Every confidence interval is estimate ± margin of error, and the margin has exactly two dials on it: how much data you collected and how confident you promised to be. They are not the same kind of dial. One of them costs money and genuinely sharpens the estimate; the other is free, changes nothing about your data, and only stretches the reach. Drag both below and watch a real interval — built from real simulated observations — widen and narrow against a fixed axis.
The interval, right now
the n observations in handx̄μ — the true valuesame procedure, different n
Margin of error against sample size
Systolic blood pressuredrag the two sliders below
What you are measuring — σ known from long experience
Dial 1 — confidence level, which sets z*
95%
Dial 2 — sample size, which sets the standard error
40
Multiplier z*1.960
Std. error σ/√n—
Margin of error—
Total width—
The square-root law
n sits under a square root, so data buys precision at a punishing exchange rate. Every row below is the margin of error you would get from the same population with a different amount of data, at the current confidence level.
Planning n before you collect anything
Read the formula backwards. If you know σ, pick the confidence level and the margin you are willing to live with, and n is forced: solve z*σ/√n = E for n. This is how sample sizes are actually chosen — before a single observation exists.
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Confidence sets the reach. n sets the ruler. Only n makes the estimate better. Sliding the confidence dial up does not gather one extra observation, does not move x̄ a hair, and does not make you any closer to μ — it simply widens the claim until it is safe enough to survive more often. Sliding n up does something categorically different: it shrinks the standard error, so x̄ itself lands nearer μ on average and the interval around it tightens. That is why "we used a 99% interval" is not a boast, while "we sampled 900 people" is.
Six things to keep straight about these two dials.(1) A wider interval is not a more accurate one. Raising confidence from 95% to 99% leaves your data untouched; it only stretches the reach by 31.4%. You have not learned more, you have claimed less. (2) Raising n does not raise your confidence level. The level is whatever you chose — 95% stays 95% at n = 10 and at n = 10,000. More data narrows the interval; it does not change the promise attached to it. (3) The margin shrinks like 1/√n, not 1/n. Doubling your data buys only about 29% off the margin. Halving the margin costs four times the data; cutting it to a tenth costs a hundred times the data. This is the single most expensive fact in study design. (4) Always round n up. n = 96.04 means 97, not 96 — 96 would leave the margin fractionally above the target you promised, and you cannot sample 0.04 of a person. (5) Neither dial touches bias. The margin of error is built out of σ/√n and nothing else, so it describes random sampling wobble only. A survey of whoever answers the phone gets tighter around the wrong number as n grows, and reports a reassuringly small margin while doing it. (6) 100% confidence is unavailable. The normal curve never reaches zero, so enclosing all the area needs an infinite reach, and "μ is somewhere between −∞ and ∞" is certainly true and completely useless. One honest caveat about this page: σ is treated as known, which is what makes z* the right multiplier and makes the planning formula solvable in one step. When σ must be estimated from the sample as s, the multiplier comes from the t distribution and depends on n too — so both dials move at once, and planning becomes an iteration rather than a formula.