Ontario · Grade 12
Mathematics of Data Management, Grade 12 (University)
- MDM4U
Ontario’s Grade 12 math course, from Counting and Probability to Culminating Data Management Investigation.
- Units
- 5
- Lessons
- 11
- Skills
- 57
- Provincial exam
- None
- Official curriculum
- Ontario Ministry of Education
Unit by unit
Every skill in the course, numbered by unit.
Unit 1 · 12 skills
Counting and Probability
Solving Probability Problems Involving Discrete Sample Spaces
Probabilities represent the likelihood of a result
I can describe how probabilities represent the likelihood of a result or a real-world event.
Skill 1.1
Discrete versus continuous sample spaces
I can distinguish a discrete sample space from a continuous one.
Skill 1.2
Theoretical probability and the probability distribution summing to 1
I can determine theoretical probabilities and recognize that they sum to 1.
Skill 1.3
Experimental probability approaching theoretical probability
I can show that experimental probability approaches theoretical probability as trials increase.
Skill 1.4
Complements and mutually exclusive events
I can determine the complement of an event and identify mutually exclusive events.
Skill 1.5
Independent, dependent, and conditional events
I can determine whether two events are independent, dependent, or conditional, and solve related problems.
Skill 1.6
Solving Problems Using Counting Principles
Permutations vs. combinations: when order matters
I can distinguish situations that need permutations from situations that need combinations, based on whether order matters.
Skill 1.7
Calculating permutations and combinations and connecting them
I can calculate permutations and combinations and connect them to each other.
Skill 1.8
Solving counting problems with standard combinatorial notation
I can solve counting problems using standard combinatorial notation.
Skill 1.9
The additive and multiplicative counting principles
I can apply the additive and multiplicative counting principles.
Skill 1.10
Connecting combinations and Pascal's triangle
I can connect combinations to Pascal's triangle.
Skill 1.11
Solving probability problems using counting principles
I can solve probability problems using counting principles for equally likely outcomes.
Skill 1.12
The official wording · 12 outcomes
- 1.1
- recognize and describe how probabilities are used to represent the likelihood of a result of an experiment (e.g., spinning spinners; draw- ing blocks from a bag that contains different- coloured blocks; playing a game with number cubes; playing Aboriginal stick-and-stone games) and the likelihood of a real-world event (e.g., that it will rain tomorrow, that an accident will occur, that a product will be defective)
- 1.2
- describe a sample space as a set that contains all possible outcomes of an experiment, and distinguish between a discrete sample space as one whose outcomes can be counted (e.g., all possible outcomes of drawing a card or tossing a coin) and a continuous sample space as one whose outcomes can be measured (e.g., all possible outcomes of the time it takes to complete a task or the maximum distance a ball can be thrown)
- 1.3
- determine the theoretical probability, P (i.e., a value from 0 to 1), of each outcome of a discrete sample space (e.g., in situations in which all outcomes are equally likely), recognize that the sum of the probabilities of the outcomes is 1 (i.e., for n outcomes, P + P + P + … + P = 1), recognize that the probabilities P form the probability distribu- tion associated with the sample space, and solve related problems
- 1.4
- determine, through investigation using class- generated data and technology-based simula- tion models (e.g., using a random-number generator on a spreadsheet or on a graphing calculator; using dynamic statistical software to simulate repeated trials in an experiment), the tendency of experimental probability to approach theoretical probability as the num- ber of trials in an experiment increases
- 1.5
- recognize and describe an event as a set of outcomes and as a subset of a sample space, determine the complement of an event, deter- mine whether two or more events are mutual- ly exclusive or non-mutually exclusive (e.g., the events of getting an even number or getting an odd number of heads from tossing a coin 5 times are mutually exclusive), and solve related probability problems [e.g., cal- culate P(~A), P(A and B), P(A or B)] using a variety of strategies (e.g., Venn diagrams, lists, formulas)
- 1.6
- determine whether two events are indepen- dent or dependent, and whether one event is conditional on another event, and solve related probability problems [e.g., calculate P(A and B), P(A or B), P(A given B)] using a variety of strategies (e.g., tree diagrams, lists, formulas)
- 1.7
- recognize the use of permutations and combi- nations as counting techniques with advan- tages over other counting techniques (e.g., making a list; using a tree diagram; making a chart; drawing a Venn diagram), distinguish between situations that involve the use of per- mutations and those that involve the use of combinations (e.g., by considering whether or not order matters)
- 1.8
- and make connections between, and calculate, permutations and combinations Sample problem: An organization with 10 members is considering two leadership models. One involves a steering committee with 4 members of equal standing. The other is an executive committee consisting of a president, vice-president, secretary, and treasurer. Determine the number of ways of selecting the executive committee from the 10 members and, using this number, the number of ways of selecting the steering committee from the 10 members.
