AP Statistics
Introduction to Elementary Statistical Methods
Mr. Molesky
Lakeville South High School
Adapted from: Instructor 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?
Yes
Under
50?
Yes
No
Total
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!