What is a sampling unit?

A single member of the population that can be selected, such as one household.

What is a sampling frame?

A list of all the sampling units in the population.

Give two advantages of a census.

It gives exact population values, and there is no sampling variability or bias from the choice of sample.

Give two advantages of a sample.

It is cheaper and quicker, and needs less data processing. It is also essential if testing is destructive.

Give two disadvantages of a sample.

It is subject to sampling variability and may not be representative, so it can be less accurate.

When is a census unsuitable?

When testing destroys the item, when the population is very large or infinite, or when time and cost are limited.

A quantity calculated only from the sample data, with no unknown parameters.

Is \bar X-\mu a statistic? Why?

No, because it contains the unknown parameter \mu, so it cannot be calculated from the sample alone.

What is a sampling distribution?

The probability distribution of a statistic over all possible samples of a given size.

Why does a statistic have a sampling distribution?

Its value changes from one random sample to the next, so it is a random variable.

How do you find a sampling distribution for a small population?

List all possible samples, find the statistic for each, and add the probabilities of equal values.

Populations, censuses and samplesEdexcel International A Level Further Maths: Flashcards

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What is a population?

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What is a population?: The whole set of items or individuals of interest.
What is a census?: A survey that observes every member of the population.
What is a sample?: A selection of members of the population, used to learn about the whole.
What is a sampling unit?: A single member of the population that can be selected, such as one household.
What is a sampling frame?: A list of all the sampling units in the population.
Give two advantages of a census.: It gives exact population values, and there is no sampling variability or bias from the choice of sample.
Give two advantages of a sample.: It is cheaper and quicker, and needs less data processing. It is also essential if testing is destructive.
Give two disadvantages of a sample.: It is subject to sampling variability and may not be representative, so it can be less accurate.
When is a census unsuitable?: When testing destroys the item, when the population is very large or infinite, or when time and cost are limited.
What is a statistic?: A quantity calculated only from the sample data, with no unknown parameters.
Is $\bar X-\mu$ a statistic? Why?: No, because it contains the unknown parameter $\mu$ , so it cannot be calculated from the sample alone.
What is a sampling distribution?: The probability distribution of a statistic over all possible samples of a given size.
Why does a statistic have a sampling distribution?: Its value changes from one random sample to the next, so it is a random variable.
How do you find a sampling distribution for a small population?: List all possible samples, find the statistic for each, and add the probabilities of equal values.

Exam questions on Populations, censuses and samples

A council keeps a register of the $4200$ households in its area. To find out how many households recycle, it selects $300$ households from the register and sends each one a questionnaire.
Give one advantage and one disadvantage of the council using a sample of $300$ households rather than a census.2 marks
The masses of bags of flour are modelled by $\mathrm{N}(\mu,\sigma^2)$ , where $\mu$ and $\sigma$ are unknown. A random sample $X_1,X_2,X_3,X_4,X_5$ of bags is taken and $\bar X$ is the sample mean.
Explain why $\bar X$ has a sampling distribution but $\mu$ does not.2 marks
A bag contains three discs numbered $1$ , $2$ and $3$ . A disc is drawn at random and replaced, and then a second disc is drawn at random. Let $M$ be the larger of the two numbers drawn (or the common value if they are equal) and let $\bar X$ be the mean of the two numbers.
Find the sampling distribution of $M$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).