- 1.10
- solve introductory counting problems involv- ing the additive counting principle (e.g., determining the number of ways of selecting 2 boys or 2 girls from a group of 4 boys and 5 girls) and the multiplicative counting princi- ple (e.g., determining the number of ways of selecting 2 boys and 2 girls from a group of 4 boys and 5 girls)
- 1.12
- solve probability problems using counting principles for situations involving equally likely outcomes Sample problem: Two marbles are drawn randomly from a bag containing 12 green marbles and 16 red marbles. What is the probability that the two marbles are both green if the first marble is replaced? If the first marble is not replaced?
Key ideas
A probability is a number between 0 and 1 that measures how likely an outcome is — 0 means impossible, 1 means certain.
A permutation counts arrangements where ORDER matters (like electing a president vs. vice-president); a combination counts selections where order does NOT matter (like choosing a committee of equals).
A discrete sample space has outcomes you can count (like the faces of a die); a continuous sample space has outcomes you can only measure (like time or distance).
A combination is a permutation divided by the number of ways to arrange the chosen group among itself (r!) — permutations and combinations are directly related.
The theoretical probability of each outcome in an equally-likely sample space is 1 divided by the number of outcomes.
Watch out for
- Thinking a probability of 0.5 means an event is 'random' rather than exactly equally likely to happen or not.
- Using a combination formula for a problem where the roles are distinct and order actually matters, or vice versa.
- Assuming every sample space involving numbers must be discrete.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 2 · 15 skills
Probability Distributions
Understanding Probability Distributions for Discrete Random Variables
Discrete random variables and generating a probability distribution
I can identify a discrete random variable and generate its probability distribution.
Skill 2.1
Calculating and interpreting expected value
I can calculate and interpret the expected value of a probability distribution.
Skill 2.2
Representing a distribution with a probability histogram
I can represent a probability distribution using a probability histogram and connect it to a frequency histogram.
Skill 2.3
The binomial probability distribution
I can recognize the conditions for a binomial distribution and calculate its probabilities.
Skill 2.4
The hypergeometric probability distribution
I can recognize the conditions for a hypergeometric distribution and calculate its probabilities.
Skill 2.5
Comparing discrete probability distributions
I can compare discrete probability distributions using numeric and graphical representations.
Skill 2.6
Solving real-world problems with probability distributions
I can solve real-world problems involving probability distributions.
Skill 2.7
Understanding Probability Distributions for Continuous Random Variables
Continuous random variables and discrete vs. continuous frequency distributions
I can identify a continuous random variable and distinguish discrete from continuous frequency distributions.
Skill 2.8
Standard deviation as a measure of spread
I can determine the mean and standard deviation of a sample of a continuous random variable.
Skill 2.9
Representing continuous data with frequency tables, histograms, and polygons
I can represent continuous data using frequency tables, histograms, and frequency polygons.
Skill 2.10
Challenges in determining a continuous frequency distribution
I can describe challenges in determining a continuous frequency distribution.
