Worked Examples
Four questions a statistic has to answer before you trust it — how big the sample was, which average it used, what it left out, and who it asked.
Figures current · August 2026
MoE ≈ 1.96 × √(p(1 − p) ÷ n)95% confidence
A survey asks 1,000 randomly chosen adults whether they have a savings account, and half say yes. The sample size sits under a square root, and that is what decides how close to the truth the 50% is likely to be.
±3.1 ptsmargin of error at n = 1,000n = 4,000 → ±1.5
→Quadrupling the sample only halves the error, because n sits under a square root — precision gets expensive fast.
mean − median = skewsame families
In the Federal Reserve's 2022 Survey of Consumer Finances, the median family had a net worth of $192,900. The mean was $1,063,700. Same families, same data, two very different summaries.
5.5×the mean is 5.5 times the medianmedian: $192,900
→A handful of very large fortunes pull the mean up until it describes almost nobody — for money, the median is the honest statistic.
−1 ≤ r ≤ +1strength, not cause
Imagine plotting a class's streaming subscriptions against their households' credit-card balances. The points climb together and r comes out near 0.93 — and neither one is causing the other.
r ≈ 0.93a very strong correlationcause: neither
→Household income moves both numbers, which makes it the confounding variable — a trend line measures how tightly two things travel together, never why.
error = sampling error + biasonly one shrinks
In 1936 the Literary Digest mailed 10 million straw ballots, got 2.4 million back, and predicted Landon would beat Roosevelt 57% to 43%. Roosevelt won 46 of the 48 states.
2.4Mresponses, and still wrongGallup: ~50,000, and right
→Sampling error shrinks when you add people; bias does not — the Digest's lists of car and telephone owners were a wealthier America than the one that voted.
Sources: Federal Reserve Survey of Consumer Finances, 2022; the 1936 Literary Digest straw poll and election result. Reviewed August 2026.
