Sample Mean Distribution (AQA A Level Maths: Statistics)

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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, begin mathsize 16px style x with bar on top end style is calculated then the distribution of the sample means, begin mathsize 16px style X with bar on top end style , 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, top enclose 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, begin mathsize 16px style x with bar on top end style = µ
    • Variance begin mathsize 16px style sigma squared over n end style
    • Standard deviation begin mathsize 16px style fraction numerator sigma over denominator square root of n end fraction end style
  • For a random variable begin mathsize 16px style X tilde straight N left parenthesis mu comma sigma squared right parenthesis end style the distribution of the sample mean would be begin mathsize 16px style stack X space with bar on top tilde N open parentheses mu comma sigma squared over n close parentheses end style
  • 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 

5-3-1-sample-means-diagram-1

Worked example

A random sample of 10 observations is taken from the population of the random variable X tilde straight N left parenthesis space 30 comma space 25 space right parenthesis and the sample mean is calculated as x with bar on top Write down the distribution of the sample mean, X with bar on top .

5-3-1-sample-means-we-1

Examiner Tip

  • Look carefully at the distribution given to determine whether the variance or the standard deviation has been given.

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.