Skill 2.11
Theoretical probability over a range for a continuous variable
I can explain that theoretical probability for a continuous variable is determined over a range, not a single value.
Skill 2.12
Properties of the normal distribution
I can describe the properties of the normal distribution and situations it models.
Skill 2.13
Connecting the normal distribution to binomial and hypergeometric distributions
I can connect the normal distribution to the binomial and hypergeometric distributions for increasing trials.
Skill 2.14
Using z-scores to solve normal distribution problems
I can use z-scores to solve probability problems involving normal distributions.
Skill 2.15
The official wording · 15 outcomes
- 2.1
- recognize and identify a discrete random vari- able X (i.e., a variable that assumes a unique value for each outcome of a discrete sample space, such as the value x for the outcome of getting x heads in 10 tosses of a coin), gener- ate a probability distribution [i.e., a function that maps each value x of a random variable X to a corresponding probability, P(X = x)] by calculating the probabilities associated with all values of a random variable, with and without technology, and represent a probabil- ity distribution numerically using a table
- 2.2
- calculate the expected value for a given probability distribution [i.e., using E(X) = ∑ xP(X = x)], interpret the expected value in applications, and make connections between the expected value and the weighted mean of the values of the discrete random variable
- 2.3
- represent a probability distribution graphical- ly using a probability histogram (i.e., a histo- gram on which each rectangle has a base of width 1, centred on the value of the discrete random variable, and a height equal to the probability associated with the value of the random variable), and make connections between the frequency histogram and the probability histogram (e.g., by comparing their shapes)
- 2.5
- recognize conditions (e.g., dependent trials) that give rise to a random variable that fol- lows a hypergeometric probability distribu- tion, calculate the probability associated with each value of the random variable (e.g., by using a tree diagram; by using combinations), and represent the distribution numerically using a table and graphically using a proba- bility histogram
- 2.6
- compare, with technology and using numeric and graphical representations, the probability distributions of discrete random variables (e.g., compare binomial distributions with the same probability of success for increasing numbers of trials; compare the shapes of a hypergeometric distribution and a binomial distribution)
- 2.7
- solve problems involving probability distri- butions (e.g., uniform, binomial, hypergeo- metric), including problems arising from real-world applications
- 2.8
- recognize and identify a continuous random variable (i.e., a variable that assumes values from the infinite number of possible outcomes in a continuous sample space), and distinguish between situations that give rise to discrete frequency distributions (e.g., counting the number of outcomes for drawing a card or tossing three coins) and situations that give rise to continuous frequency distributions (e.g., measuring the time taken to complete a task or the maximum distance a ball can be thrown)
- 2.9
- recognize standard deviation as a measure of the spread of a distribution, and determine, with and without technology, the mean and standard deviation of a sample of values of a continuous random variable
- 2.10
- represent, using intervals, a sample of values of a continuous random variable numerically using a frequency table and graphically using a frequency histogram and a frequency poly- gon, recognize that the frequency polygon approximates the frequency distribution, and determine, through investigation using tech- nology (e.g., dynamic statistical software, graphing calculator), and compare the effec- tiveness of the frequency polygon as an approximation of the frequency distribution for different sizes of the intervals
- 2.11
- describe challenges associated with determin- ing a continuous frequency distribution (e.g., the inability to capture all values of the vari- able, resulting in a need to sample; uncer- tainties in measured values of the variable), and recognize the need for mathematical models to represent continuous frequency distributions
- 2.12
- recognize that theoretical probability for a continuous random variable is determined over a range of values (e.g., the probability that the life of a lightbulb is between 90 hours and 115 hours), that the probability that a continuous random variable takes any single value is zero, and that the probabilities of ranges of values form the probability distri- bution associated with the random variable
- 2.13
- recognize that the normal distribution is commonly used to model the frequency and probability distributions of continuous ran- dom variables, describe some properties of the normal distribution (e.g., the curve has a central peak; the curve is symmetric about the mean; the mean and median are equal; approximately 68% of the data values are within one standard deviation of the mean and approximately 95% of the data values are within two standard deviations of the mean), and recognize and describe situations that can be modelled using the normal distribution
- 2.14
- make connections, through investigation using dynamic statistical software, between the normal distribution and the binomial and hypergeometric distributions for increasing numbers of trials of the discrete distributions (e.g., recognizing that the shape of the hyper- geometric distribution of the number of males on a 4-person committee selected from a group of people more closely resembles the shape of a normal distribution as the size of the group from which the committee was drawn increases)
- 2.15
- recognize a z-score as the positive or negative number of standard deviations from the mean to a value of the continuous random variable, and solve probability problems involving normal distributions using a variety of tools and strategies (e.g., calculating a z-score and reading a probability from a table; using tech- nology to determine a probability), including problems arising from real-world applications
Key ideas
A discrete random variable assigns a number to each outcome of a discrete sample space (like 'number of heads' in 10 coin tosses).
