Sample Mean Distribution (OCR A Level Maths A): 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,  is calculated then the distribution of the sample means,  , 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, 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,  = µ

  • Variance 

  • Standard deviation 

• For a random variable  the distribution of the sample mean would be 

• 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