Definition: The margin of error is the amount added and subtracted from a sample estimate to form a confidence interval, reflecting sampling variability at a chosen confidence level.
What is the margin of error?
It equals a critical value times the standard error: ME = z* times SE. A bigger sample shrinks the standard error. A higher confidence level makes the critical value larger and widens the margin.
How do you plan a sample size?
For a proportion at 95% confidence, ME = 1.96 times square root of p(1 – p)/n. Solve for n. When p is unknown, use p = 0.5, the most cautious choice.
Worked example: to get a margin of error of at most 0.03, n is at least (1.96/0.03)2 times 0.25. That is 4268.4 times 0.25 = 1067.1, so round up to n = 1068.
Halving the margin would require four times as many people.
What mistakes occur?
Rounding down the sample size, and thinking the margin accounts for biased sampling. It covers only random sampling variation. Mathbot always rounds up and notes that a poor design cannot be rescued by a small margin.