K1, 3.21.1 - The AQA car data set and populations

K1, 3.21.1 - The AQA car data set and populations

Identify the group behind a statistical claim and read the real AQA car workbook with confidence. You will distinguish measurements from category codes, interpret its selection rules and decide what a blank or zero can mean before calculating.

Populations, samples and informal inference

Can a data set containing thousands of cars tell you about every car in England? The number of rows is only part of the answer: first ask which cars could appear in it.

You already know

A data set records observations about a stated group. A percentage compares a part with its whole using 100×part/whole100\times\text{part}/\text{whole}; both counts must refer to the same group. Not in the formula booklet: learn it.

Start with a smaller investigation. A college wants to know how its student drivers park. It has 620620 student drivers and asks 4040 of them. The group the college wants to describe and the group it actually asks have different roles.

Population, sample and frame

The population is the entire group about which an investigation seeks information. A sample is the group from which information is obtained. A census obtains information from every member of the defined population. A sampling frame is the list or accessible set from which a sample is selected. For example, a complete list of this college's student drivers is a frame for those drivers; a list of campus permit holders misses drivers who park elsewhere.

The population is set by the question. “All students at this college” and “all students who drive to this college” are different populations. A parameter describes a population; a statistic is calculated from a sample. A sample mean journey time is a statistic used to estimate a population mean journey time.

Informal inference uses sample evidence to suggest something about the population. It needs caution about selection, non-response and ordinary variation between samples. It does not turn a sample percentage into an exact population percentage.

Worked example 1: interpreting a sample percentage

Question. A college has 620620 student drivers. Of 4040 surveyed drivers, 2626 usually park on campus. Identify the population and sample, calculate the sample percentage and give a cautious inference.

Plan. Separate the target group from the respondents, then divide the number reporting campus parking by the number surveyed.

Population size=620all this college’s student driversSample size=40the surveyed driversSample percentage=2640×100=65%use respondents as denominator\begin{aligned} \text{Population size} &= 620 && \text{all this college's student drivers} \\ \text{Sample size} &= 40 && \text{the surveyed drivers} \\ \text{Sample percentage} &= \frac{26}{40}\times100 = 65\% && \text{use respondents as denominator} \end{aligned}

Why this step? The 2626 responses came from 4040 people, so dividing by 620620 would answer a different question.

Answer. The population is all 620620 student drivers at this college and the sample is the 4040 surveyed drivers. The statistic is 65%65\%. If the selection and responses represent the drivers fairly, about 65%65\% of the population may usually park on campus.

Check. 0.65×40=260.65\times40=26. The result describes student drivers here; it gives no direct information about other colleges.

Change only the number of respondents who report campus parking. The population and sample sizes stay fixed, so you can see exactly which statistic changes.

Worked example 2: a different response count

Question. The same survey has 3030, rather than 2626, of its 4040 drivers reporting campus parking. Calculate the statistic and assess the statement “exactly 75%75\% of all 620620 drivers park on campus.”

Plan. Recalculate the sample percentage and distinguish an estimate from a population fact.

Sample percentage=3040×100=75%changed response countEstimated number of drivers=0.75×620=465estimate, not a count made\begin{aligned} \text{Sample percentage} &= \frac{30}{40}\times100 = 75\% && \text{changed response count} \\ \text{Estimated number of drivers} &= 0.75\times620 = 465 && \text{estimate, not a count made} \end{aligned}

Why this step? Even a correctly calculated statistic is based on the selected respondents, so it need not equal the unknown population parameter.

Answer. 75%75\% is the sample percentage. It suggests about 465465 of the college's student drivers might park on campus, if the sample is suitable. The word “exactly” is unsupported.

Check. The number 465465 was inferred; only 4040 drivers were surveyed. A different fair sample could give a different percentage.

Calculating a percentage is ordinary arithmetic; the rule 100×part/whole100\times\text{part}/\text{whole} is not in the formula booklet: learn it. The important statistical decision is which whole belongs in the denominator.

Watch out

Wrong: “30/62030/620 is the proportion of surveyed drivers who park on campus.” Only 4040 were surveyed. Right: use 30/4030/40 for the sample proportion, and describe a population claim as an inference whose reliability depends on the sample.

Use the same distinction in the next investigation, with a new target population.

The AQA car release and its sampling frame

The actual AQA car workbook contains 38273827 records, but those records passed several deliberate filters before they reached your spreadsheet.

This is the car large data set used for AQA A-level Mathematics examinations from 20202020 onwards. It replaces the earlier Family Food data set. The source is an extract from the UK Department for Transport Stock Vehicle Database, 20172017, which concerns vehicles registered in England. The workbook has four sheets: Introduction, Car data, Definition of fields and Field Values.

Download the unchanged official workbook or the raw CSV extraction. The CSV preserves source cells and saved random helpers; it is not a cleaned replacement release.

Original filterRecords retained in the AQA extract
MakeBMW, FORD, TOYOTA, VAUXHALL and VOLKSWAGEN: the five most frequently registered makes in the original database
First-registration date33–99 June 20022002, or 66–1212 June 20162016
Manufacture year20022002 or 20162016 respectively, matching the registration year
Current registered keeper's regionLondon, North West or South West
Vehicle categoryM1 passenger vehicles, described in the workbook as cars with no more than eight seats

The two date ranges are particular weeks, not every registration in either year. The region belongs to the current registered keeper, not necessarily to the first owner, manufacturer or usual driver. The 20172017 stock extract also cannot include a 20022002 car that had already disappeared from that stock database.

There are 998998 London-keeper records, giving

9983827×100≈26.1%\frac{998}{3827}\times100\approx26.1\%

This percentage describes the extract. It does not establish the London share of all cars in England: other regions, makes and registration dates are missing from the frame. There are no Nissan cars in this extract, for example; there is no Nissan subgroup waiting to be revealed by a clever filter.

A surprising consequence is that checking every row can still leave a claim unsupported. A census of all 38273827 records has no sampling uncertainty about that extract, but none of the excluded cars has appeared. Taking a larger random sample within the frame also cannot bring them back.

Key idea

Attach a conclusion to the population the frame can describe. Random selection improves selection within a frame; it does not remove the frame's original exclusions.

When a question asks for a limitation, name the relevant restriction and connect it to the proposed comparison. “There are only five makes” becomes more useful when you say which missing make the researcher wanted to study.

Continue this lesson.

You’re previewing NovaLearn without an account. Create a free account to read the rest.

Create free account