Collecting and Using Data
Every question with a fully worked solution.
- Collecting and Using Data - Video - Statistics - Types of Data Watch
Theory
Every statistical investigation starts with a question and a plan for collecting data. This Year 10A maths page revises how to pose a question and a hypothesis, classify data as numerical or categorical, choose between a census and a sample, use random, systematic and stratified sampling, and estimate for a population from a sample. It revises Year 7–9 work that the rest of the Year 10 statistics topic builds on.
A variable is anything recorded about each member of a group. A numerical variable has number values: discrete if counted (number of texts sent) and continuous if measured (time, mass). A categorical variable has labels: nominal if the categories have no order (eye colour) and ordinal if they can be ranked (very, somewhat, not at all). Postcodes and jersey numbers are labels, so they are categorical.
The population is the whole group being studied. A census collects data from every member. A sample collects data from part of the population, chosen to represent all of it. The sample size is the number in the sample.
A representative sample is like the whole population. A biased sample leans one way, usually because it leaves out a whole sub-group, such as asking about gym membership outside a gym. A self-selected sample, such as an online poll, is made up of people who chose to answer, so people with strong views are over-represented. A huge number of replies does not fix bias.
Different samples from the same population give slightly different results, even with the same method. Larger random samples vary less, so their estimates are more reliable.
A statistical investigation follows a cycle: pose a question, plan what data to collect and from whom, collect the data, display and analyse it, then answer the question. A hypothesis is a statement, not a question, that the data can show to be true or false, such as “Year 10 students who sleep more get higher marks”.
| Sampling 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 |
| stratified | a random sample from each group, in proportion to its size | group sizes must be known |
| self-selected | people choose to take part (phone-in, online poll) | usually biased: only keen people reply |
How to plan a statistical investigation
- Pose the question and, if you can, a hypothesis to test.
- Choose the variables to record and classify each one: numerical (discrete or continuous) or categorical (nominal or ordinal).
- Name the population. Decide on a census or a sample.
- Choose a fair sampling method (random, systematic or stratified) and a sample size.
- Collect, display and analyse the data, then answer the question and say how reliable the answer is.
How to classify a variable
- Are the values numbers that behave like numbers? If yes, numerical; if not, categorical.
- Numerical: counted is discrete, measured is continuous.
- Categorical: a natural order is ordinal, no order is nominal.
(i) Write a hypothesis Leo could test.
(ii) Classify each of the three variables.
(i) A hypothesis is a statement, not a question, that the data can show to be true or false. For example: “Students who live further from school take longer to travel there.” To test it, Leo should also record each student's travel time.
(ii) Distance: measured, so numerical, continuous.
Way of travel: labels, so categorical, nominal.
Times late: counted, so numerical, discrete.
The way of travel has no natural order: walking is not “more” than catching the bus.
(i) How many should be chosen from each group?
(ii) Why is this better than choosing \(20\) from each group?
(i) Find what fraction of the club the sample is, then take that fraction of each group.
| \(\text{fraction}\) | \(=\) | \(\dfrac{60}{500} = 0.12\) |
| \(\text{juniors}\) | \(=\) | \(0.12 \times 200 = \mathbf{24}\) |
| \(\text{adults}\) | \(=\) | \(0.12 \times 225 = \mathbf{27}\) |
| \(\text{seniors}\) | \(=\) | \(0.12 \times 75 = \mathbf{9}\) |
Check: \(24 + 27 + 9 = 60\). Within each group the members are then chosen at random.
(ii) The groups are different sizes. Choosing \(20\) from each would give the \(75\) seniors as much say as the \(225\) adults, so seniors would be over-represented and adults under-represented.
(i) What is the sampling interval?
(ii) The first loaf chosen is the \(23\)rd made that day. Which loaves are the \(2\)nd and \(3\)rd in the sample, and which is the last?
(iii) Why should the starting loaf be chosen at random?
| \(\text{(i) interval}\) | \(=\) | \(\dfrac{3600}{48} = \mathbf{75}\) |
(ii) Add the interval each time.
| \(2\text{nd}\) | \(:\) | \(23 + 75 = \mathbf{98}\text{th}\) |
| \(3\text{rd}\) | \(:\) | \(98 + 75 = \mathbf{173}\text{rd}\) |
| \(48\text{th}\) | \(:\) | \(23 + 47 \times 75\) |
| \(\) | \(=\) | \(\mathbf{3548}\text{th}\) |
The last loaf is \(47\) intervals after the first, not \(48\).
(iii) If the bakery always started at the same place, for example the first loaf of each batch, the sample might always land on loaves made at one point in the process. A random start gives every loaf the same chance of being checked.
(i) Estimate the number of households in the town that own a dog.
(ii) A second random sample of \(120\) finds \(36\) with a dog. Give the new estimate and explain why the two estimates differ.
(iii) Combine the samples for a better estimate.
| \(\text{(i)}\ \dfrac{42}{120} \times 9000\) | \(=\) | \(\mathbf{3150}\) |
| \(\text{(ii)}\ \dfrac{36}{120} \times 9000\) | \(=\) | \(\mathbf{2700}\) |
Each random sample contains different households, so the results vary from sample to sample. Neither estimate is wrong.
(iii) Combine: \(42 + 36 = 78\) dogs in \(120 + 120 = 240\) households.
| \(\dfrac{78}{240} \times 9000\) | \(=\) | \(\mathbf{2925}\) |
The combined sample of \(240\) is larger, so its estimate is more reliable. It is still an estimate, not an exact count.
Common pitfalls
Frequently asked questions
What is the difference between a census and a sample?
A census collects data from every member of the population. A sample collects data from only part of it, chosen to represent the whole.
What makes a sample biased?
A sample is biased when some part of the population has little or no chance of being chosen, or when people choose themselves, so the results lean one way.
What are the four main sampling methods?
Simple random (everyone has an equal chance), systematic (every kth member of a list), stratified (a random share from each group) and self-selected (people choose to take part, which is usually biased).
How do you estimate a population number from a sample?
Find the fraction of the sample with the feature and multiply by the population size. If \(60\) of \(250\) households have solar panels, then about \(\dfrac{60}{250} \times 12\,500 = 3000\) of \(12\,500\) households do.
What is the difference between a question and a hypothesis?
A question asks what is true, such as “Do taller students have longer arm spans?”. A hypothesis states what you expect, such as “Taller students have longer arm spans”, so the data can support it or show it is false.