Mathematics of Data Management

This unit opens Mathematics of Data Management (MDM4U) with the mathematics of chance. You will start with what a probability means: a number from 0 to 1 that can come from reasoning (theoretical), from data (experimental) or from judgment (subjective), and you will learn to express chance as odds. You will describe sample spaces with lists, tables and tree diagrams, tell discrete sample spaces from continuous ones, build probability distributions (such as the sum of two dice) and check that they add to 1. Then you will see the law of large numbers at work in experiments and design your own simulations with dice, random digits and spreadsheets.

The second half treats events as sets. You will use set notation, complements (including the powerful "at least one" strategy), Venn diagrams with two and three sets, mutually exclusive and non-mutually exclusive events and the addition rules, independent and dependent events with tree diagrams, and conditional probability from formulas and two-way tables. The last lesson combines everything in multi-step problems, including why an accurate test for a rare condition can still give many false positives.

Remember from earlier grades: in Grades 7 and 8 you compared theoretical and experimental probability, used tree diagrams for independent and dependent events, and found P(A or B) with Venn diagrams. This unit makes those ideas precise and takes them much further. You will need a scientific calculator; a spreadsheet program (or a free online one) is helpful for simulations but not required. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Coming next: Permutations and Combinations gives you fast counting methods for large sample spaces, and Discrete Probability Distributions builds on the distributions introduced here.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand A. Counting and Probability, overall expectation A1 (solving problems involving the probability of an event or a combination of events for discrete sample spaces): specific expectations A1.1 to A1.6. Overall expectation A2 (counting principles) is taught in Permutations and Combinations. Odds are not named in the expectations but are included as a standard application of A1.1 and A1.3. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

How many different licence plates, lottery tickets, committees, card hands or routes to school are possible? In this unit you will learn to count without listing, and then use those counts to find probabilities.

You will start with the multiplicative and additive counting principles and indirect counting (total minus what you don't want). Then you will use factorial notation and permutations P(n, r) for arrangements where order matters, handle conditions (objects together or apart, restricted positions) and count arrangements with repetition and with identical objects. Next come combinations C(n, r) for selections where order does not matter, including committees, "at least" problems and hands of cards, and a lesson on deciding which tool fits. You will see why the entries of Pascal's triangle are combinations, count routes on a city grid in three ways, and use the binomial theorem. The last two lessons turn counts into probabilities for equally likely outcomes: marbles drawn with and without replacement, quality-control samples, lotteries and coin flips.

Remember from earlier courses: in Sequences and Series (Grade 11 Functions, MCR3U) you built Pascal's triangle, counted grid routes with it and expanded (a + b)ⁿ. In the unit Probability (just before this one) you met sample spaces, theoretical probability, complements and Venn diagrams. This unit explains why Pascal's triangle counts, and gives you the tools to find n(A) and n(S) when the sample space is far too large to list. You will need a scientific calculator with n!, nPr and nCr keys; every answer can also be reached by hand.

Coming next: in Discrete Probability Distributions you will use these combinations to build the binomial and hypergeometric distributions.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand A. Counting and Probability, overall expectation A2 (solving problems involving the application of permutations and combinations to determine the probability of an event): specific expectations A2.1 to A2.5. Overall expectation A1 is taught in the unit Probability. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you move from single probabilities to whole probability distributions. A random variable turns each outcome of a chance experiment into a number, such as the number of heads in 3 tosses or the number of defective bulbs in a package. You will build distribution tables, draw probability histograms and compare them with frequency histograms from real experiments, and find expected values, the long-run averages that tell you whether a game, a raffle or a lottery is fair and how businesses plan.

Then you meet three important models. The uniform distribution covers equally likely values. The binomial distribution counts successes in independent trials, such as free throws or defective parts. The hypergeometric distribution counts successes when sampling without replacement, such as cards in a hand or dead batteries in a sample. You will compare these models, simulate them with dice, cards, random digits and spreadsheets, and use them to solve real problems: hotel cancellations, airline overbooking, quality control, group testing and everyday decisions.

Builds on: the units Probability and Permutations and Combinations in this course (sample spaces, independence, and C(n, r)), and Pascal's triangle and the binomial expansion from Grade 11 Functions (MCR3U, Sequences and Series). Leads to: Continuous Distributions and the Normal Distribution, where large binomial and hypergeometric distributions are approximated by the normal curve.

You will need a scientific calculator; a spreadsheet (Google Sheets or Excel) is very helpful but every answer can be found by hand. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

In this unit you move from probabilities you can count to probabilities you must measure. You will start by telling discrete random variables (counted: 0, 1, 2, ...) apart from continuous ones (measured: time, mass, length), and you will learn the standard deviation, the measure of spread that the rest of the unit depends on, for raw and for grouped data. You will organize continuous data in frequency tables, histograms and frequency polygons, see how the class width changes the picture, and discover why continuous variables need mathematical models: smooth density curves with total area 1, where probability is area and any single exact value has probability 0.

