Statistics guide, Unit 1, step 2 of 22Previous step Next step

Step 2: Data and How It’s Collected

By the EconTutorials team. 10-minute read with 6 practice questions.

SyllabusAP Stats 1.10–1.12

In 30 seconds

Data comes in tables: each row is an individual, each column is a variable. Variables are categorical or quantitative (discrete or continuous).
The four levels of measurement are nominal, ordinal, interval and ratio. Only ratio data has a true zero, and the level decides which calculations are allowed.
Good samples let chance decide: simple random, stratified, cluster or systematic. Convenience and voluntary-response samples are almost always biased, however large.
Only a randomised experiment can show cause and effect. Observational studies show association, which a confounding variable may explain.

Video lesson coming soonA short animated lesson for this step will appear here.

Individuals, variables and data tables

Think of the class register your teacher keeps. Each line is one student. Across the line are details: age, course, how many hours they study, whether they own a car. That register is a data table, and almost all data you meet in this course arrives in this shape.

Key termIndividuals and variables: the individuals (also called cases or observations) are the people or objects described and form the rows; a variable is a characteristic recorded for each individual and forms the columns.

Student ID	Age	Programme	Study hours / week	Owns a car?	Grade band
S-014	19	Economics	12.5	No	B
S-021	21	Business	7.0	Yes	C
S-036	18	Economics	18.0	No	A
S-052	23	Statistics	15.5	Yes	A

This table shows four individuals (the four students) and six variables (the six columns). Every individual has one value for each variable, which is what makes it a tidy data table.

Types of variables and levels of measurement

The most useful distinction in descriptive statistics is between two kinds of variable, because it decides which summaries and graphs you are even allowed to use.

Key termCategorical vs quantitative: a categorical variable puts each individual into a group (Programme, Owns a car?); a quantitative variable records a numerical amount you can sensibly do arithmetic on (Age, Study hours).

Quantitative variables come in two flavours. A discrete variable takes separated values, usually from counting (number of siblings, text messages sent yesterday). A continuous variable can take any value in a range, usually from measuring (height, body temperature, time to run 5 km).

Categorical variables are summarised with counts and shown in a bar chart. Quantitative variables get histograms, which you meet in Step 3.

Common mistakeThinking every number is quantitative. A postal code, a bus route number or a footballer’s shirt number are labels. Ask “does arithmetic on it make sense?”, not “is it written in digits?” The average of two postal codes is nonsense, so postal code is categorical.

The four levels of measurement

Statisticians go one step further and sort variables into four levels. Each level adds something to the one before.

Level	What it adds	Examples
Nominal	Named categories, no order	Eye colour, programme, blood type
Ordinal	Ordered categories, gaps not equal	Grade band A/B/C, T-shirt size S/M/L, medal
Interval	Equal gaps, but no true zero	Temperature in °C, calendar year
Ratio	Equal gaps and a true zero	Height, weight, study hours, income, exam mark

Key termTrue zero: a zero that means “none of the quantity”, so ratios make sense; 20 kg really is twice 10 kg, but 20 °C is not twice as hot as 10 °C because 0 °C does not mean “no temperature”.

The level decides your toolkit. Nominal data can be counted but never averaged. Ratio data unlocks everything: means, standard deviations and ratios. Ask “what level is this?” before you calculate anything.

Common mistakeAveraging survey scores as if they were ratio data. If Strongly disagree = 1 and Strongly agree = 5, it is tempting to report “a mean of 3.8”. But the distance from 1 to 2 need not equal the distance from 4 to 5. Treat such averages with caution and lean on the median or the full spread of responses.

Census, sample and sampling frame

Why not just ask everyone? Sometimes it costs too much or is impossible: a battery maker cannot test every battery until it dies, or there would be none left to sell. A well-drawn sample of a few thousand can describe millions accurately, as long as the list you sample from and the way you pick are both sound.

Key termSampling frame: the actual list from which the sample is drawn; if it leaves out part of the population, bias is built in before a single answer is collected.

