Estimating Probability

60% Boys

Which hospital is more likely to have 60% males born on any given day?  Answer: the smaller one.  Sample size affects how reliably any given sample approximates the population being estimated.  Small samples are more likely to give one an incorrect estimate.

Two quick examples make this clear.  Suppose you want to know how 100,000 registered Libertarian Green National Socialists are going to vote in the upcoming union election.  If you ask all 100,000 how they plan to vote on a particular issue, your estimate will be wrong only to the extent that people lie to pollsters, or people change their mind between now and the election.  If you ask only 10 people about that same issue, your guess is obviously much less reliable. Small sample sizes are much less reliable.  The second example is more intuitive. Suppose you want to toss coins in the air and see how many come up all “heads”; which sample size is more likely to give you all heads, and no tails?  Two coins, four coins, six coins, eight coins, or ten coins?  The obvious, intuitive answer is two coins.  A small sample size gives you more extreme and unrepresentative answers.