CIE A Level Maths: Probability & Statistics 2

Revision Notes

1.2.2 Distribution of Sample Means

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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 sigma squared over n
    • Standard deviation fraction numerator sigma over denominator square root of n end fraction
  • For a random variable X tilde straight N left parenthesis mu comma sigma squared right parenthesis the distribution of the sample mean would be stack X space with bar on top tilde N open parentheses mu comma sigma squared over n close parentheses
  • 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 N space left parenthesis 30 space comma space 25 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

Exam Tip

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

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 begin mathsize 16px style X with bar on top end style  also follows a normal distribution
  • If the variable X for the population does not follow a normal distribution then the sample mean distribution begin mathsize 16px style X with bar on top end style does not necessarily follow a normal distribution
    • If the sample size is small then begin mathsize 16px style X with bar on top end style 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 X with bar on top 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 squared  then the Central Limit Theorem states that begin mathsize 16px style X with bar on top end style can be approximated by the normal distribution straight N open parentheses mu comma sigma squared over n close parentheses 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 begin mathsize 16px style X with bar on top end style will be normal automatically
    • This is important as you might be asked whether the Central Limit Theorem was needed

Worked example

The integers 1 to 29 are written on counters and placed in a bag. The expected value when one is picked at random is 15 and the variance is 70. Susie randomly picks 40 integers, returning the counter after each selection.

Find the probability that the mean of Susie’s 40 numbers is less than 13. Explain whether it was necessary to use the Central Limit Theorem in your calculation.

1-2-2-central-limit-theorem-we-solution

Exam Tip

  • If asked to explain whether it was necessary to use the Central Limit Theorem, check whether the population follows a normal distribution, if it does not then check the size of the sample. If your answer is yes comment on both of these things.

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.