Sampling Theory

The Logic of Sampling

The idea that 1,000 people can represent 330 million seems counterintuitive, but the mathematics is well-established. The Central Limit Theorem states that the mean of a randomly drawn sample clusters around the population mean, and that this approximation tightens as the sample grows. With a truly random sample of 1,000, you can estimate a population proportion with a margin of error of roughly +/- 3 percentage points, 95% of the time.

The critical word is 'random.' The mathematical guarantee depends on every member of the population having an equal chance of being selected. In practice, perfectly random sampling is impossible: you cannot reach people without phones, people who never answer unknown numbers, or people who refuse to participate. Every deviation from randomness introduces potential bias, which is why the art of polling is as important as the science.

Worked Example: Where +/-3 Comes From

The margin of error for a proportion at the 95% confidence level is approximately 1.96 times the square root of p(1-p)/n. Pollsters use p = 0.5 because that maximizes the uncertainty, giving the most conservative (widest) margin.

For n = 1,000: the square root of (0.5 x 0.5 / 1,000) = square root of 0.00025 = 0.0158. Multiply by 1.96 and you get about 0.031, or +/-3.1 percentage points. That is exactly the '+/- 3 points' you see quoted on national polls.

Because n sits under a square root, precision improves slowly:

Notice that going from 1,000 to 4,000 (four times the cost) only cuts the margin in half. That diminishing return is why most national polls stop near 1,000 respondents.

Common Misconceptions

The most common misconception is that sample size should be proportional to population size. In fact, a sample of 1,000 is equally valid for estimating preferences in a city of 100,000 or a country of 300 million, because the formula above depends on the absolute sample size n, not its proportion to the population.

Another misconception is that doubling the sample size halves the margin of error. As the table shows, the actual relationship follows a square-root law: to halve the margin of error, you need to quadruple the sample. Going from 1,000 to 4,000 respondents reduces the margin from roughly +/-3 to +/-1.5 points. This is why most polls settle at around 1,000 respondents: the cost of additional precision increases steeply while the payoff shrinks.