E. Culminating Data Management Investigation · Culminating Data Management Investigation

Posing a Significant Question

E1.1About 65 minutes

Learning goals

I can explain what the culminating investigation is and how it brings together the whole course.
I can tell the difference between a topic, an issue and an investigable question.
I can judge whether a question is significant, measurable, feasible and ethical.
I can turn a broad interest into a precise question about a population, variables and data.
I can decide whether my problem is about statistics, probability or both.

Words to know

culminating investigation
A project at the end of the course in which you pose your own problem, collect or locate data, analyse it with the course's tools, and report and defend your conclusions.

significant problem
A question worth answering: it matters to someone, it has no obvious answer, and quantitative data can shed real light on it.

investigable question
A question precise enough that you know exactly which data would answer it: the population, the variables and how they will be measured are all clear.

quantitative data
Data that are numbers or counts (hours of sleep, number of wins, percentages) that can be summarized with statistics.

population of interest
The whole group your question is about, for example all Grade 12 students at one school.

feasible
Possible to complete with the time, data, tools and permissions you actually have.

Your own question, your own answer

All course long you have used other people's questions: What is the probability of a full house? How strong is the link between study time and marks? Now it is your turn. The culminating data management investigation asks you to pose a problem that matters to you, find or collect real data, analyse it with the tools from this course, and present and defend what you found.

The Ontario curriculum describes the project like this: you design and carry out an investigation that brings together the knowledge and skills of the whole course, then communicate your findings and give constructive critiques of other students' investigations. The problem can be a single problem on probability and statistics, or two smaller problems, one on probability and one on statistics.

This unit walks through every stage, and it models one complete investigation from start to finish so you can see what each stage looks like when it is done well. Then each lesson's activity moves your project forward one step.

Where the course tools come in

The investigation uses every sibling unit of MDM4U:

Organization of Data for Analysis: populations, samples, sampling methods, bias, surveys, ethics, Statistics Canada data.
One-Variable Data Analysis: mean, median, standard deviation, quartiles, box plots, histograms, margin of error.
Two-Variable Data Analysis and Validity: scatter plots, correlation, regression, residuals, cause and effect, judging claims in the media.
Probability, Permutations and Combinations, Discrete Probability Distributions and Continuous Distributions and the Normal Distribution: probability models, expected value, binomial and normal models.

You do not need every tool. You need the tools that fit your question, used correctly and explained clearly.

Two kinds of culminating problem. A statistics problem studies data about a group; a probability problem studies a random process. Many good projects combine them.

Topic, issue, question

Most students start with a topic (sleep, hockey, climate, music). A topic is not yet a question. Narrow it in two steps:

Issue: what do you wonder about the topic? Do part-time jobs affect how much teens sleep?
Investigable question: make every word precise. Whose sleep (which population)? Which sleep (school nights or weekends)? How is a job measured (yes or no, or hours per week)?

A strong question names the population, the variables and the comparison or relationship you will look at.

Narrowing a topic to an investigable question. Only the bottom box tells you exactly which data to collect.

Is it significant? Five tests

Use these tests on any question before you commit to it.

Significant: does the answer matter to someone (students, families, a school, a business, a community)? Is the answer not obvious in advance?
Quantitative: will the answer come from numbers you can summarize and model? "What is the best music?" is opinion; "Which streaming genres grew fastest in Canada?" uses data.
Measurable: can every variable be measured clearly and consistently?
Feasible: can you get enough data in the time you have, from people or sources you can actually reach? Secondary data from a reliable source (the curriculum names Statistics Canada's database as an example) often make a big question feasible.
Ethical: can you collect the data without harming, pressuring or identifying anyone? Questions about health, money, family or identity need extra care, and some are best answered only with published secondary data.

The curriculum gives two sample starting points: primary data from a game of chance you design, and secondary data from a reliable database. Both are excellent choices.

Worked example

The model investigation, Stage 1: choosing the question

Throughout this unit we follow one investigation by an example student, Amira, at a fictional school we call Pinecrest Secondary School. All of her data in this unit are made up for teaching, but they are realistic, and every calculation is real.

Amira works 15 hours a week at a grocery store and often feels tired. She wants to turn this into a culminating question.

Topic: sleep and part-time work.
Issue: "Do part-time jobs cut into teenagers' sleep?"
Make the population precise: Grade 12 students at Pinecrest (about 240 students). She can reach them, and the answer matters to her classmates and the school's guidance staff.
Make the variables precise: sleep = usual hours of sleep on a school night (a continuous, numerical variable); work = usual hours of paid work per week (numerical, with 0 meaning no job). From hours she can also make a categorical variable, job: yes or no.
Write the question: "Do Grade 12 students at Pinecrest who have part-time jobs sleep less on school nights than those who do not, and among students with jobs, is working more hours associated with less sleep?"
Run the five tests: significant (affects health and school success, and the answer is not obvious: maybe students who work are simply more organized); quantitative and measurable (hours); feasible (one short survey); ethical (an anonymous survey with no names, and no sensitive questions).

Amira's question names a population (Grade 12 students at Pinecrest), two variables (school-night sleep, weekly work hours) and two analyses (a comparison of two groups and a relationship between two numerical variables). It passes all five tests.

Worked example

Example 2: Fixing weak questions

Explain what is wrong with each question and rewrite it. (a) "Is social media bad?" (b) "How many people in Canada like poutine?" (c) "What is the probability of winning at a casino?"

