Sampling Distributions (Edexcel International A Level Maths: Statistics 2)

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Sampling Distributions

What key words do I need to know about sampling?

  • The population refers to the whole set of things which you are interested in
    • For example: if a vet wanted to know how long a typical French bulldog slept for in a day then the population would be all the French bulldogs in the world
  • A sampling unit is an individual member of the population
    • For example: a French bulldog would be a sampling unit in the above population
  • A sample refers to a subset of the population which is used to collect data from
    • For example: the vet might take a sample of French bulldogs from different cities and record how long they sleep in a day
  • A sampling frame is a list of all members of the population
    • For example: a list of employees’ names within a company
  • A population parameter is a numerical value which describes a characteristic of the population
    • These are usually unknown
    • For example: the mean (μ) height of all 16-year-olds in the UK
  • A sample statistic is a value computed using data from the sample
    • These are used to estimate population parameters
    • For example: the mean (x with bar on top) height of 200 16-year-olds from randomly selected cities in the UK

What are the differences between a census and sampling?

  • A census collects data about all the members of a population
    • For example: the Government in England does a national census every 10 years to collect data about every person living in England at the time
  • The main advantage of a census is that it gives fully accurate results
  • The disadvantages of a census are:
    • It is time consuming and expensive to carry out
    • It can destroy or use up all the members of a population when they are consumables (imagine a company testing every single firework)
  • Sampling is used to collect data from a subset of the population
  • The advantages of sampling are:
    • It is quicker and cheaper than a census
    • It leads to less data needing to be analysed
  • The disadvantages of sampling are:
    • It might not represent the population accurately
    • It could introduce bias

What is a statistic and its sampling distribution?

  • A statistic is a random variable that is calculated by only using the data from a sample
    • It does not contain any unknown values such as population parameters
    • fraction numerator straight capital sigma x over denominator n end fraction minus X with bar on top squared would be a statistic as straight X with bar on top is the mean of the sample
    •  fraction numerator straight capital sigma x over denominator n end fraction minus mu squared would not be a statistic as μ is an unknown population parameter
  • Common examples for a statistic include:
    • Sample mean, sample median, range of the sample, sample variance
  • The sampling distribution of a statistic gives all the possible values of the statistic along with their associated probabilities

How do I find the sampling distribution of a statistic?

  • STEP 1: List all the possible samples
    • Some samples can have the same combination of members but still list each one separately
      • AAB and ABA both contain two A's and one B
      • It can be helpful to group these together
    • Consider how many possible samples there are
      • If there are 2 different types of members in a population and you take a sample of 3 then there are 8 possible samples (23 or 2 × 2 × 2)
  • STEP 2: Calculate the value of the statistic for each possible sample
    • This is why grouping the samples with the same combination is helpful
  • STEP 3: Calculate the probability of each sample being picked
    • The samples will be independent
    • If the population is described as being large then you can treat the sampling as if it were with replacement
      • The probability of selecting a type of member will be constant regardless of how many has already been selected
  • STEP 4: Construct the distribution table
    • Add together the probabilities of the samples that result in the same value of the statistic

Worked example

A money box contains a large number of £5, £10 and £20 notes.  50% are £5 notes, 30% are £10 notes and 20% are £20 notes.

A random sample of two notes is taken from the money box.

(a)    List all the possible samples.

(b)    Find the sampling distribution of the mean.

(a)    List all the possible samples.


(b)    Find the sampling distribution of the mean.



Exam Tip

  • Remember the probabilities in a sampling distribution should add up to 1. You can use this to either check your answer or as a shortcut to find the final probability once you have the others.

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Author: Paul

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.