The Sampling Distribution of the Mean

Starting point: what you already know

• Mean and standard deviation measure the centre and spread of data. Variance is the standard deviation squared.
• Population and sample: the population is every item. A sample is the part you actually measure. Greek letters $\mu$ (mean) and $\sigma$ (standard deviation) describe the population.
• The normal distribution (Mathematics Advanced): bell shaped, with about 68%, 95% and 99.7% of values within 1, 2 and 3 standard deviations of the mean.
• z-scores: $z=\frac{x-\mu }{\sigma }$ says how many standard deviations a value is from the mean.
• The binomial distribution (previous chapter): mean $np$ and variance $np(1-p)$.

The sample mean is a random variable

Take a random sample of size $n$ and work out its mean, $\bar{X}$. Take another sample and you get a slightly different $\bar{X}$. So $\bar{X}$ changes from sample to sample: it is a random variable with its own distribution, called the sampling distribution of the mean.

• The sample mean is aimed at the right place, $\mu$.
• Bigger samples wobble less. To halve the standard error, you need four times the sample size.

The central limit theorem

Here is the surprising part. Even if single values are not bell shaped (lopsided, flat, anything), the sample mean is close to normal once $n$ is large enough. A common guide is $n\ge 30$.

Quick summary

• The sample mean $\bar{X}$ is a random variable.
• $E(\bar{X})=\mu$ and the standard error is $\frac{\sigma }{\sqrt{n}}$.
• Central limit theorem: for large $n$, $\bar{X}$ is close to normal, whatever the population looks like.
• Use $z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$ with the 68, 95, 99.7 rule or a normal table.
• Sample proportions: mean $p$, standard deviation $\sqrt{\frac{p(1-p)}{n}}$.

Quiz