AP Statistics Curriculum 2007

This is a General Advanced-Placement (AP) Statistics Curriculum E-Book

Preface

This is an Internet-based E-Book for advanced-placement (AP) statistics educational curriculum. The E-Book is initially developed by the UCLA Statistics Online Computational Resource (SOCR), however, any statistics instructor, researcher or educator is encouraged to contribute to this effort and improve the content of these learning materials.

Format

Follow the instructions in this page to expand, revise or improve the materials in this E-Book.

Chapter I: Introduction to Statistics

The Nature of Data & Variation

No mater how controlled the environment, the protocol or the design, virtually any repeated measurement, observation, experiment, trial, study or survey is bound to generate data that varies because of intrinsic (internal to the system) or extrinsic (due to the ambient environment) effects. How many natural processes or phenomena in real life can we describe that have an exact mathematical closed-form description and are completely deterministic? How do we model the rest of the processes that are unpredictable and have random characteristics?

Uses and Abuses of Statistics

Statistics is the science of variation, randomness and chance. As such, statistics is different from other sciences, where the processes being studied obey exact deterministic mathematical laws. Statistics provides quantitative inference represented as long-time probability values, confidence or prediction intervals, odds, chances, etc., which may ultimately be subjected to varying interpretations. The phrase Uses and Abuses of Statistics refers to the notion that in some cases statistical results may be used as evidence to seemingly opposite theses. However, most of the time, common principles of logic allow us to disambiguate the obtained statistical inference.

Design of Experiments

Design of experiments is the blueprint for planning a study or experiment, performing the data collection protocol and controlling the study parameters for accuracy and consistency. Data, or information, is typically collected in regard to a specific process or phenomenon being studied to investigate the effects of some controlled variables (independent variables or predictors) on other observed measurements (responses or dependent variables). Both types of variables are associated with specific observational units (living beings, components, objects, materials, etc.)

Statistics with Tools (Calculators and Computers)

All methods for data analysis, understanding or visualization are based on models that often have compact analytical representations (e.g., formulas, symbolic equations, etc.) Models are used to study processes theoretically. Empirical validations of the utility of models are achieved by plugging in data and actually testing the models. This validation step may be done manually, by computing the model prediction or model inference from recorded measurements. This however is possible by hand only for small number of observations (<10). In practice, we write (or use existent) algorithms and computer programs that automate these calculations for better efficiency, accuracy and consistency in applying models to larger datasets.

Chapter II: Describing, Exploring, and Comparing Data

Types of Data

There are two important concepts in any data analysis - population and sample. Each of these may generate data of two major types - quantitative or qualitative measurements.

Summarizing data with Frequency Tables

There are two important ways to describe a data set (sample from a population) - Graphs or Tables.

Pictures of Data

There are many different ways to display and graphically visualize data. These graphical techniques facilitate the understanding of the dataset and enable the selection of an appropriate statistical methodology for the analysis of the data.

Measures of Central Tendency

There are three main features of populations (or sample data) that are always critical in understanding and interpreting their distributions - Center, Spread and Shape. The main measures of centrality are mean, median and mode(s).

Measures of Variation

There are many measures of (population or sample) spread, e.g., the range, the variance, the standard deviation, mean absolute deviation, etc. These are used to assess the dispersion or variation in the population.

Measures of Shape

The shape of a distribution can usually be determined by just looking at a histogram of a (representative) sample from that population frequency plots, dot plots or stem and leaf displays may be helpful.

Statistics

Variables can be summarized using statistics - functions of data samples.

Graphs & Exploratory Data Analysis

Graphical visualization and interrogation of data are critical components of any reliable method for statistical modeling, analysis and interpretation of data.

Chapter III: Probability

Probability is important in many studies and disciplines because measurements, observations and findings are often influenced by variation. In addition, probability theory provides the theoretical groundwork for statistical inference.

Fundamentals

Some fundamental concepts of probability theory include random events, sampling, types of probabilities, event manipulations and axioms of probability.