The second half is about the most important model in statistics, the normal distribution. You will learn its properties, the empirical rule (68%, 95%, 99.7%), z-scores, and how to read a z-table forwards (probabilities) and backwards (percentiles, cut-offs, unknown means and standard deviations). Every lesson that needs the table includes the values it uses, and the lesson Using the z-Table to Find Normal Probabilities has the full table. You will see how binomial and hypergeometric distributions take on the normal shape as the number of trials grows, use the normal approximation with a continuity correction, and finish by solving real problems: maple seedlings, syrup bottles, warranties, quality control and survey claims.

Remember from earlier: the Grade 9 data unit (MTH1W) taught mean, median, range, quartiles, box plots and histograms. The units Probability, Permutations and Combinations and Discrete Probability Distributions in this course gave you random variables, expected value and the binomial and hypergeometric distributions. You will need a scientific calculator; a spreadsheet is helpful for checking but every answer can be found by hand with the z-table.

Coming next: Organization of Data for Analysis and One-Variable Data Analysis apply these tools to real data sets, and Two-Variable Data Analysis and Validity and the Culminating Data Management Investigation build on them.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand B. Probability Distributions, overall expectation B2 (understanding probability distributions for continuous random variables): specific expectations B2.1 to B2.8. Overall expectation B1 is taught in Discrete Probability Distributions. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

Every statistical study is only as good as its data. In this unit you will learn how statisticians collect and organize data before any analysis begins. You will see how data are used in medical research, government, business and data mining, and why two studies of the same question can disagree. You will learn why all data vary, how to classify data in many ways (categorical or numerical, nominal or ordinal, discrete or continuous, primary or secondary, experimental or observational, microdata or aggregate), and how to tell one-variable from multivariable situations.

The middle of the unit is about populations and samples: the census versus a sample, the sampling frame, five random sampling techniques (simple random, systematic, stratified, cluster and multistage) and the many kinds of bias (sampling, non-response, response, measurement and cultural). You will then design data collection properly: experiments built on control, randomization and replication (including block and matched-pairs designs, placebos and blinding), surveys with clear, fair questions that encourage honest answers, and research that is ethical, including the principles that guide research with First Nations, Inuit and Métis communities. Finally, you will organize data in tables, spreadsheets and databases, and find, judge, clean and cite secondary data from Statistics Canada and other reliable sources.

Remember from Grade 9: in Mathematics (MTH1W) you met large data sets and data sovereignty, primary and secondary data, populations and samples, some sampling methods and types of bias, and you planned a small data-collection project. This unit goes much deeper and adds the design of experiments, survey design, ethics and database skills.

How it fits: the sibling units One-Variable Data Analysis and Two-Variable Data Analysis and Validity analyse the kinds of data you learn to collect here, and the Culminating Data Management Investigation asks you to plan and carry out a study of your own. You will need a calculator and, for some activities, a free spreadsheet program (Google Sheets, LibreOffice Calc or Excel). Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand C. Organization of Data for Analysis: overall expectation C1 (understanding data concepts: C1.1 to C1.3) and overall expectation C2 (collecting and organizing data: C2.1 to C2.5). The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you will learn to analyse, interpret and draw conclusions from one-variable data: data about a single attribute, such as the lengths of trout in a lake, the commute times of a class or the answers to a poll. You will organize data into frequency distributions, choose graphs that suit the type of data (bar graphs, circle graphs, histograms, frequency polygons, ogives, stem-and-leaf plots and box plots), and describe the shape of a distribution. You will calculate and interpret measures of central tendency (mean, median, mode, weighted mean) and measures of spread (range, IQR, variance and standard deviation, for both populations and samples), by hand, from frequency tables and grouped data, and with a spreadsheet or graphing calculator. You will locate individual values with quartiles, percentiles, percentile ranks and z-scores, identify outliers, and use the normal distribution to model suitable data. Then you will see why different samples give different results, interpret statistics reported with a margin of error and confidence level ("accurate to within 3 percentage points, 19 times out of 20"), and find the sample size a poll needs. Finally, you will compare related data sets, write and present justified conclusions, and learn how statistical summaries and graphs can be used to misrepresent data.

Remember from Grade 9: in Principles of Mathematics (MTH1W), unit Data, Statistics and Mathematical Modelling, you found means, medians, modes, ranges, quartiles and IQRs, drew box plots, histograms and stem-and-leaf plots, and described shapes. This unit reviews those skills quickly and goes much further. You will need a scientific calculator; a free spreadsheet (such as Google Sheets) is very helpful. Every lesson has worked examples, a hands-on activity, plenty of practice with full answers and a short quiz.

