A national poll asks about a thousand people and claims to speak for hundreds of millions. That works for the same reason a cook can taste one spoonful of a well-stirred pot: randomness does the stirring. The math says how far a poll can drift by luck alone. It says nothing about a pot that was never stirred.
Each dot is a person in a made-up population. Green dots would answer "yes". The true share p is the fraction of green dots. A poll picks n people at random, without picking anyone twice, and reports its sample share: the fraction of them who said yes. Run many polls and stack up their answers. That stack is the sampling distribution, and its width is how much a single poll can miss by chance.
The standard error is the typical distance between a poll's share and the true share. The first factor is the whole story for most polls. The second is the finite-population correction: if you sample a big fraction of the N people, there is less left to be wrong about. When n is a tiny fraction of N it is almost exactly 1.
A spoonful tells you how salty the soup is whether the pot holds one liter or a hundred, as long as it is stirred. Random sampling is the stirring. What matters is how many people you ask, and the error shrinks with the square root of that number: four times the sample for half the error.
The bell shape is no accident. A poll's count is a sum of many small yes-or-no draws, and sums like that pile up into a normal curve. 006 · Probability builds the same curve from coin flips.
A poll reports its share plus or minus a margin of error. The share ± margin is a range called a confidence interval. Here 100 separate polls each draw their own interval. The gold line is the truth, which a real pollster never gets to see. Intervals that fail to touch it turn red. Count them.
How to read this: each vertical bar is one poll. The dot is its result and the bar stretches its ± margin either side. The faint gold band is the truth ± one margin.
p̂ ("p-hat") is the poll's own share; the pollster can't use the true p because it is unknown. z = 1.96 because 95% of a normal curve lies within 1.96 standard errors of its center. p(1−p) is largest at p = 50%, so pollsters plan for that worst case. Round 1,067.1 up and about 1,068 people guarantee ±3%.
The 95% describes the method. If you ran this polling method again and again, about 95 of every 100 intervals it produced would contain the true share. Any single interval either contains the truth or it doesn't. You just can't tell which from inside one poll.
It also covers only one kind of error: the luck of who got picked. The margin assumes every person had the same chance of being asked and of answering. Lab 3 breaks that assumption.
First, a trap with honest polls. In a close race the headline number is the lead, the gap between two candidates, and the lead is shakier than either share. Then the bigger problem: bias, error that pushes every poll the same way. The margin of error never includes it, and asking more people doesn't shrink it. The town below is hypothetical: 40% younger people who mostly support a ballot measure, 60% older people who mostly oppose it, and exactly 50% support overall.
How to read this: the dots are the people contacted. The two rows below are estimates from those who answered: the raw poll, and the same poll weighted so each age group counts as much as it should.
In one poll, every extra vote for A tends to be one fewer for B, so their errors move in opposite directions and the gap swings about twice as far as either share. In a two-way race the margin of the lead is exactly twice the reported margin. With minor candidates it is a little less. A rough rule: a lead smaller than about twice the stated margin is not clear.
Wg is each group's known share of the population, from a census. p̂g is the share saying yes among that group's respondents. Weighting gives every group its true voice no matter how many of them answered. Its margin is wider than the raw one because the scarce group's few answers count for more:
Weighting repairs a skew only along the traits you weight on, and only if the people who answered within each group think like the people who didn't. Self-selection, as in a call-in or website poll, breaks that: the people who answer are the people who care. A leading question changes the answers themselves. No sample size or formula can undo either one.
Weighting by groups is also how a split by subgroup can flip an overall result. D02 · Simpson's paradox follows that thread.