AP Statistics Introduction to Elementary Statistical Methods Mr. Kent Wheeler High School Adapted from: Instructor Molesky, Andre Bucove and Statistics in Action: Watkins, Scheaffer, Cobb

Statistics vs. Mathematics Statistical Thinking differs from other mathematical thinking in important ways: The role of CONTEXT The logic of Statistical INFERENCE

Role of Context: Mathematics Mathematicians rely on context for motivation and for sources of problems for research. The ultimate focus of mathematical thinking is on abstract patterns. Context must be “boiled off” to reveal the pure structure. In Mathematics, context obscures structure.

Role of Context: Statistics Statisticians look for patterns, but whether those patterns have meaning or value depends on the context. In Statistics, context provides meaning. We must not “plow through” the story to get to the “real stuff” like formulas and number crunching...the story IS the “real” stuff!

Role of Context: Statistics In Statistics, we must think carefully about context: to answer what a result means in the context of a particular application to suggest questions that need to be answered and the appropriate methods to be utilized to answer them

Logic of Statistical Inference Most math courses are based on concepts built through structured proof. The use of statistical inference requires us to use “what if” reasoning that includes consideration of uncertainty and variability. Inferential results do not prove or disprove, they only provide evidence whether an observation may be due to chance.

What Does this Mean for You? In other math courses, calculating the correct number is often the goal. In statistics, simply calculating is rarely the goal; you must justify your answer and your approach...including stating assumptions, showing graphs and computations, and writing a conclusion in context.

What Does this Mean for You? In other math courses, the common structure between problems is emphasized. In statistics, the shared patterns or structure are important, but you must also consider how one problem differs from another.

What Does this Mean for You? In other math courses, study focuses on “drill and skill” with limited use of technology. In this course, the use of technology is encouraged and will allow you to focus on understanding and communicating statistical concepts in context. Drill and skill and memorization will not be enough to master the topics.

A Case Study of Statistics in Action Adapted from Chapter 1 of “Statistics In Action”

This Activity’s Goal... The purpose of this activity is to give you a head start on the thinking and concepts you will encounter this year. We will explore the basic ideas of: Exploring Data - uncovering and summarizing patterns. Making Inferences - deciding whether or not an observation could be due to chance.

Martin v. Westvaco In 1991, Westvaco Corp. underwent 5 rounds of layoffs. In the end, only 22 of 50 engineers in the envelope division still had their jobs. The average age of those workers fell from 48 to 46 years old. Robert Martin, age 55, was laid off and sued Westvaco for age discrimination.

Martin v. Westvaco Question: Were older workers discriminated against during the layoffs? Martin’s lawyers used statistics to answer the question. The analysis considered all 50 employees in the envelope division, with separate analyses for salaried and hourly workers.

Data Exploration

Data Exploration An Exploratory Data Analysis is an informal, open-ended examination of data for patterns. The goal is to uncover and summarize patterns in data and to formulate basic questions about the data. The tools include graphs and numeric summaries.

Westvaco Data Refer to Display 1.1: The Data in Martin v. Westvaco. Cases - subjects or objects of statistical examination {eg, Westvaco employees} Variables - characteristics recorded for each case. What variables are listed here? Note the variability in each column.

Variability Variability makes it difficult to see the patterns in data. If there were no variability, conclusions would be obvious and there would be no need for Statistics. Statistical Methods were designed to cope with variability.

A Definition of Statistics “Statistics is the Science of learning from data in the presence of variability.”

Patterns in Data The variability that exists is easy to see in the columns, but the patterns are not so obvious. The distribution of the variable shows the pattern - what the values are, how spread out they are, how often each occurs, and any unusual values. {Note the SOCS}

Visualizing the Distribution We will be studying a number of data displays...some of which will be familiar. A dotplot is one display that can be used to visualize a distribution.

What Do You See? In statistics, we must consider “what we see” before considering “why we are seeing it” Answer D3, D4, and D5, focusing on what you are seeing...

Discussion D3

Discussion D4

Discussion D5

Discussion D6 Interpret the following tables summarizing the ages and employment status of Salaried workers at Westvaco. Laid Off? Under 50? Yes No Total % Yes

Discussion D7 and D8 D7: Do these patterns seem “real”? That is, if Westvaco really did lay off workers at random, what would you expect to see? D8: Does the data suggest older workers were more likely to be laid off? What considerations might justify laying off older workers?

Inference

Inference Unlike an exploratory data analysis, inference follows strict rules and focuses on judging whether patterns in data can be attributed to chance, or should be investigated under another explanation.

What do the Patterns Mean? Can we infer from the dotplots that Westvaco has some explaining to do? Could we observe those patterns if there was no discrimination going on? To answer these questions, we must investigate what patterns would occur by chance...

Simulation Using the data from Round 2, we will simulate randomly selecting three workers. We will calculate the average age of our workers and explore the pattern of averages in repeated simulations. We will estimate the likelihood of observing an average of 58 or more...why?

Discussion of Activity The following display shows the results of 1000 simulations...how likely is it to get an average of 58 or more?

Where Do We Go From Here? This exploration illustrated some of the topics we will be exploring this year. You will learn the methods behind Collecting, Exploring, Interpreting, and making Inferences from Data. Hopefully, you will become expert statistical thinkers and will perform well on the AP Exam!