Quadruple the Sample to Halve the Interval

Where the square root sends the budget

Precision gets expensive faster than intuition expects. Going from a margin of 5.37 to 2.68 costs ninety extra observations; going from 2.68 to 1.34 costs another 360. Each halving multiplies the cost by four, which is why serious studies argue about the width they actually need rather than asking for the narrowest one affordable — and why a survey that doubles its sample and reports a dramatically better result is usually reporting something other than precision. Decide the width the decision requires, then solve for n; the reverse order produces a number nobody can defend.

z assumes a normal sampling distribution, not normal data

The 1.96 comes from the normal distribution, and the assumption it needs is about the distribution of sample means, not about the individual observations. The central limit theorem does most of the work: the mean of a big enough sample is approximately normal even when the raw data is badly skewed, which is why this calculation survives on income, waiting times and click rates. Two situations break it — a small sample from a skewed population, where thirty is nowhere near enough, and data with heavy tails, where a single extreme value moves both the mean and the standard deviation and the interval quietly widens around the wrong centre.

Frequently asked questions

Should I use z or t?

Strictly, t whenever the population standard deviation is unknown and estimated from the sample — which is nearly always. The practical difference shrinks fast with sample size: at n = 30 the t multiplier is 2.045 against z's 1.96, so the margin here would be 5.60 instead of 5.37, about four per cent wider. Below roughly n = 30 the gap matters and t is the right choice; above it, the two answers agree closely enough that the assumptions about your data will fail long before the multiplier does.

Two intervals overlap — does that mean no difference?

No, and this is one of the most common errors in reading charts. Overlapping intervals can still come from a difference that a direct test finds significant, because the right comparison is a confidence interval on the difference between the two means, not a visual inspection of two separate intervals. The reverse holds too: non-overlapping intervals do imply a significant difference, so the rule is only safe in one direction. When the question is whether two groups differ, compute the interval for the difference and read that.