Statistics and Probability Strand

Because of the importance of statistics and probability in everyday life, virtually every set of standards and recommendations for the secondary school curriculum includes a strand in statistics and probability. The National Council of Teachers of Mathematics in the Principles and Standards for School Mathematics (PSSM) recommends an increased emphasis on data analysis and probability from kindergarten through grade 12. Most state standards, as well as the College Board, recommend an increased emphasis on statistics and probability. Instructional programs should enable all students to

• formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them

• select and use appropriate statistical methods to analyze data

• develop and evaluate inferences and predictions that are based on data

• understand and apply basic concepts of probability (PSSM, p.324)

Why not wait for college to teach statistics and probability? Because most students are 18 years old before they even begin college. At that age they can vote, buy cigarettes and lottery tickets, join the army, get married, and make independent medical decisions. They can't afford to delay learning how to evaluate probabilistic situations and how to use data to make intelligent decisions.

Each course in the Core-Plus Mathematics curriculum includes one unit on statistics and one on probability, which are described briefly below.

Course 1

Unit 2 - Patterns in Data develops student ability to make sense of real-world data through use of graphical displays, measures of center, and measures of variability.

Topics include: distributions of data and their shapes, as displayed in dot plots, histograms, and box plots; measures of center including mean and median, and their properties; measures of variability including interquartile range and standard deviation, and their properties; and percentiles and outliers.

Unit 8 - Patterns in Chance develops student ability to solve problems involving chance by constructing sample spaces of equally-likely outcomes or geometric models and to approximate solutions to more complex probability problems by using simulation.

Topics include: sample spaces, equally-likely outcomes, probability distributions, mutually exclusive (disjoint) events, Addition Rule, simulation, random digits, discrete and continuous random variables, Law of Large Numbers, and geometric probability.

Course 2

Unit 4 - Regression and Correlation develops student understanding of the characteristics and interpretation of the least squares regression equation and of the use of correlation to measure the strength of the linear association between two variables.

Topics include: interpreting scatterplots; least squares regression, residuals and errors in prediction, sum of squared errors, influential points; Pearson's correlation coefficient and its properties, lurking variables, and cause and effect.

Unit 8 - Probability Distributions further develops student ability to understand and visualize situations involving chance by using simulation and mathematical analysis to construct probability distributions.

Topics include: Multiplication Rule, independent and dependent events, conditional probability, probability distributions and their graphs, waiting-time (or geometric) distributions, expected value, and rare events.

Course 3

Unit 1 - Reasoning and Proof develops student understanding of formal reasoning in statistical contexts and of basic principles that underlie those reasoning strategies.

Topics include: inductive and deductive reasoning strategies; principles of logical reasoning—Affirming the Hypothesis and Chaining Implications; design of experiments including the role of randomization, control groups, and blinding; sampling distribution, randomization test, and statistical significance.

Unit 4 - Samples and Variation extends student understanding of the measurement of variability, develops student ability to use the normal distribution as a model of variation, introduces students to the binomial distribution and its use in decision making, and introduces students to the probability and statistical inference involved in control charts used in industry for statistical process control.

Topics include: normal distribution, standardized scores, binomial distributions (shape, expected value, standard deviation), normal approximation to a binomial distribution, odds, statistical process control, control charts, and the Central Limit Theorem.

Course 4

Unit 5 - Exponential Functions, Logarithms, and Data Modeling extends student understanding of exponential and logarithmic functions to the case of natural exponential and logarithmic functions, solution of exponential growth and decay problems, and use of logarithms for linearization and modeling of data patterns.

Topics include: exponential functions with rules in the form f(x) = Ae^kx, natural logarithm function, linearizing bivariate data and fitting models using log and log-log transformations.

Unit 9 - Binomial Distributions and Statistical Studies extends student understanding of the binomial distribution, including its exact construction and how the normal approximation to the distribution of the sample proportion is used in statistical inference.

Topics include: binomial probability formula; shape, expected value, and standard deviation of the distribution of the sample proportion, p-hat; design of sample surveys including random sampling and stratified random sampling; measurement (response) bias; sample selection bias; variability in sampling; con€ dence intervals; margin of error; and test of significance of a proportion.