent the moment you use x̄ to compute s — the deviations from x̄ must sum to zero, so only n − 1 of them are free to vary. (4) t requires a roughly normal population when n is small. With σ unknown the t interval is exact only if the underlying values are normal. Try the strongly right-skewed setting at n = 5 and watch even the correct t interval fall short of its promise; push n up and watch the Central Limit Theorem gradually repair it. (5) Never use z* with s. It is the single most common mistake in this chapter and the coverage panel shows exactly what it costs: an interval that looks satisfyingly narrow and quietly misses μ more often than you promised. The narrowness is the tell, not the reward. (6) None of this covers bias. Every formula here is built out of random sampling wobble alone. A convenience sample, a mis-calibrated instrument, or a leading question produces a tight interval confidently centered on the wrong number — and collecting more data only tightens it further.