A census collects data from every individual in the population; a sample collects from a subset. Suppose a city samples names from the property-ownership register to find out what residents think. Renters are missing from the frame entirely, and they may have systematically different views. This is called undercoverage, and no amount of careful random sampling inside a flawed frame can repair it.

Methods of sampling

Once the frame is sound, how do you choose who is in the sample? The gold standard is to let chance, not a person, decide.

Key termSimple random sample (SRS): a sample of size \( n \) chosen so that every possible group of \( n \) individuals has an equal chance of being the one selected.

Three other designs are common, each trading a little simplicity for lower cost or more precision:

Stratified: split the population into similar groups called strata (for example, year groups), then take an SRS from each stratum. Every group is guaranteed to appear.
Cluster: split the population into many similar groups called clusters (for example, tutorial classes), randomly pick a few whole clusters and measure everyone in them. Cheaper to reach, but riskier if clusters differ.
Systematic: put the frame in order and take every \( k \)-th individual after a random start.

The contrast to remember: stratified samples within every group; cluster samples whole groups.

A convenience sample takes whoever is easiest to reach, such as the first 30 people outside the library. It is fast and cheap but almost always biased, because “easy to reach” is rarely “representative”.

Worked example

A university has 4,000 students. A researcher wants a sample of 200. Describe how she would take a systematic sample, and how many she would take from each of three equal-sized year groups in a stratified sample.

Step 1Systematic gap: \( k = \dfrac{4000}{200} = 20 \), so she takes every 20th student on the list.

Step 2She picks a random start between 1 and 20, say 7, then selects students \( 7, 27, 47, \ldots \)

Step 3Stratified, with three equal year groups: \( \dfrac{200}{3} \approx 66.7 \), so she would take about 67 from two years and 66 from the third (\( 67 + 67 + 66 = 200 \)), each chosen by SRS within its year.

Both methods use chance, so both allow honest inference. The stratified version also guarantees that every year group is represented.

Bias: when a sample lies

In 1948 the Chicago Tribune printed the famous headline “DEWEY DEFEATS TRUMAN”. Truman won. The pollsters had used quota sampling: interviewers had to fill quotas (so many men, so many city-dwellers) but chose the individuals themselves. They tended to approach people who were easier to reach and a little better off, a group that leaned towards Dewey. They also stopped polling too early to catch late deciders.

Even with a plan, bias can leak back in. Learn these four names:

Key termFour common biases: undercoverage (part of the population cannot be selected), nonresponse (people chosen don’t reply and differ from those who do), response bias (people answer inaccurately because of sensitive topics, leading wording or the interviewer) and voluntary response (people opt in, so strong opinions dominate).

A classic case: in 1976 the advice columnist Ann Landers asked readers whether, given the chance again, they would have children. About 10,000 wrote in and around 70% said no. This was a voluntary-response sample. Unhappy parents were far more motivated to write than contented ones. When pollsters later asked a randomly selected sample, a large majority said they would have children again.

Common mistake“But my sample is huge.” A large voluntary-response or convenience sample is still biased. Size shrinks random error, not systematic error. When an online poll boasts about its sample size, first ask how those people got in.

Observational studies versus experiments

Suppose you notice that towns with more ice-cream sales also have more drownings. Is ice cream dangerous? No. Hot weather raises both ice-cream sales and swimming, and so drownings. The weather is linked to both variables, creating an association with no cause between them.

Key termObservational study vs experiment: in an observational study you measure individuals without intervening; in an experiment you deliberately impose a treatment and compare results, ideally assigning individuals to treatments at random.

Three more terms go with this. The explanatory variable is the suspected cause (ice-cream sales). The response variable is the effect you measure (drownings). A confounding variable is a lurking third factor tied to both (temperature), so you cannot tell which is really driving the response.

Random assignment to treatment and control groups tends to balance out every lurking variable, even ones you never thought to measure. So if the groups end up different, you can pin that difference on the treatment. Observational studies can show strong, real associations, but association is not causation.

Exam tipIn “name the method” or “name the bias” questions, give the name and a one-line justification that points to the detail in the question (“every 50th item after a random start, so systematic”). When criticising a causal claim from observational data, name a specific confounder and explain how it is linked to both variables. Use “is associated with”, never “causes”.

