Introducing Statistics: What Can We Learn from Data? Study Guide (Statistics 1.1)

What can we learn from data?

Statistics is the study of how to learn from data when the answer is not the same for everyone. If you ask one student how long they slept last night, you get one number. If you ask how long the students in your school typically sleep, there is no single answer. Different students sleep for different lengths of time. This variability is why we need statistics.

A statistical investigation starts with a question about a large group. We usually cannot ask or measure everyone in that group, so we collect data from part of it. Then we use what we found to say something about the whole group, and we say it in plain words about the real situation.

This guide follows one fictional study from start to finish, then gives you two more worked examples.

The running example. Sandmere College (a fictional college) has 2,400 students. The student council wants to know whether a late-night bus on Fridays would be used. It asks 150 students, and 87 of them say they would use the bus.

The parts of a statistical study

A statistical study gathers data from a sample so that it can answer an investigative question about the whole population the sample comes from. Each part has a name.

Part	Meaning	Sandmere College
Investigative question	The question the study is designed to answer	“What proportion of Sandmere College students would use a late-night bus on Fridays?”
Population	All the items or individuals of interest	All students at Sandmere College
Population size, N	How many items or individuals are in the population	N = 2,400
Sample	The part of the population that data are actually collected from	The 150 students who were asked
Sample size, n	How many items or individuals are in the sample	n = 150
Datum	One piece of information about one item or individual	One student’s answer, for example “Yes”
Data set	The whole collection of data	All 150 answers

A few points to notice:

• N and n are different symbols. Capital N is for the population. Lower-case n is for the sample. A sample is a subset of the population, so n is less than N. Here n = 150 and N = 2,400, so the council asked 150 ÷ 2,400 = 6.25% of the students.
• “Data” is plural. One answer is a datum. Many answers are data. The full collection is a data set.
• A population does not have to be people. It can be apples, phone chargers, days, trees or messages. It is whatever the question is about.

Why use a sample?

Why not just ask every one of the 2,400 students? Studies use a sample when collecting data from the whole population is not possible or not sensible. Common reasons are:

• The population is too large. Asking 2,400 students takes far longer than asking 150.
• Reaching everyone is too difficult or costly. Some students are hard to contact; some never reply.
• Measuring destroys the item. To find how long a battery lasts, you run it until it is flat. Testing every battery would leave none to sell.

Recording information from every member of the population has its own name, a census. You will meet it again in Topic 1.10.

Seeing the whole process

What makes a good investigative question?

The investigative question drives the whole study. It decides who is in the population, what data you collect and what you will do with them. A valid investigative question has three features.

1. It needs a statistical investigation to answer it. The answer must depend on data that vary from one individual to another. “What time does the first lesson start at Sandmere College?” has one fixed answer; you just look it up. “How long do Sandmere College students typically take to travel to college?” needs data from many students, because travel times differ.

2. It has a clear, defined purpose. It says which population it is about and what you want to find out. “Do students like buses?” is too vague: which students, which buses, and what counts as “like”?

3. It asks for data you can collect and analyse. You must be able to measure or record the answer for each individual, in practice. “Would students use a late-night bus on Fridays?” can be asked as a yes/no question. “How much do students care about transport?” cannot be measured without first saying how.

Fix the question before you look at the results. The investigative question is chosen at the start. You must not change it because of what the data show. If you could keep rewording the question until the data gave an interesting answer, the “finding” would be produced by your choices, not by the population. A reader would be misled about how strong the evidence is.

Weak question	Problem	Better question
“How tall is Leila?”	One person; one measurement answers it	“What is the typical height of first-year students at Sandmere College?”
“Is the canteen good?”	“Good” is not defined; no clear data	“What proportion of Sandmere College students buy lunch at the canteen at least three days a week?”
“Do people sleep enough?”	Population and measurement unclear	“What is the typical number of hours Sandmere College students sleep on a night before a weekday?”

Topic 1.10 returns to investigative questions and breaks them into parts that guide data collection, the analysis and the conclusion. For now, focus on the three features and on fixing the question first.

Writing “in context”

Every part of a study, and every number calculated from it, matches something in the real situation. Saying which real thing each number describes is what “in context” means. A statement in context names:

• who or what (the individuals in the sample or population),
• what was measured or recorded, and
• the units (minutes, grams, a percentage of the students asked).

Compare these for the Sandmere study:

• Not in context: “The answer is 58%.”
• In context: “87 of the 150 Sandmere College students asked (58%) said they would use a late-night bus on Fridays.”