(a) Vague and value-laden. "Bad" for what? Measured how? Rewrite: "Is the number of hours Grade 11 and 12 students at my school spend on social media on a school day related to their usual hours of sleep?"
(b) Not feasible as stated (you cannot survey Canada), and "like" is unclear. Rewrite as a secondary-data question about something measured, or narrow it: "What proportion of students at my school choose poutine when it is on the cafeteria menu?" (using sales counts the cafeteria already keeps, with permission).
(c) Too broad (which game?) and it invites a gambling activity. Rewrite as a probability problem you can model and simulate safely: "In a dice game I design for a school carnival, what is the expected number of prize tickets paid out per play, and does the game make a profit for the carnival over 300 plays?"

Good questions are specific (population, variables, measurement), feasible and ethical. Each rewrite says exactly what data would answer it.

Worked example

Example 3: Statistics, probability or both?

Classify each project. (a) Comparing the mean July temperatures in Ottawa from 1970 to 2020 using Environment and Climate Change Canada data. (b) Finding whether a new card game you invented is fair. (c) Surveying students about how many hours they practise music, then using a normal model to estimate the percentage who practise more than 5 hours a week.

(a) Data about a real-world variable over time: statistics (secondary data).
(b) A random process with outcomes you can count: probability (theory with counting, plus primary data from playing many rounds).
(c) Survey data summarized with statistics, then a probability model (the normal distribution) applied to the data: both.

(a) statistics, (b) probability, (c) both.

Choose a question you care about

You will live with your question for several weeks. Pick something you are curious about and can explain to a friend in one sentence. Before you commit, write down the data you would need and where you would get them. If you cannot name a data source, the question is not ready yet.

Just for fun

A statistician's favourite way to start a conversation: "Interesting question. How would you measure that?" (It works at parties about 30% of the time, with a large margin of error.)

Hands-on activity

Your investigation, step 1: brainstorm and narrow

You need: paper or a notebook, and 20 minutes. A family member or friend to talk to is helpful.

List six topics you are curious about (sports, music, money, weather, games, school, transportation, food...).
For each, write one issue that starts with "Does...", "How much..." or "Is there a relationship between...".
Pick your two favourites. For each, write an investigable question that names the population, the variables and the comparison or relationship.
Run the five tests (significant, quantitative, measurable, feasible, ethical) on each. Note any problems.
Explain each question to your family member in one sentence. Ask: "Would you want to know the answer?" and "What would you guess?"
Choose one question (or one statistics question and one probability question) and write it at the top of a new page: this is your project log.

What to look for: a final question in which a reader can tell exactly what data you will collect. Example: "Among Ontario's 49 census divisions, is the median age related to the percentage of people who commute by public transit?" (secondary data from the census).

Your turn

What two options does the curriculum allow for the problem in the culminating investigation?

Is "basketball" a topic, an issue or an investigable question? Explain.

Rewrite "Do teens text too much?" as an investigable question.

Name the population and the variables in: "Is the number of goals a team scores in a season related to its number of wins, for all teams in the National Hockey League in the 2024-25 season?"

Why is "What is the best pizza topping?" not a good culminating question?

Give one example of a probability problem you could investigate with primary data.

What does "feasible" mean for an investigation?

More practice: warm-up

True or false: a good question has an obvious answer before you start.

Which is quantitative: hours of sleep, or favourite colour?

Name one reliable source of secondary data in Canada.

In Amira's question, what is the population?

What should go at the top of your project log?

More practice: on your own

A student asks, "Do people who drink coffee get better marks?" List two things that must be made precise.

Explain why "Do students cheat on tests?" raises ethical problems as a survey question, and suggest a safer approach.

Amira could have asked "Does working cause students to sleep less?" Why is her chosen wording ("sleep less", "associated with") better for a survey study?

Classify as statistics, probability or both: "Using five years of Statistics Canada data, estimate the probability that a randomly chosen Ontario resident aged 15 or older was employed."

A student wants to survey every student in Canada aged 15 to 19. Suggest a feasible version of the same question.

Challenge

Write a single culminating question that needs both a probability model and a statistical analysis of real data, and say which tools you would use.

A student's question is "Is the average house price in Ontario too high?" Explain why it fails the tests and transform it into two different investigable questions, one comparing groups and one about a relationship.

Explain the trade-off between a question that is very significant and one that is very feasible, using an example.

Check your understanding

Quiz

Which of these is an investigable question?

The curriculum allows the culminating problem to be:

A question is feasible when:

In Amira's investigation, "usual hours of sleep on a school night" is:

Which project is mainly a probability problem?

Why is "Do students cheat?" a poor survey question for a student project?

For parents and teachers

Big idea (E1.1): a culminating investigation starts with a significant problem that real data can answer. The hardest and most important step is turning a broad topic into a precise question that names a population, variables and an analysis.

How to help: Be the friendly listener. Ask your teen to explain their question in one sentence, then ask: "Who exactly is this about? How would you measure that? Where will the data come from? Is there any way this could make someone uncomfortable?" Resist offering your own topic; curiosity is the engine of the project.

About the model investigation: the student "Amira" and Pinecrest Secondary School are fictional, and all of the data shown in this unit are made up for teaching. The calculations are real, so your teen can check every number.

Common difficulties:

Questions that are opinions ("best", "too much", "bad").
Questions that are too big to be feasible ("all Canadians").
Choosing a sensitive topic (health, money, family) for a self-made survey; steer these toward published secondary data.
Wording that claims cause ("does X cause Y?") when the data will only show association.