Propspectus:
THE LOGIC OF STATISTICAL REASONING
The Logic of Statistical Reasoning (LSR) fills a gap among statistics texts for
social science students. The current population of texts vary somewhat in length, style of
writing, graphics format and the kinds of examples used. But they replicate earlier texts
in that they assume that the ground to be covered in the introductory course has not
changes significantly over the last two or three decades. While statistical texts haven’t
changed, the quantitative methods used in social research have. The result is that while
students may leave the introductory course with some understanding of statistical
inference they are ill-equipped to read the research literature. We have addressed that
problem in LSR, and in the process produced a text that is pedagogically stronger than
most on the market.
Like many other texts, LSR assumes that the typical student in the introductory
social statistics course has very limited mathematical background. But unlike other texts,
LSR emphasizes critical thinking about the logic underpinning various statistical
methods. The goal of the book is to help the student become a thoughtful reader of
quantitative analysis. Our experience demonstrates that students are capable of such
critical thinking even when they lack mathematical sophistication.
LSR uses three strategies to encourage critical thinking about statistical methods:
--LSR emphasizes models as a fundamental metaphor in all quantitative analysis, from
the simplest hypothesis tests through regression. The idea of the model becomes the
basis that structures for critical thinking. And the models allow students to connect the
statistical procedures to the substantive concerns that motivate them.
--LSR uses repetition with variation to underpin its pedagogy. All hypothesis tests and
confidence intervals and all measures of goodness of fit are treated as variations on
theme. This builds confidence and helps students see connections. LSR also uses a
consistent set of examples in every chapter so that students develop familiarity with the
substance of the examples and also learn that there is more than one way to approach the
same research question.
-- LSR does not elaborate on tools rarely used in contemporary research. Rather the
emphasis on models as a master metaphor allows LSR to move briskly to regression,
which is the basis of nearly all statistical analysis in the social sciences.
The implementation of these strategies gives LSR in a new and much stronger
pedagogical approach than existing texts. It reflects both the reality of statistical practice
the students will encounter in other courses and their jobs, and it reflects the realities of
how students with limited mathematical backgrounds use statistics.
Competition. There more than a dozen texts targeting the introductory course in statistics
for sociology and other social science students, so detailed comparison with all of them is
not practical. LSR is as clearly written as the best of them and has a better pedagogic
strategy. A crucial issue for a new text is how well its outline parallels that of texts
currently being used, since similarity encourages adoption. However, the best selling
texts (Frankfort-Nachmias & Guerrero, Healey and Levin & Fox) each have somewhat
different outlines so no new text can match all of them. The outline of LSR is closest to
that of Healey.
Market. LSR is intended for introductory social statistics courses that are taught in all
sociology departments. We have chosen recurring examples that also make it suitable for
students in criminal justice, health sciences and environmental studies, and the text has
been tested with these audiences as well as with sociology students.
Status. The full manuscript of LSR, including examples and exercises, is complete. This
text has been used in draft several times in introductory social statistics courses. We are
open to developing supplemental materials but have not yet done so.
THE LOGIC OF STATISTICAL REASONING
Thomas Dietz & Linda Kalof
Michigan State University
July 2005
TABLE OF CONTENTS
PREFACE:
A STRATEGY FOR APPROACHING QUANTITATIVE ANALYSIS
Statistics is hard
Statistics is important
Quantitative analysis as craftwork
The discourse of science
A strategy for learning
What have we learned?
CHAPTER 1: AN INTRODUCTION TO QUANTITATIVE ANALYSIS
What is statistics?
Models to explain variation
Explaining variation
The use of statistical methods
Types of error
Error in models
Sampling error
Randomization error
Measurement error
Perceptual error
Comparison to random numbers
Assumptions
What have we learned?
Applications
Advanced Topic: A third and more technical definition of statistics
Advanced Topic: Ways of drawing probability samples
Feature: Tea tasting and random numbers
Feature: Diversity in Statistics: Profiles of African American, Mexican American and
Women Statisticians
CHAPTER 2: SOME BASIC CONCEPTS
Key Concepts
Variables
Levels of measurement
Nominal
Ordinal
Interval
Ratio
Tools for working with measurement levels
Scaling
Other terms
Labeling variables
Constants
Functions
Units of analysis
Data structure
Sample size and sample selection
What have we learned?
Applications
Exercises
CHAPTER 3: DISPLAYING DATA ONE VARIABLE AT A TIME
Graphic display of nominal and ordinal data
Pie chart
Bar chart
Dot plot histogram
Graphic display of continuous data
One-way scatterplot
Cleveland dotplot
Histogram
Stem and leaf diagram
Skew and mode
Rules for graphing
What have we learned?
Applications
Exercises
Advanced Topic: Tukey’s new way of tabulating
Feature: John W. Tukey
CHAPTER 4: DESCRIBING DATA
Descriptive statistics and exploratory analysis
Measures of central tendency
The mean
Deviations from the mean
Effect of outliers
The median
Deviations from the median
Effect of outliers
The trimmed mean
The mode
Measures of variability
Variance
Standard deviation
Median absolute deviation from the median
Interquartile range
Relationship among the measures
Boxplot
Side-by-side boxplot
What have we learned?