A continuous random variable can take any value in a range (like time or height), unlike a discrete variable's countable values.
A probability distribution maps each value of the random variable to its probability.
Standard deviation measures how spread out the data values are around the mean — a small standard deviation means values cluster tightly.
Expected value E(X) = the sum of each value times its probability — it's the long-run average outcome, and connects to the weighted mean.
Watch out for
- Confusing the random variable itself (a number) with the outcome it describes (an event).
- Treating rounded continuous measurements (like height to the nearest cm) as if they were truly discrete.
- Expecting the expected value to always equal one of the possible outcomes, rather than a weighted average that might not match any single outcome.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 3 · 8 skills
Organization of Data for Analysis
Understanding Data Concepts
The role of data in statistical studies
I can describe the role of data in statistical studies and applications.
Skill 3.1
Why variability is inherent in data, and one-variable vs. multi-variable situations
I can explain why variability is inherent in data and distinguish one-variable from multi-variable situations.
Skill 3.2
Distinguishing types of statistical data
I can distinguish different types of statistical data and give examples of each.
Skill 3.3
Collecting and Organizing Data
Principles of primary data collection
I can describe principles of primary data collection and criteria for reliable data.
Skill 3.4
Population vs. sample, and sampling techniques
I can distinguish population from sample and describe sampling techniques.
Skill 3.5
How sample bias affects study results
I can describe how sample bias affects the results of a study.
Skill 3.6
Designing effective surveys and experiments
I can describe characteristics of an effective survey and design a questionnaire or experiment.
Skill 3.7
Collecting and organizing data from primary and secondary sources
I can collect data from primary or secondary sources and organize it to answer a question.
Skill 3.8
The official wording · 8 outcomes
- 3.1
- recognize and describe the role of data in statistical studies (e.g., the use of statistical techniques to extract or mine knowledge of relationships from data), describe examples of applications of statistical studies (e.g., in medical research, political decision making, market research), and recognize that conclu- sions drawn from statistical studies of the same relationship may differ
- 3.2
- recognize and explain reasons why variability is inherent in data (e.g., arising from limited accuracy in measurement or from variations in the conditions of an experiment; arising from differences in samples in a survey), and distinguish between situations that involve one variable and situations that involve more than one variable
- 3.3
- distinguish different types of statistical data (i.e., discrete from continuous, qualitative from quantitative, categorical from numerical, nominal from ordinal, primary from secondary, experimental from observational, microdata from aggregate data) and give examples
- 3.4
- determine and describe principles of primary data collection (e.g., the need for randomiza- tion, replication, and control in experimental studies; the need for randomization in sample surveys) and criteria that should be consid- ered in order to collect reliable primary data (e.g., the appropriateness of survey questions; potential sources of bias; sample size)
- 3.5
- explain the distinction between the terms population and sample, describe the character- istics of a good sample, explain why sampling is necessary (e.g., time, cost, or physical con- straints), and describe and compare some sampling techniques (e.g., simple random, systematic, stratified, convenience, voluntary)
- 3.6
- describe how the use of random samples with a bias (e.g., response bias, measurement bias, non-response bias, sampling bias) or the use of non-random samples can affect the results of a study
- 3.7
- describe characteristics of an effective survey (e.g., by giving consideration to ethics, priva- cy, the need for honest responses, and possi- ble sources of bias, including cultural bias), and design questionnaires (e.g., for determin- ing if there is a relationship between a person’s age and their hours per week of Internet use, between marks and hours of study, or between income and years of education) or experiments (e.g., growth of plants under different condi- tions) for gathering data
- 3.8
- collect data from primary sources, through experimentation, or from secondary sources (e.g., by using the Internet to access reliable data from a well-organized database such as E-STAT; by using print sources such as news- papers and magazines), and organize data with one or more attributes (e.g., organize data about a music collection classified by artist, date of recording, and type of music using dynamic statistical software or a spreadsheet) to answer a question or solve a problem
Key ideas
Statistical techniques extract knowledge and relationships from data across fields like medicine, politics, and market research.
