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

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

(a)    List all the possible samples.

(b)    Find the sampling distribution of the mean.