Sampling distributions

A statistic is a random variable

Draw another sample and the mean changes. Any number computed from a sample, a statistic, is itself random, and its distribution over repeated samples is called its sampling distribution. This is the idea underneath every error bar and p-value: they all describe how a statistic would vary if the data collection were repeated, not how the data vary.

The standard deviation of a sampling distribution is the standard error. For a mean it is $\sigma/\sqrt{n}$, estimated by $s/\sqrt{n}$; for a proportion $\hat p$ it is $\sqrt{p(1-p)/n}$. When $\sigma$ must be estimated from the data, the standardised mean follows a Student-$t$ distribution with $n-1$ degrees of freedom instead of a normal, which has fatter tails for small $n$ and approaches the normal as $n$ grows. The $t$-statistic measures how many standard errors the sample mean is from a hypothesised value.

• Sampling distribution: how a statistic varies over repeated samples.
• Standard error: the standard deviation of the sampling distribution; it shrinks like $1/\sqrt{n}$.
• $t$-distribution: use it for a standardised mean when $\sigma$ is estimated, with $n-1$ degrees of freedom.

Lab · what's graded

• The standard error of a mean10 pts
• The standard error of a proportion15 pts
• A t-statistic15 pts
• Which distribution15 pts
• Facts about sampling distributions15 pts
• How many trades15 pts
• Standard deviation versus standard error15 pts