Randomization, replication, and control are key principles for gathering reliable primary data.
Different studies of the same relationship can reach different conclusions.
A population is the entire group of interest; a sample is a smaller subset used to make inferences about the population.
Variability arises naturally from measurement limits, experimental conditions, and sample differences.
Watch out for
- Assuming a single statistical study provides a final, unquestionable answer.
- Assuming a large sample size alone guarantees reliable data, regardless of how it was collected.
- Assuming that if a sample doesn't match the population exactly, something went wrong with the sampling.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 4 · 13 skills
Statistical Analysis
Analysing One-Variable Data
Numerical summaries of one-variable data
I can determine numerical summaries of one-variable data.
Skill 4.1
Positions within a data set using quartiles, percentiles, and z-scores
I can determine the position of a data point using quartiles, percentiles, and z-scores.
Skill 4.2
Graphical summaries of one-variable data
I can generate graphical summaries of one-variable data appropriate to the data type.
Skill 4.3
Margin of error and confidence level
I can interpret margin of error and confidence level, and connect them to sample size.
Skill 4.4
Interpreting, comparing, and concluding from one-variable statistical summaries
I can interpret, compare, and draw justified conclusions from one-variable statistical summaries.
Skill 4.5
Analysing Two-Variable Data
Two-variable data, correlation coefficient, and numerical summaries
I can determine numerical summaries for two-variable data, including the correlation coefficient.
Skill 4.6
Types of relationships between two variables
I can distinguish cause-and-effect, common-cause, and accidental relationships between two variables.
Skill 4.7
Graphical summaries of two-variable data
I can generate graphical summaries of two-variable data appropriate to the data type.
Skill 4.8
Linear regression and the fit of an individual data point
I can perform linear regression and determine the fit of an individual data point.
Skill 4.9
Interpreting, comparing, and concluding from two-variable statistical summaries
I can interpret, compare, and draw justified conclusions from two-variable statistical summaries.
Skill 4.10
Evaluating Validity
How the media and advertising use and misuse statistics
I can explain how the media and advertising industry use and misuse statistics.
Skill 4.11
Assessing the validity of conclusions presented in the media
I can assess the validity of conclusions presented in the media.
Skill 4.12
Applications of data management in occupations and university programs
I can describe applications of data management in occupations and university programs.
