Sampling Distributions (Edexcel IAL Maths)

Spec point — 4.1

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 ($\overline{x}$) 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
  • $\frac{Σx}{n}-\overline{X}^{2}$ would be a statistic as $\overline{X}$ is the mean of the sample
  • $\frac{Σx}{n}-μ^{2}$ 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.

Answer:

Exam Hint

• 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.