A sample hands you one number — a point estimate like x̄ — and that number is almost certainly not μ. So you don't report it bare. You report it with a margin of error: a reach, built out of the standard error, wide enough that intervals made this way capture μ as often as you promised. Draw real samples below, watch each interval get built out of its own data, and then watch a few hundred of them at once to see exactly what that margin does — and does not — cover.
One sample → one interval
the n values you drewx̄ — point estimateμ — the true valuezoom window
The same procedure, run again and again
Resting heart ratepress “Draw a new sample” to build an interval
What you are measuring — σ known from long experience
Confidence level — sets the multiplier z*
95%
25
Point estimate x̄—
Multiplier z*1.960
Std. error σ/√n1.800
Margin of error3.528
Where z* comes from
The multiplier is not a style choice. It is read off the standard normal curve: z* is the number of standard errors you must reach in each direction to enclose the middle C% of the area. Move the confidence slider and watch the shaded middle grow.
What extra confidence costs
Confidence and precision are traded against each other at a fixed exchange rate. Every bar below is a margin of error at the current n — same data, same standard error, only the promise changed.
estimate ± (multiplier × standard error). That is the shape of almost every confidence interval you will ever build — for a mean, for a proportion, for a difference between two groups, for a slope. Only the three ingredients get swapped out. The estimate comes from your data. The standard error says how much an estimate like yours typically wobbles from sample to sample. The multiplier says how many of those wobbles you want to reach out, and it is set by the confidence level alone. Learn the shape once and every interval in the rest of the book is a substitution exercise.
Four things the margin of error does not cover.(1) It is not a probability statement about this interval. Once the numbers are computed, μ is either inside or it is not — there is no chance left in it. The 95% belongs to the procedure: repeat the whole study endlessly and 95% of the intervals it produces will contain μ. That is precisely what the stack of intervals above is showing you. (2) It says nothing about individual values. Look at the two panels in the first chart: the interval is a claim about where μ sits, and it is far narrower than the data. It is not the range where 95% of hearts, commutes or bolts fall. (3) It does not cover bias. The margin of error is built entirely out of σ/√n, which describes random sampling wobble and nothing else. Volunteers only, one neighborhood, whoever picks up the phone, a scale reading 2 g heavy, a leading question — none of that is in the formula. A biased study just produces a tight interval centered on the wrong number, and raising n tightens it further. (4) It assumes the sample was actually random. Break that and the arithmetic still runs, still prints an answer, and still means nothing. One honest caveat on this page: σ is treated as known, which is why z* is the right multiplier here. When you must estimate σ from the sample itself, the multiplier comes from the t distribution instead — a slightly longer reach to pay for the extra thing being guessed.