Skill 4.13
The official wording · 13 outcomes
- 4.1
- recognize that the analysis of one-variable data involves the frequencies associated with one attribute, and determine, using technol- ogy, the relevant numerical summaries (i.e., mean, median, mode, range, interquartile range, variance, and standard deviation)
- 4.2
- determine the positions of individual data points within a one-variable data set using quartiles, percentiles, and z-scores, use the normal distribution to model suitable one- variable data sets, and recognize these processes as strategies for one-variable data analysis
- 4.3
- generate, using technology, the relevant graphical summaries of one-variable data (e.g., circle graphs, bar graphs, histograms, stem-and-leaf plots, boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)
- 4.4
- interpret, for a normally distributed popula- tion, the meaning of a statistic qualified by a statement describing the margin of error and the confidence level (e.g., the meaning of a statistic that is accurate to within 3 percentage points, 19 times out of 20), and make connec- tions, through investigation using technology (e.g., dynamic statistical software), between the sample size, the margin of error, and the confidence level (e.g., larger sample sizes create higher confidence levels for a given margin of error)
- 4.5
- interpret statistical summaries (e.g., graphical, numerical) to describe the characteristics of a one-variable data set and to compare two related one-variable data sets (e.g., compare the lengths of different species of trout; compare annual incomes in Canada and in a third-world country; compare Aboriginal and non-Aboriginal incomes); describe how statis- tical summaries (e.g., graphs, measures of central tendency) can be used to misrepresent one-variable data; and make inferences, and make and justify conclusions, from statistical summaries of one-variable data orally and in writing, using convincing arguments
- 4.6
- recognize that the analysis of two-variable data involves the relationship between two attributes, recognize the correlation coefficient as a measure of the fit of the data to a linear model, and determine, using technology, the relevant numerical summaries (e.g., summary tables such as contingency tables; correlation coefficients)
- 4.7
- recognize and distinguish different types of relationships between two variables that have a mathematical correlation (e.g., the cause- and-effect relationship between the age of a tree and its diameter; the common-cause rela- tionship between ice cream sales and forest fires over the course of a year; the accidental relationship between the consumer price index and the number of known planets in the universe)
- 4.8
- generate, using technology, the relevant graphical summaries of two-variable data (e.g., scatter plots, side-by-side boxplots) based on the type of data provided (e.g., categorical, ordinal, quantitative)
- 4.9
- determine, by performing a linear regression using technology, the equation of a line that models a suitable two-variable data set, deter- mine the fit of an individual data point to the linear model (e.g., by using residuals to iden- tify outliers), and recognize these processes as strategies for two-variable data analysis
- 4.10
- interpret statistical summaries (e.g., scatter plot, equation representing a relationship) to describe the characteristics of a two- variable data set and to compare two related two-variable data sets (e.g., compare the relationship between Grade 12 English and mathematics marks with the relationship between Grade 12 science and mathematics marks); describe how statistical summaries (e.g., graphs, linear models) can be used to misrepresent two-variable data; and make inferences, and make and justify conclusions, from statistical summaries of two-variable data orally and in writing, using convincing arguments
- 4.11
- interpret statistics presented in the media (e.g., the UN’s finding that 2% of the world’s population has more than half the world’s wealth, whereas half the world’s population has only 1% of the world’s wealth), and explain how the media, the advertising indus- try, and others (e.g., marketers, pollsters) use and misuse statistics (e.g., as represented in graphs) to promote a certain point of view
- 4.12
- assess the validity of conclusions presented in the media by examining sources of data, including Internet sources (i.e., to determine whether they are authoritative, reliable, unbiased, and current), methods of data collection, and possible sources of bias (e.g., sampling bias, non-response bias, cultural bias in a survey question), and by questioning the analysis of the data (e.g., whether there is any indication of the sample size in the analysis) and conclusions drawn from the data (e.g., whether any assumptions are made about cause and effect)
- 4.13
- gather, interpret, and describe information about applications of data management in occupations (e.g., actuary, statistician, busi- ness analyst, sociologist, medical doctor, psychologist, teacher, community planner), and about university programs that explore these applications
Key ideas
Mean, median, mode, range, interquartile range, variance, and standard deviation are the standard numerical summaries of a single attribute's data.
The correlation coefficient measures how well a linear model fits two-variable data, ranging from -1 (perfect negative) to +1 (perfect positive).
