Confidence Interval vs Margin of Error

A confidence interval turns a single estimate into a range of plausible parameter values by attaching an explicit allowance for sampling error. Every interval in AP Statistics has the same shape:

The product of the last two pieces is the margin of error, and the standard error estimates how much the statistic varies from one sample to the next.

Say 520 of 1000 randomly sampled voters back a measure. The estimate is $\hat{p} = 0.52$ (p hat), the standard error is $\sqrt{\frac{0.52(0.48)}{1000}} = 0.0158$, and a 95 percent interval uses the critical value $z^* = 1.960$ (z star). The margin of error is $1.960 \times 0.0158 = 0.031$, so the interval runs from 0.489 to 0.551, or 48.9 percent to 55.1 percent.

The interpretation that loses marks is what the 95 percent attaches to. There is not a 95 percent chance that the true proportion lies between 0.489 and 0.551. The true proportion is a fixed number: it is either inside those endpoints or it is not, and no probability is left over once the sample is drawn. What varies from sample to sample is the interval itself, since a fresh sample of 1000 voters produces different endpoints. The 95 percent describes the method, which captures the parameter in about 95 percent of all possible samples. The interval also says nothing about where 95 percent of individual voters sit.

Width is the other half of the idea. The margin of error shrinks like $1/\sqrt{n}$, so quadrupling the sample from 1000 to 4000 only halves it, from 0.0310 to 0.0155. Raising confidence widens the interval instead ($z^* = 1.645$ at 90 percent, $2.576$ at 99 percent), so precision and confidence trade against each other at any fixed sample size.

One reading comes free. This interval contains 0.50, so the data do not rule out an even split, which is the same verdict a two-sided test of $H_0: p = 0.5$ at $\alpha = 0.05$ returns on these counts.

The margin of error is a product of two pieces. The critical value comes from the confidence level; the standard error comes from the data and the sample size. Multiply them and you have the reach on each side of the estimate, so the interval is $\text{estimate} \pm \text{margin of error}$ and the margin is exactly half the interval's width. For a proportion it is $z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

A poll of 1000 randomly selected adults finds 47% support. The standard error is $\sqrt{\frac{0.47(0.53)}{1000}} = 0.01578$ and the 95% critical value is $z^* = 1.960$, so the margin of error is $1.960 \times 0.01578 = 0.0309$, about 3.1 percentage points, and the reported interval runs from 43.9% to 50.1%. That is where a poll's plus or minus 3 points comes from.

"The margin of error tells you how far off the poll could be" is the reading to kill. It measures one source of error only: the variability from surveying a random sample rather than everybody. Nonresponse, a sampling frame that misses part of the population, leading question wording, and respondents who change their minds all sit outside it, and any of them can move a result by more than 3 points. A survey of self-selected volunteers has a perfectly computable margin of error and no useful accuracy, which is why undercoverage and nonresponse bias are named separately.

Two smaller slips are common. A margin of 3.1 percentage points is not 3.1 percent of 47%, which would be 1.5 points. And the margin is not a wall: a 95% method is built to miss about 1 time in 20, so a true value outside 43.9% to 50.1% is not evidence the poll was run badly.

Shrinking it is expensive. The margin falls with the square root of the sample size, so quadrupling a poll from 1000 respondents to 4000 takes 3.1 percentage points down to 1.5, not to 0.8. Raising the confidence level pushes it back up. To hit a target margin you solve the formula for $n$, which is a sample size calculation.

The most-watched monthly figure that carries one of these is the US unemployment rate, which is estimated from a household survey rather than counted from everyone: the unemployment rate.