Rules for Computing Probabilities

There are many important rule for computing probabilities of composite events. These include conditional probability, statistical independence, multiplication and addition rules, the law of total probability and the Bayesian rule.

Probabilities Through Simulations

Many experimental setting require probability computations of complex events. Such calculations may be carried out exactly, using theoretical models, or approximately, using estimation or simulations.

Counting

There are many useful counting principles (including permutations and combinations) to compute the number of ways that certain arrangements of objects can be formed. This allows counting-based estimation of probabilities of complex events.

Chapter IV: Probability Distributions

There are two basic types of processes that we observe in nature - discrete and continuous. We begine by discussing several important discrete random processes, their distributions, expectations, variances and applications. In the next chapter, we will discuss their continuous counterparts.

Random Variables

To simplify the calculations of probabilities, we will define the concept of a random variable which will allows ut to study uniformly various processes, using the same mathamatical and computational techniques.

Bernoulli & Binomial Experiments

The Bernoulli and Binomial processes provide the simplest models for discrete random experiments. Here we also define the expectation and the variance for any discrete random variable or process.

Geometric, HyperGeometric & Negative Binomial

Overview TBD

Poisson Distribution

Overview TBD

Chapter V: Normal Probability Distribution

The Standard Normal Distribution

Overview TBD

Nonstandard Normal Distribution: Finding Probabilities

Overview TBD

Nonstandard Normal Distribution: Finding Scores (critical values)

Overview TBD

Chapter VI: Relations Between Distributions

The Central Limit Theorem

Overview TBD

Law of Large Numbers

Overview TBD

Normal Distribution as Approximation to Binomial Distribution

Overview TBD

Poisson Approximation to Binomial Distribution

Overview TBD

Binomial Approximation to HyperGeometric

Overview TBD

Normal Approximation to Poisson

Overview TBD

Chapter VII: Estimates and Sample Sizes

Estimating a Population Mean: Large Samples

Overview TBD

Estimating a Population Mean: Small Samples

Overview TBD

Estimating a Population Proportion

Overview TBD

Estimating a Population Variance

Overview TBD

Chapter VIII: Hypothesis Testing

Fundamentals of Hypothesis Testing

Overview TBD

Testing a Claim about a Mean: Large Samples

Overview TBD

Testing a Claim about a Mean: Small Samples

Overview TBD

Testing a Claim about a Proportion

Overview TBD

Testing a Claim about a Standard Deviation or Variance

Overview TBD

Chapter IX: Inferences from Two Samples

Inferences about Two Means: Dependent Samples

Overview TBD

Inferences about Two Means: Independent and Large Samples

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Comparing Two Variances

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Inferences about Two Means: Independent and Small Samples

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Inferences about Two Proportions

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Chapter X: Correlation and Regression

Correlation

Overview TBD

Regression

Overview TBD

Variation and Prediction Intervals

Overview TBD

Multiple Regression

Overview TBD

Chapter XI: Non-Parametric Inference

Differences of Means of Two Paired Samples

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Differences of Means of Two Independent Samples

Overview TBD

Differences of Medians of Two Paired Samples

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Differences of Medians of Two Independent Samples

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Differences of Proportions of Two Independent Samples

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Differences of Means of Several Independent Samples

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Differences of Variances of Two Independent Samples

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Chapter XII: Multinomial Experiments and Contingency Tables

Multinomial Experiments: Goodness-of-Fit

Overview TBD

Contingency Tables: Independence and Homogeneity

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Chapter XIII: Statistical Process Control

Control Charts for Variation and Mean

Overview TBD

Control Charts for Attributes

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Chapter XIV: Survival/Failure Analysis

Overview TBD

Chapter XV: Multivariate Statistical Analyses

Multivariate Analysis of Variance

Overview TBD

Multiple Linear Regression

Overview TBD

Logistic Regression

Overview TBD

Log-Linear Regression

Overview TBD

Multivariate Analysis of Covariance

Overview TBD

Chapter XVI: Time Series Analysis

Overview TBD

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