Statistics can be manipulated through misleading graph scales, weak correlations presented as strong, or cherry-picked data.
Quartiles, percentiles, and z-scores each describe where one specific data point sits relative to the rest of the data set.
A mathematical correlation between two variables does NOT automatically mean one causes the other — it could be common-cause or purely accidental.
Watch out for
- Using the mean as the single best summary regardless of whether the data is skewed (where the median may be more representative).
- Interpreting a correlation coefficient near zero as meaning 'no relationship at all,' when it may mean 'no LINEAR relationship' (a curved one could still exist).
- Assuming a statistic reported in a credible-seeming source is automatically presented fairly.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Unit 5 · 9 skills
Culminating Data Management Investigation
Designing and Carrying Out a Culminating Investigation
Stage 1 — Posing a significant problem and doing background research
I can pose a significant, data-answerable problem and conduct background research on it.
Skill 5.1
Stage 2 — Designing a plan to study the problem
I can design a plan to study my problem, including variables, population, and data-collection procedures.
Skill 5.2
Stage 3 — Gathering and organizing investigation data
I can gather and organize the data for my investigation.
Skill 5.3
Stage 4 — Interpreting, analysing, and summarizing the data
I can interpret, analyse, and summarize the data from my investigation.
Skill 5.4
Stage 5 — Drawing conclusions and evaluating the strength of the evidence
I can draw conclusions from my analysis, evaluate the evidence's strength, and identify limitations.
Skill 5.5
Stage 6 — Compiling a clear, well-organized, detailed report
I can compile a clear, well-organized, and detailed report of my investigation.
Skill 5.6
Presenting and Critiquing the Culminating Investigation
Stage 7 — Presenting the investigation summary to peers
I can present a summary of my culminating investigation to an audience of peers within a time limit.
Skill 5.7
Stage 8 — Answering questions and responding to critiques
I can answer questions about my investigation and respond to critiques.
Skill 5.8
Stage 9 — Critiquing the mathematical work of others constructively
I can critique the mathematical work of others in a constructive manner.
Skill 5.9
The official wording · 9 outcomes
- 5.1
- pose a significant problem of interest that requires the organization and analysis of a suitable set of primary or secondary quantita- tive data (e.g., primary data collected from a student-designed game of chance, secondary data from a reliable source such as E-STAT), and conduct appropriate background research related to the topic being studied
- 5.2
- design a plan to study the problem (e.g., iden- tify the variables and the population; develop an ethical survey; establish the procedures for gathering, summarizing, and analysing the primary or secondary data; consider the sam- ple size and possible sources of bias)
- 5.3
- gather data related to the study of the problem (e.g., by using a survey; by using the Internet; by using a simulation) and organize the data (e.g., by setting up a database; by establishing intervals), with or without technology
- 5.4
- interpret, analyse, and summarize data related to the study of the problem (e.g., generate and interpret numerical and graphical statistical summaries; recognize and apply a probability distribution model; calculate the expected value of a probability distribution), with or without technology
- 5.5
- draw conclusions from the analysis of the data (e.g., determine whether the analysis solves the problem), evaluate the strength of the evidence (e.g., by considering factors such as sample size or bias, or the number of times a game is played), specify any limitations of the conclusions, and suggest follow-up pro- blems or investigations
- 5.6
- compile a clear, well-organized, and detailed report of the investigation
- 5.7
- present a summary of the culminating investi- gation to an audience of their peers within a specified length of time, with technology (e.g. presentation software) or without technology
- 5.8
- answer questions about the culminating inves- tigation and respond to critiques (e.g., by elaborating on the procedures; by justifying mathematical reasoning)
- 5.9
- critique the mathematical work of others in a constructive manner
Key ideas
A strong investigation question requires organizing and analysing real quantitative data — not just an opinion question.
A good presentation distills the full investigation into its key problem, method, findings, and conclusion within the given time.
