Sampling and Types of Data
Data is either categorical (nominal or ordinal) or numerical (discrete or continuous), and it comes from a whole population, called a census, or from a sample. This NSW Year 9 Advanced maths subtopic classifies data and compares sampling methods (NESA outcome MA5‑DAT‑P‑01).
You will classify variables, choose a hypothesis that data can test, and decide when a census or a sample is needed. You will name simple random, systematic, stratified and self-selected samples, choose a stratified sample in proportion, see why a small sample is less reliable, estimate population totals from samples, and estimate the fish in a dam by tagging. It is part of Year 9 Advanced Investigating Data.
Every question with a fully worked solution.
- Sampling and Types of Data - Video - Types of Data Watch
Theory
Every statistical inquiry starts with two decisions: what kind of data you are collecting, and who you collect it from. This Year 9 Advanced maths page shows how to classify variables, choose between a census and a sample, use the four main sampling methods, and turn a sample into an estimate for the whole population. It is useful for students heading towards Standard or Advanced maths.
A variable is something recorded about each person or object. It is categorical if its values are labels, and numerical if its values are numbers you could add or average.
Categorical data are nominal when the categories have no natural order (eye colour, favourite fruit, phone number) and ordinal when they do (small, medium, large; strongly disagree to strongly agree).
Numerical data are discrete when they are counted, so only certain values are possible (number of siblings), and continuous when they are measured and can take any value in a range (arm span, time, temperature).
The population is the whole group you want to know about. A census collects data from every member; a sample uses part of the population, chosen to represent it.
A hypothesis is a statement that the data can test, such as 'taller students jump further'. A question or an opinion is not a hypothesis.
| Method | How the sample is chosen | Watch out for |
|---|---|---|
| simple random | every member has the same chance (names drawn, random numbers) | needs a full list of the population |
| systematic | every \(k\)th member of a list, from a random start | a pattern in the list that repeats every \(k\) |
| stratified | split into groups (strata), then a random sample from each, in proportion | group sizes must be known |
| self-selected | people choose to take part (phone-in, online poll) | usually biased: only keen people reply |
Classifying a variable
- Is it a number you could add or average? No: categorical. Yes: numerical.
- Categorical: do the categories have a natural order? Ordinal if yes, nominal if no.
- Numerical: is it counted (discrete) or measured (continuous)?
Choosing and using a sample
- Census or sample? Use a sample when the population is large, a census is too slow or costly, or testing destroys the item (light globes, crash tests).
- Choose a method that gives every part of the population a fair chance. For groups of different sizes, use a stratified sample.
- Systematic: find \(k\), pick a random start from \(1\) to \(k\), then keep adding \(k\). The \(n\)th item is start \(+ (n - 1)k\).
- Estimate the population count with the sample proportion, and say it is an estimate.
- Review: is the sample big enough, and could the way it was chosen favour one answer?
(i) Classify each variable.
(ii) Write a hypothesis Jai's data can test.
(i) Ask: is it a number you can add or average? If so, is it counted or measured? If not, do the categories have an order?
| Variable | Type |
|---|---|
| distance | numerical, continuous (measured) |
| buses caught | numerical, discrete (counted) |
| way of travel | categorical, nominal (no order) |
| enjoyment | categorical, ordinal (ordered) |
| postcode | categorical, nominal |
A postcode is written with digits, but adding or averaging postcodes means nothing, so it is a label, not a number.
(ii) A hypothesis is a statement that the data can check, for example: students who live further from school take longer to travel there. Jai would also need to record each student's travel time to test it.
(i) How many should be chosen from each group?
(ii) Why is this better than surveying the first \(40\) members to arrive at training?
(i) The club has \(150 + 162 + 48 = 360\) members, so the sample is \(\dfrac{40}{360} = \dfrac{1}{9}\) of each group.
| \(\text{juniors}\) | \(:\) | \(\dfrac{1}{9} \times 150 = 16.7 \to \mathbf{17}\) |
| \(\text{seniors}\) | \(:\) | \(\dfrac{1}{9} \times 162 = \mathbf{18}\) |
| \(\text{veterans}\) | \(:\) | \(\dfrac{1}{9} \times 48 = 5.3 \to \mathbf{5}\) |
Check: \(17 + 18 + 5 = 40\ \checkmark\). Within each group, choose the members at random.
(ii) Every group is represented in its true proportion. The first \(40\) at training would leave out members who rarely train, and might be mostly one age group.
(i) Find the sampling interval.
(ii) The first book chosen is number \(19\). Which are the \(2\)nd, \(3\)rd and last books?
(iii) \(9\) of the \(75\) books are damaged. Estimate the number of damaged books in the library.
| \(\text{interval}\) | \(=\) | \(\dfrac{2400}{75} = \mathbf{32}\) |
(ii) Add \(32\) each time. The last is the \(75\)th book, which is \(74\) steps after the first.
| \(2\text{nd}\) | \(:\) | \(19 + 32 = \mathbf{51}\) |
| \(3\text{rd}\) | \(:\) | \(51 + 32 = \mathbf{83}\) |
| \(75\text{th}\) | \(:\) | \(19 + 74 \times 32 = \mathbf{2387}\) |
(iii) Apply the sample proportion to the whole library.
| \(\dfrac{9}{75} \times 2400\) | \(=\) | \(\mathbf{288}~\text{books}\) |
This is an estimate: a different starting number would give a slightly different sample and answer.
(i) Estimate the number of turtles in the lake.
(ii) State two assumptions.
(iii) If \(12\) of the \(60\) had been tagged, would the estimate be larger or smaller?
(i) The tagged fraction in the second catch should match the tagged fraction in the lake. Let \(N\) be the number of turtles.
| \(\dfrac{48}{N}\) | \(=\) | \(\dfrac{8}{60}\) |
| \(8N\) | \(=\) | \(48 \times 60 = 2880\) |
| \(N\) | \(=\) | \(\mathbf{360}~\text{turtles}\) |
(ii) The tagged turtles have mixed evenly through the lake; no turtles were born, died, arrived or left in between; tags do not fall off or change how easily a turtle is caught.
(iii) Smaller. A larger tagged fraction means the \(48\) tagged turtles are a bigger share of the lake: \(N = \dfrac{48 \times 60}{12} = 240\).
Common pitfalls
Frequently asked questions
What is the difference between discrete and continuous data?
Discrete data are counted, so only separate values are possible, like \(0, 1, 2\) goals. Continuous data are measured and can take any value in a range, like a height of \(162.4\) cm.
What is the difference between nominal and ordinal data?
Both are categorical. Ordinal categories have a natural order, such as small, medium, large. Nominal categories do not, such as colours or countries.
When should you use a sample instead of a census?
When the population is very large, a census would cost too much or take too long, or testing destroys the item.
What is a stratified sample?
The population is split into groups, such as year groups, and a random sample is taken from each group in proportion to its size.
How do you estimate a population from a sample?
Find the fraction of the sample with the feature and multiply it by the size of the population.
How does capture and recapture work?
Tag some animals and release them. Later, catch another group and count the tagged ones. The tagged fraction of the catch is assumed to equal the tagged fraction of the whole population.