Distribution of Sample Means (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Revision Note

Sample Mean Distribution

What is the distribution of the sample means?

• For any given population it can often be difficult or impractical to find the true value of the population mean, µ

  • The population could be too large to collect data using a census or

  • Collecting the data could compromise the individual data values and therefore taking a census could destroy the population

  • Instead, the population mean can be estimated by taking the mean from a sample from within the population

• If a sample of size $n$  is taken from a population, $X$, and the mean of the sample, $\overline{x}$ is calculated then the distribution of the sample means, $\overline{X}$ , is the distribution of all values that the sample mean could take

• If the population, $X$,  has a normal distribution with mean, µ , and variance, σ²  , then the mean expected value of the distribution of the sample means, $\overline{X}$ would still be µ but the variance would be reduced

  • Taking a mean of a sample will reduce the effect of any extreme values

  • The greater the sample size, the less varied the distribution of the sample means would be

• The distribution of the means of the samples of size taken from the population, will have a normal distribution with:

  • Mean, $\overline{x}$ = µ

  • Variance $\frac{{\sigma }^2}{n}$

  • Standard deviation $\frac{\sigma }{\sqrt{n}}$

• For a random variable $X~\mathrm{N}(\mu ,{\sigma }^2)$ the distribution of the sample mean would be $\overline{X}~N\left(\mu ,\frac{{\sigma }^2}{n}\right)$

• The standard deviation of the distribution of the sample means depends on the sample size, $n$

  • It is inversely proportional to the square root of the sample size

  • This means that the greater the sample size, the smaller the value of the standard deviation and the narrower the distribution of the sample means 

Central Limit Theorem

Does the distribution of the sample means always follow a normal distribution?

• If the variable $X$ for the population follows a normal distribution then the sample mean distribution $\overline{X}$  also follows a normal distribution

• If the variable $X$ for the population does not follow a normal distribution then the sample mean distribution $\overline{X}$ does not necessarily follow a normal distribution

  • If the sample size is small then $\overline{X}$ can not be assumed to follow a normal distribution

  • If the sample size is big enough then we can use the Central Limit Theorem to approximate $\overline{X}$ using a normal distribution

What is the central limit theorem?

• If a random sample of size n is taken from a population with mean µ  and variance ${\sigma }^2$  then the Central Limit Theorem states that $\overline{X}$ can be approximated by the normal distribution $\mathrm{N}\left(\mu ,\frac{{\sigma }^2}{n}\right)$ provided n is large enough

  • Notice the variance is still divided by the size of the sample

• We usually say n is large enough if it is at least 30

• This is a powerful theorem as it allows us to use the normal distribution for even when the population itself does not follow a normal distribution

• If the population follows a normal distribution then the Central Limit Theorem is not needed as $\overline{X}$ will be normal automatically

  • This is important as you might be asked whether the Central Limit Theorem was needed