Related units in this course: Organization of Data for Analysis (types of data, sampling and bias), Continuous Distributions and the Normal Distribution (the normal distribution as a probability model), Two-Variable Data Analysis and Validity (relationships between variables and statistics in the media) and the Culminating Data Management Investigation, where you will use all of these tools on a question of your own.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand D. Statistical Analysis, overall expectation D1 (analysing one-variable data using numerical and graphical summaries): specific expectations D1.1 to D1.5. Overall expectations D2 and D3 are taught in Two-Variable Data Analysis and Validity. The mathematical process expectations (problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing and communicating) are woven through every lesson.

In this unit you will study how two variables are related, and how to judge the claims people make about such relationships. You will describe scatter plots by form, direction and strength, then calculate the correlation coefficient r by hand, see why the formula works, and learn what r² tells you and when r misleads (curves, outliers, small samples, restricted ranges). For categorical data you will build contingency tables and compare conditional percentages; for a category and a number you will compare groups with side-by-side box plots.

Next you will fit the least-squares line with the formulas behind the technology, interpret its slope and intercept, and check it with residuals, residual plots, the standard error, outliers and influential points. You will make and judge predictions (interpolation and extrapolation) and compare related relationships, such as math marks with science marks and with English marks. You will sort correlations into types of relationships (cause and effect, reverse cause, common cause, accidental and presumed), meet lurking variables and Simpson's paradox, and learn why only a randomized experiment can settle questions of cause.

The last part turns these tools on the world around you: how the media and advertisers use and misuse statistics (truncated axes, percent vs percentage points, relative vs absolute risk, missing groups), a checklist for judging the validity of a claim (the curriculum's "Big Increase in Profits" example), and careers and university programs that use data management.

Remember from Grade 9: in Mathematics (MTH1W) you made scatter plots, found r and lines of best fit with technology, met residuals and learned that correlation does not prove causation. This unit adds the formulas, the reasoning behind them, and much more careful judgement. It builds on the sibling units Organization of Data for Analysis (types of data, sampling, bias) and One-Variable Data Analysis (box plots, standard deviation, margin of error), and uses expected value from Discrete Probability Distributions. You need a scientific calculator; Desmos or a spreadsheet is very helpful for checking. Every lesson has worked examples, diagrams, a hands-on activity, practice with full answers and a short quiz.

Coming next: in the Culminating Data Management Investigation you will pose your own question, gather data and analyse it with the tools of the whole course.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand D. Statistical Analysis: overall expectation D2 (analysing two-variable data), specific expectations D2.1 to D2.5, and overall expectation D3 (evaluating validity), specific expectations D3.1 to D3.3. Overall expectation D1 is taught in One-Variable Data Analysis. The mathematical process expectations are woven through every lesson.

This is the capstone of Mathematics of Data Management. You will design and carry out your own culminating investigation: pose a significant question, do background research and write hypotheses, plan a sample and an ethical survey (or find reliable secondary data, or design a game of chance), gather and clean the data, analyse it with the statistics and probability tools of the whole course, draw conclusions with honest limitations, and then write a report, give a timed presentation, answer questions and critique other students' investigations.

To show what each stage looks like when it is done well, the unit models one complete investigation from start to finish. A student we call Amira asks whether Grade 12 students with part-time jobs sleep less on school nights, surveys a random sample at a fictional school (Pinecrest Secondary School), and analyses a clearly labelled example data set of 40 responses (made up for teaching) with box plots, standard deviations, a normal model, correlation, regression, a contingency table and a shuffle simulation. Her second, smaller problem is a dice game for a school carnival, analysed with expected value and the binomial distribution. Every calculation in the model is real, so you can check each step yourself. Each lesson then ends with a step for your own investigation.

How it fits: this unit uses every sibling unit of MDM4U: Probability, Permutations and Combinations, Discrete Probability Distributions, Continuous Distributions and the Normal Distribution, Organization of Data for Analysis, One-Variable Data Analysis, and Two-Variable Data Analysis and Validity. You will need a calculator, a spreadsheet program (Google Sheets, LibreOffice Calc or Excel), and presentation software or poster paper. Many schools start the culminating investigation early in the semester and work on it alongside the other units; the lessons here can be used that way too.

Curriculum: The Ontario Curriculum, Grades 11 and 12: Mathematics (2007, revised), Mathematics of Data Management, Grade 12, University Preparation (MDM4U), strand E. Culminating Data Management Investigation: overall expectation E1 (designing and carrying out a culminating investigation: E1.1 to E1.5) and overall expectation E2 (presenting and critiquing: E2.1 to E2.4). The investigation may address a single problem on probability and statistics, or two smaller problems, one on probability and one on statistics. The mathematical process expectations are woven through every lesson.