Why Study Statistics and Probability
An introduction to the real-world applications of probability and statistics. Examples include: insurance companies, sports figures, computer gamers, small business owners, and meteorologists.
Collecting Data for Study: Sampling
An introduction to some important considerations when collecting data to use for statistics and probability studies. Explains the many different types of sampling, the difference between a population and a representative sample, how to identify a sample that is representative of a given population, how to ensure a sample is properly randomized, and the concerns of undersampling. Note that this chapter is about the concepts involved with identifying and creating valid samples for data collection, later lessons will return to many of these topics and re-evaluate them from a more mathematically rigorous standpoint.
Common Types of Samples
A study of specific methods of collecting sample data. Students will become familiar with many different methods of identifying representative samples, including stratified and cluster sampling, and non-probability sampling. The primary goal of lessons in this set is to encourage students to correlate different sampling methods with appropriate applications. Students should become comfortable with the notion that although most sampling should be as random as possible, some samples are entirely non-random for good reason.
Evaluating and Displaying Data
A detailed examination of myriad data visualization methods. Beginning with a discussion of how to organize and evaluate data, lessons progress through the creation and evaluation of: histograms, frequency polygons, scatter plots, box-and-whisker plots, pie charts, and bar graphs. Each type of data visualization is approached from both creation and evaluation standpoints, so the student will be able to use data from others' graphs and apply it to his/her own, well-designed presentation.
A detailed, and somewhat advanced, study of measures of central tendency and variability. Beginning with arithmetic, geometric and harmonic means, lessons then progress through median and mode before moving on to variability measures. Each new measure is illustrated with applications and sample theoretical uses before the next is introduced. Studies of variability include calculating and applying variance and standard deviation, and finding and applying the coefficient of variation.
An introduction to the concept of probability evaluation. Concepts progress from defining statistical and experimental probability and inclusive and exclusive events (including the general definitions of intersection and union) to conditional probability and identifying the complement. As each new concept is introduced, entertaining examples of use such as playing cards and dropping buttered slices of bread are provided for students to practice the application of various types of probability study.
An examination of random variables, both discrete and continuous, and the related generation of probability distributions. Students become familiar with probability distribution graphs and tables, and the relationship between probability density functions and distribution visualizations. Later concepts introduce expected values and the variance of random variables, and finally examine transformations of random variables (operations on random variables).
Combinations and Permutations
An examination of the two basic divisions of combinatorics: permutations, which are groupings of items where the order of the items is important, and combinations, where only the identity of the items is important, regardless of the order in which they appear. Students study both kinds of groupings, and learn to calculate the number of possible unique ways to combine or arrange items of all kinds, whether allowing or denying repeated or indistinguishable items.
The Normal Distribution
Concepts introduce the application of the normal distribution, and introduce the Empirical rule and the concept and calculation of z-scores. Z-score probabilities are introduced, and students practice associating z-scores with probabilities via reference tables and technology. The concept of the sampling distribution of the sample mean is examined, along with the Central Limit Theorem. Students practice using the normal distribution to approximate the output of binomial random variables.
Predicting and Testing
An introduction to the concepts of hypothesis testing and predicting values. The concept of the null and alternative hypothesis are introduced and practiced. The tails of a curve are described and students practice identifying the applicable portions of a normal curve when identifying ‘above’, ‘below’, and ‘between’ values probabilities. Critical values and critical regions and their association with alpha values is examined. Students learn about confidence intervals and learn the meaning of confidence level. The similarities between T and Z testing are reviewed and students practice using both to test hypotheses in word problems.
Linear Regression and Chi-Squared
An introduction to the concept of linear regression and Pearson’s correlation coefficient. Students learn to calculate a line of best fit using the least squares method. After learning to create contingency tables, students practice extracting data from contingency tables to calculate the Chi-Squared statistic. Finally the students learn to use chi-square to run tests of goodness of fit and variable independence.
A study of logical argument and inductive and deductive reasoning. Methods of evaluating argument validity such as Euler diagrams and rewriting premises and conclusions are introduced and practiced. Students learn about structural and content fallacies and how to identify hidden premises and practice recognizing valid and invalid forms of argument.