Applications
Exercises
Advanced Topic:
Squared deviations and their relationship to the mean
Advanced Topic:
Breakdown point
Feature: Models, error, and the mean: A brief history
CHAPTER 5:
PLOTTING RELATIONSHIPS AND CONDITIONAL DISTRIBUTIONS
Scatterplot
Time series graph
Bivariate plot for categorical or grouped independent variables
An historical example
What have we learned?
Applications
Exercises
Advanced Topic: Scatterplot matrices
Advanced Topic: Ordinary least squares
Advanced Topic: Smoothers
CHAPTER 6: CAUSATION AND MODELS OF CAUSAL EFFECTS
Causation and correlation
Causation in non-experimental data
Explanatory and Extraneous Variables
Causal notation
Assessing causality: Elaboration and controls
Another example
An example with continuous variables
What have we learned?
Applications
Exercises
Advanced Topic: More elaborate causal notation
Advanced Topic: Further ideas in assessing causality
Feature
Feature
Feature
What is an experiment?
What is correlation?
Causal notation and measurement
CHAPTER 7: PROBABILITY
What is random?
Probability
Equal probability and independence
Independent events
Random variables and models of error
Probability distributions
The Normal distribution and the law of large numbers
Random variables and data analysis
Sampling error
Randomization error
Measurement error
Left-out variables
Permutation errors
What have we learned?
Feature
Feature
Feature
Feature
A brief note on Thomas Bayes
Making decisions about uncertainty
Cumulative probability distributions
Kinds of samples and probability
CHAPTER 8: SAMPLING DISTRIBUTIONS AND INFERENCE
The logic of inference
The sampling experiment
The sampling distribution
The expected value and the standard error
The law of large numbers
Small samples and estimating the population variance
The t distribution
What have we learned?
Advanced Topic: Sampling with replacement and sampling from large populations
Advanced Topic: Bootstrapping
Advanced Topic: Small sample versus large sample estimators
Advanced Topic: Properties of estimators (bias, efficiency, mean square error,
consistency, robustness, maximum likelihood estimators)
Feature: Who invented the Central Limit Theorem?
Feature: Pearson to Gosset to Fisher: The coevolution of statistical and scientific
thinking
Feature: William Sealy Gosset
CHAPTER 9:
USING SAMPLING DISTRIBUTIONS: CONFIDENCE INTERVALS
Confidence intervals using the Central Limit Theorem
Constructing the confidence interval for large samples
Sampling experiments
Confidence intervals using the t distribution
Size of confidence intervals
Graphing confidence intervals
Confidence intervals for dichotomous variables
Rough confidence intervals
What have we learned?
Applications
Exercises
CHAPTER 10:
USING SAMPLING DISTRIBUTIONS: HYPOTHESIS TESTS
The logic of hypothesis tests
The formal approach
Steps in testing a hypothesis
An example
An example with a small sample
One-sided and two-sided tests
Small sample tests
Hypotheses about differences in means
A small sample difference in means test
Models for differences in means
Limits to hypothesis tests
What have we learned?
Applications
Exercises
Advanced Topic:
Advanced Topic:
Advanced Topic:
Thinking about the hypothesis test as a model
The more correct, less simple formulas
Confidence intervals and hypothesis tests
CHAPTER 11:
THE SUBTLE LOGIC OF ANALYSIS OF VARIANCE
A review of the two-group example
More than two groups example
A model
Partitioning variance
Inference in analysis of variance (the F-test)
A step-by-step example
What have we learned?
Applications
Exercises
Advanced Topic:
Advanced Topic:
Feature:
Feature:
What is Normally-distributed?
F versus t and z
Analysis of variance the hard way
How to conduct many tests with the same data
CHAPTER 12:
GOODNESS OF FIT AND MODELS OF FREQUENCY TABLES
Chi-square applied to a frequency table
Assumptions for chi-square
Chi-square and the association between two qualitative variables
Beyond 2 x 2 tables
What we have learned?
Applications
Exercises
CHAPTER 13:
BIVARIATE REGRESSION AND CORRELATION
Straight Lines
Fitting a straight line to data
Lines as summaries
Error in fit
Ordinary least squares
OLS and conditional distributions
Calculating the OLS line
Calculating B
Calculating A
^
Calculating Y
Calculating E
Goodness of fit
Fit and error
Partitioning sums of squares or variances
R and R2
Pearson’s correlation coefficient
Interpreting Regression Lines
Interpreting A
Interpreting B
Interpreting R2
Interpreting r
Interpreting E
Inference in regression: A basic approach
Working with B
Working with A
Working with R2
What have we learned?
Applications
Exercises
Advanced Topic: Breakdown point
Advanced Topic:
Advanced Topic:
Advanced Topic:
Advanced Topic:
Pearson’s r and B coefficients

Confidence intervals for predictions
Thinking about inference in regression
Feature: The invention of regression
CHAPTER 14:
BASICS OF MULTIPLE REGRESSION
Two independent variables
Terminology and notation
Geometric view
Algebraic view
Calculations
Interpretation
Interpreting a
Interpreting b
Interpreting R2
Relationship between bivariate and multiple regression
Inference
Tests regarding values for a and b
Test for R2
Partitioning variance and statistical control
Multiple regression
Other features of multiple regression
 coefficients
Collinearity
Sample size
Statistical control
A three independent variable example
What have we learned?
Applications
Exercises
Advanced Topic: Adjusted R2
Advanced Topic: Residuals and statistical control