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, , 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:
- 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
A random sample of 10 observations is taken from the population of the random variable and the sample mean is calculated as . Write down the distribution of the sample mean, .
- Look carefully at the distribution given to determine whether the variance or the standard deviation has been given.