5.3.1 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