Sampling Distributions (Edexcel International A Level Maths): Revision Note

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 () 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

  • would be a statistic as  is the mean of the sample

  •   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