Width of a Confidence Interval vs Margin of Error

The width of a confidence interval is its upper bound minus its lower bound. Every interval on this course has the form statistic plus or minus a margin of error, so the width is exactly $2 \times ME = 2 \times (\text{critical value}) \times (\text{standard error})$. Because the interval is symmetric about the statistic, the width and the margin of error carry the same information in different units, and either one recovers the other.

The interval 12 to 18 has width $18 - 12 = 6$, so its margin of error is $6/2 = 3$ and its center, the point estimate, is $(12 + 18)/2 = 15$. Given only the endpoints you can always rebuild both.

Sample size enters through a square root, since every standard error on the course divides by $\sqrt{n}$. Holding the statistic and the confidence level fixed, the width is proportional to $1/\sqrt{n}$. So "I doubled my sample, so the interval should be half as wide" is wrong. Doubling $n$ multiplies the width by $1/\sqrt{2} = 0.7071$, so a width of 6 becomes 4.243, about 29% narrower, and not the 3 that guess predicts: the real width is 41% larger than the guess. Halving the width takes four times the data, since $1/\sqrt{4} = 0.5$ exactly, which does give 3.

The confidence level is the other multiplier, and it acts on the critical value directly rather than through a square root. At a fixed $n$, moving from 95% to 99% multiplies the width by $2.576/1.960 = 1.314$, about 31% wider, and moving down to 90% multiplies it by $1.645/1.960 = 0.839$. That is the whole trade: confidence is bought with width, and width is bought with data.

One caution about comparing widths across two real samples. The $1/\sqrt{n}$ rule holds with the statistic held fixed, and a fresh sample gives a fresh $\hat{p}$ or $\bar{x}$, so the observed width will only be near the predicted one. And a narrow interval is a claim about precision alone, never about whether the center is in the right place.

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.