Background research grounds the investigation in what's already known about the topic.
Responding well to critique means elaborating on your procedures and justifying your mathematical reasoning, not just defending the conclusion.
A good plan identifies the variables and population up front, and anticipates sample size and bias before any data is collected.
Watch out for
- Choosing a question that's really an opinion question rather than one requiring genuine data analysis.
- Trying to present every detail of the full report instead of the key highlights within the time limit.
- Jumping straight to data collection without first identifying what population and variables the investigation targets.
Every skill’s key ideas, worked examples and checks are in MapleMind.Practise this unit
Plan the term
Pick your term and the days you study. Each unit gets its share of the weeks and comes back a week later for a review. The last two weeks go over the whole course.
Which course?
Ontario has 6 math courses for Grade 12. Here they are side by side.
- Advanced Functions, Grade 12 (University)MHF4U4 units · 72 skillsNo provincial examWhat it coversExponential and Logarithmic Functions · Trigonometric Functions · Polynomial and Rational Functions · Characteristics of Functions
- Calculus and Vectors, Grade 12 (University)MCV4U3 units · 54 skillsNo provincial examWhat it coversRate of Change · Derivatives and Their Applications · Geometry and Algebra of Vectors
- Foundations for College Mathematics, Grade 12 (College)MAP4C4 units · 59 skillsNo provincial examWhat it coversMathematical Models · Personal Finance · Geometry and Trigonometry · Data Management
- Mathematics for College Technology, Grade 12 (College)MCT4C4 units · 57 skillsNo provincial examWhat it coversExponential Functions · Polynomial Functions · Trigonometric Functions · Applications of Geometry
- Mathematics for Work and Everyday Life, Grade 12 (Workplace)MEL4E3 units · 52 skillsNo provincial examWhat it coversReasoning With Data · Personal Finance · Applications of Measurement
- Mathematics of Data Management, Grade 12 (University)This courseMDM4U5 units · 57 skillsNo provincial examWhat it coversCounting and Probability · Probability Distributions · Organization of Data for Analysis · Statistical Analysis · Culminating Data Management Investigation
What a university, college or program asks for is theirs to say: check its admission page, and your school’s course guide.
The official curriculum
This page follows Ontario’s own curriculum for Mathematics of Data Management, Grade 12 (University).
Textbooks
The textbooks Ontario Ministry of Education lists for Mathematics of Data Management, Grade 12 (University):
- McGraw-Hill Ryerson Data Management 12
- McGraw-Hill Ryerson Mathematics: Mathematics of Data Management
From the Trillium List, read October 27, 2026. ontario.ca
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Course Companion
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Questions about Mathematics of Data Management, Grade 12 (University)
What is Mathematics of Data Management, Grade 12 (University)?
Ontario’s Grade 12 math course: 5 units and 57 skills, from Counting and Probability to Culminating Data Management Investigation.
What units are in Mathematics of Data Management, Grade 12 (University)?
1. Counting and Probability, 2. Probability Distributions, 3. Organization of Data for Analysis, 4. Statistical Analysis and 5. Culminating Data Management Investigation.
Is there a provincial exam for Mathematics of Data Management, Grade 12 (University)?
No. Ontario gives no provincial exam for Mathematics of Data Management, Grade 12 (University).
What textbook does Mathematics of Data Management, Grade 12 (University) use?
The textbooks Ontario Ministry of Education lists for Mathematics of Data Management, Grade 12 (University): McGraw-Hill Ryerson Data Management 12, McGraw-Hill Ryerson and McGraw-Hill Ryerson Mathematics: Mathematics of Data Management, McGraw-Hill Ryerson.
Can MapleMind help with Mathematics of Data Management, Grade 12 (University)?
Yes. MapleMind teaches every skill of Mathematics of Data Management, Grade 12 (University), step by step, with practice and checks, and its tutor answers questions about any of them. It’s free with daily limits, on iPhone, Android and the web.
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