Practise: 6 questions

Try each one on paper first, then check the worked answer.

EasyQ1. Classify each variable as categorical or quantitative (and if quantitative, discrete or continuous): (a) number of siblings; (b) nationality; (c) weight of a parcel in kg. Show worked answer

(a) Quantitative, discrete: it is a count.
(b) Categorical: it places people into groups.
(c) Quantitative, continuous: it is a measurement that can take any value in a range.

EasyQ2. Give the level of measurement for: (a) finishing position in a race (1st, 2nd, 3rd); (b) temperature of a room in °C; (c) jersey number of a player; (d) annual income in dollars. Show worked answer

(a) Ordinal: ordered, but the gap between 1st and 2nd need not equal the gap between 2nd and 3rd.
(b) Interval: equal gaps, but 0 °C is not “no temperature”.
(c) Nominal: a label; averaging jersey numbers is meaningless.
(d) Ratio: equal gaps and a true zero (zero dollars means no income).

MediumQ3. Name the sampling method or bias in each case: (a) a researcher randomly selects 6 of 90 schools and surveys every pupil in those 6; (b) a website poll lets anyone click to vote; (c) a phone survey reaches only landline numbers, while many young adults have only mobiles. Show worked answer

(a) Cluster sampling: whole randomly chosen schools are fully surveyed.
(b) Voluntary-response bias: people choose themselves, so strong opinions dominate.
(c) Undercoverage: the frame (landlines) leaves out mobile-only young adults, so they can never be selected.

MediumQ4. Of 2,000 surveys about workplace dissatisfaction mailed out, only 180 are returned. What percentage replied, and what bias is present? Show worked answer

Response rate \( = \dfrac{180}{2000} \times 100 = 9\% \).
This is nonresponse bias: 91% did not reply, and those who did are likely to differ from those who didn’t (for example, the more dissatisfied may be keener to respond).

MediumQ5. People who take vitamin supplements have fewer illnesses, so researchers conclude supplements prevent illness. Identify a plausible confounder and explain why the causal claim is not justified. Show worked answer

Confounder: general health-consciousness. People who take supplements also tend to exercise, eat well and see doctors more, and these habits reduce illness on their own. Because the study only observed people, rather than randomly assigning supplements, the lower illness rate may come from those habits, not the supplements. Without random assignment you can only say supplements are associated with fewer illnesses.

ExamQ6. Design a randomised experiment to test whether a new study app improves exam scores. Name the explanatory and response variables, describe the groups and explain why random assignment matters. [5 marks] Show worked answer

Mark scheme (1 mark each):
1. Explanatory variable: use of the app (yes/no).
2. Response variable: exam score.
3. Treatment group uses the app; control group does not; everything else is kept the same.
4. Students are assigned to groups at random, for example by coin flip or random numbers, and mean scores are compared.
5. Random assignment spreads lurking variables such as motivation evenly across both groups, so any difference in scores can be attributed to the app.

Check yourself

1. A pollster divides voters into age bands and randomly samples within each band. This is…

Sampling randomly within every group is stratified; cluster sampling takes whole groups.

2. Which level of measurement has equal gaps and a true zero?

Ratio data, such as height or income, has a true zero, so “twice as much” makes sense.

3. Which type of study can establish cause and effect?

Only random assignment balances confounders, so only an experiment can support a causal claim.

Your score: 0 out of 3

The short version

Rows are individuals, columns are variables. Variables are categorical or quantitative; ask whether arithmetic on them makes sense.
Nominal, ordinal, interval, ratio: the level decides which summaries are legitimate, and only ratio has a true zero.
A bad sampling frame causes undercoverage. Prefer random designs (simple random, stratified, cluster, systematic) over convenience or voluntary response.
Undercoverage, nonresponse, response bias and self-selection cannot be fixed by a bigger sample.
Only randomised experiments show causation. The full chapter in Statistics Made Simple has 40 practice questions with a complete answer key.