The second statement also makes clear that 58% describes the sample. It is our best clue about all 2,400 students, but another sample of 150 would probably give a slightly different percentage. Whether we can trust a sample to represent the population depends on how it was chosen, which you will study in Topics 1.11 and 1.12.

Worked example 1: identifying the parts of a study

Question. Larch Hill Orchard (fictional) harvested 18,000 apples this season. The manager wants to know the mean mass of the apples, to decide how to price boxes. She weighs 60 of the apples. Their total mass is 9,720 g.

(a) State the investigative question. (b) Identify the population, N, the sample and n. (c) Give an example of one datum and describe the data set. (d) Calculate the mean mass of the apples weighed and state it in context. (e) Give a reason the manager weighed a sample.

(a) Investigative question. “What is the mean mass of the apples harvested at Larch Hill Orchard this season?”

(b) Population and sample.

1. Population: all apples harvested at Larch Hill Orchard this season. N = 18,000.
2. Sample: the 60 apples that were weighed. n = 60.
3. Check: the sample is part of the population, and n = 60 is less than N = 18,000.

(c) Datum and data set. One datum is the mass of one apple, for example “158 g”. The data set is the list of 60 masses, one for each apple weighed. Notice that the individuals here are apples, not people.

(d) Mean. Mean = total mass ÷ number of apples = 9,720 ÷ 60 = 162 g.

In context: the 60 apples weighed from this season’s Larch Hill harvest had a mean mass of 162 grams.

(e) Reason for a sample. Weighing 18,000 apples one by one would take far too long. A sample of 60 can be weighed quickly and still gives useful information about the whole harvest.

Check. Every answer names apples, this orchard and this season. The mean has units (grams). The 162 g describes the 60 apples weighed; it is an estimate for all 18,000, not a guaranteed value.

Worked example 2: improving an investigative question

Question. Brantwick (a fictional town of about 31,000 adults) opened a new cycle lane on Harbour Road. A councillor proposes the investigative question: “Is the new cycle lane good?”

(a) Explain why this is not a valid investigative question. (b) Write a better one. (c) The council asks 400 adult residents, and 64 say they cycled on the lane at least once in the past month. State this result in context. (d) After seeing the result, the councillor wants to change the question to “Do residents who cycled on the lane like it?” Explain why the council should not do this.

(a) What is wrong. The question has no defined purpose: “good” could mean safe, popular, cheap or quick. It does not say whose opinion counts or what data to collect. So no data set could answer it.

(b) A better question. “What proportion of adults living in Brantwick cycled on the Harbour Road lane at least once in the past month?”

1. It names the population: adults living in Brantwick.
2. It says what to record for each person: cycled on the lane in the past month, yes or no.
3. It needs a statistical investigation: the answer differs from person to person, and the result is a proportion of many people.

(c) In context. Calculate: 64 ÷ 400 = 0.16 = 16%. In context: 64 of the 400 Brantwick adults asked (16%) said they had cycled on the Harbour Road lane at least once in the past month.

(d) Why not change the question. The investigative question must be fixed before the analysis. Changing it because the first result looked disappointing lets the data choose the question. The council could keep switching until it found a result that suited it, and the final “finding” would give a misleading picture of the town. A new question about cyclists’ opinions would need its own study, planned in advance, with its own population (cyclists) and its own data.

Check. The rewritten question passes all three tests: it needs data that vary, it has a clear purpose and population, and the answer for each person can be recorded.

Common misconceptions

• “The sample and the population are the same thing.” The population is everyone or everything the question is about. The sample is only the part you collected data from.
• Mixing up N and n. Capital N is the population size; lower-case n is the sample size. If you write n = 2,400 for Sandmere College, you are saying everyone was asked.
• “Any question with a number for an answer is statistical.” “How many students are in my class?” has a number for an answer, but you just count; no variability is involved. A statistical question expects the answer to vary across individuals.
• “A population must be people.” Apples, chargers, days and birds can all form a population.
• “It’s fine to change the question once you see the data.” The question is fixed first. Changing it afterwards makes results look stronger than they are.
• “The sample result is the population value.” 58% of the students asked is not exactly 58% of all students. Another sample would probably give a slightly different value.
• “In context means repeating the number.” “The mean is 162” is not in context. Say what was measured, on which individuals, and the units.
• “Data is one piece of information.” One piece is a datum; data is the plural.

Where this leads

Next, Topic 1.2 names the characteristic you record about each individual (a variable) and separates numbers that describe a population from numbers that describe a sample. Read Topic 1.2: Variables. Now try the practice questions, then use the revision notes and the checklist to consolidate.