A: Counting and Probability

A.1: solve problems involving the probability of an event or a combination of events for discrete sample spaces;

A.1.1: recognize and describe how probabilities are used to represent the likelihood of a result of an experiment (e.g., spinning spinners; drawing 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)

Binomial Probabilities
Geometric Probability
Independent and Dependent Events
Probability Simulations
Theoretical and Experimental Probability

A.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)

Independent and Dependent Events

A.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 base 1) + (P base 2) + (P base 3) +... + (P base n) = 1), recognize that the probabilities P form the probability distribution associated with the sample space, and solve related problems

Binomial Probabilities

A.1.4: determine, through investigation using class-generated data and technology-based simulation 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 number of trials in an experiment increases (e.g., “If I simulate tossing two coins 1000 times using technology, the experimental probability that I calculate for getting two tails on the two tosses is likely to be closer to the theoretical probability of ¼ than if I simulate tossing the coins only 10 times”)

Geometric Probability

A.1.6: determine whether two events are independent 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)

Independent and Dependent Events

A.2: solve problems involving the application of permutations and combinations to determine the probability of an event.

A.2.1: recognize the use of permutations and combinations as counting techniques with advantages 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 permutations and those that involve the use of combinations (e.g., by considering whether or not order matters), and make connections between, and calculate, permutations and combinations

Binomial Probabilities
Permutations and Combinations

A.2.2: solve simple problems using techniques for counting permutations and combinations, where all objects are distinct, and express the solutions using standard combinatorial notation [e.g., n!, P(n, r),(n factorial r factorial)]

Binomial Probabilities
Permutations and Combinations

A.2.4: make connections, through investigation, between combinations (i.e., n choose r) and Pascal’s triangle [e.g., between (2 factorial r factorial) and row 3 of Pascal’s triangle, between (n factorial 2 factorial) and diagonal 3 of Pascal’s triangle]

Binomial Probabilities

B: Probability Distributions

B.1: demonstrate an understanding of discrete probability distributions, represent them numerically, graphically, and algebraically, determine expected values, and solve related problems from a variety of applications;

B.1.1: recognize and identify a discrete random variable 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), generate 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 probability distribution numerically using a table

Polling: City

B.1.2: calculate the expected value for a given probability distribution [i.e., using E(X)= Sigma 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

Lucky Duck (Expected Value)
Polling: City

B.1.4: recognize conditions (e.g., independent trials) that give rise to a random variable that follows a binomial probability distribution, calculate the probability associated with each value of the random variable, represent the distribution numerically using a table and graphically using a probability histogram, and make connections to the algebraic representation P(X=x)= (n factorial x factorial) (p to the x power) ((1 - p) to the (n - x) power)

Binomial Probabilities

B.1.7: solve problems involving probability distributions (e.g., uniform, binomial, hyper-geometric), including problems arising from real-world applications

Binomial Probabilities
Polling: City

B.2: demonstrate an understanding of continuous probability distributions, make connections to discrete probability distributions, determine standard deviations, describe key features of the normal distribution, and solve related problems from a variety of applications.

B.2.1: 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)

Polling: City

B.2.2: 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

Polling: City
Real-Time Histogram
Sight vs. Sound Reactions

B.2.6: recognize that the normal distribution is commonly used to model the frequency and probability distributions of continuous random 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 (e.g., birth weights of males or of females, household incomes in a neighbourhood, baseball batting averages)

Sight vs. Sound Reactions

C: Organization of Data for Analysis

C.1: demonstrate an understanding of the role of data in statistical studies and the variability inherent in data, and distinguish different types of data;

C.1.1: 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 conclusions drawn from statistical studies of the same relationship may differ (e.g., conclusions about the effect of increasing jail sentences on crime rates)

Polling: City
Real-Time Histogram

C.1.2: 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

Polling: City
Populations and Samples

C.2: describe the characteristics of a good sample, some sampling techniques, and principles of primary data collection, and collect and organize data to solve a problem.

C.2.1: determine and describe principles of primary data collection (e.g., the need for randomization, replication, and control in experimental studies; the need for randomization in sample surveys) and criteria that should be considered in order to collect reliable primary data (e.g., the appropriateness of survey questions; potential sources of bias; sample size)

Correlation
Describing Data Using Statistics
Polling: City
Polling: Neighborhood
Populations and Samples
Real-Time Histogram

C.2.2: explain the distinction between the terms population and sample, describe the characteristics of a good sample, explain why sampling is necessary (e.g., time, cost, or physical constraints), and describe and compare some sampling techniques (e.g., simple random, systematic, stratified, convenience, voluntary)

Polling: City
Polling: Neighborhood
Populations and Samples

C.2.3: 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

Polling: Neighborhood
Populations and Samples

C.2.4: describe characteristics of an effective survey (e.g., by giving consideration to ethics, privacy, the need for honest responses, and possible sources of bias, including cultural bias), and design questionnaires (e.g., for determining 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 conditions) for gathering data

Polling: Neighborhood

D: Statistical Analysis

D.1: analyse, interpret, and draw conclusions from one-variable data using numerical and graphical summaries;

D.1.1: recognize that the analysis of one-variable data involves the frequencies associated with one attribute, and determine, using technology, the relevant numerical summaries (i.e., mean, median, mode, range, interquartile range, variance, and standard deviation)

Box-and-Whisker Plots
Describing Data Using Statistics
Mean, Median, and Mode
Sight vs. Sound Reactions
Stem-and-Leaf Plots

D.2.2: 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 onevariable data sets, and recognize these processes as strategies for one-variable data analysis

Sight vs. Sound Reactions

D.1.3: 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)

Box-and-Whisker Plots
Histograms
Stem-and-Leaf Plots

D.1.4: interpret, for a normally distributed population, 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 connections, 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)

Polling: City

D.1.5: 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 statistical 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

Box-and-Whisker Plots
Describing Data Using Statistics
Graphing Skills
Mean, Median, and Mode
Stem-and-Leaf Plots

D.2: analyse, interpret, and draw conclusions from two-variable data using numerical, graphical, and algebraic summaries;

D.2.1: 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)

Correlation
Solving Using Trend Lines

D.2.3: 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)

Graphing Skills

D.2.4: determine, by performing a linear regression using technology, the equation of a line that models a suitable two-variable data set, determine the fit of an individual data point to the linear model (e.g., by using residuals to identify outliers), and recognize these processes as strategies for two-variable data analysis

Correlation
Least-Squares Best Fit Lines
Solving Using Trend Lines
Trends in Scatter Plots

D.2.5: 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

Solving Using Trend Lines
Trends in Scatter Plots

D.3: demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations.

D.3.2: 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)

Polling: City

E: Culminating Data Management Investigation

E.1: design and carry out a culminating investigation that requires the integration and application of the knowledge and skills related to the expectations of this course;

E.1.2: design a plan to study the problem (e.g., identify the variables and the population; develop an ethical survey; establish the procedures for gathering, summarizing, and analysing the primary or secondary data; consider the sample size and possible sources of bias)

Correlation
Describing Data Using Statistics
Polling: City
Polling: Neighborhood
Populations and Samples
Real-Time Histogram

E.1.3: 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

Correlation
Describing Data Using Statistics
Polling: City
Stem-and-Leaf Plots

E.1.4: 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

Box-and-Whisker Plots
Polling: City
Real-Time Histogram

E.1.5: 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 problems or investigations

Polling: Neighborhood