Advance Researcha and Statistic Guidence.pdf

Daniel J. Denis
SPSS Data Analysis for Univariate,
Bivariate, and Multivariate Statistics

This edition first published 2019
© 2019 John Wiley & Sons, Inc.
Printed in the United States of America
Set in 10/12pt Warnock by SPi Global, Pondicherry, India
Names: Denis, Daniel J., 1974– author.
Title: SPSS data analysis for univariate, bivariate, and multivariate statistics / Daniel J. Denis.
Description: Hoboken, NJ : Wiley, 2019. | Includes bibliographical references and index. |
Identifiers: LCCN 2018025509 (print) | LCCN 2018029180 (ebook) | ISBN 9781119465805 (Adobe PDF) |
ISBN 9781119465782 (ePub) | ISBN 9781119465812 (hardcover)
Subjects: LCSH: Analysis of variance–Data processing. | Multivariate analysis–Data processing. |
Mathematical statistics–Data processing. | SPSS (Computer file)
Classification: LCC QA279 (ebook) | LCC QA279 .D45775 2019 (print) | DDC 519.5/3–dc23
LC record available at https://lccn.loc.gov/2018025509
Library of Congress Cataloging‐in‐Publication Data

Preface ix
1 Review of Essential Statistical Principles 1
1.1
Variables and Types of Data 2
1.2
Significance Tests and Hypothesis Testing 3
1.3
Significance Levels and Type I and Type II Errors 4
1.4
Sample Size and Power 5
1.5 Model Assumptions 6
2 Introduction to SPSS 9
2.1
How to Communicate with SPSS 9
2.2
Data View vs. Variable View 10
2.3
Missing Data in SPSS: Think Twice Before Replacing Data! 12
3 Exploratory Data Analysis, Basic Statistics, and Visual Displays 19
3.1 Frequencies and Descriptives 19
3.2
The Explore Function 23
3.3
What Should I Do with Outliers? Delete or Keep Them? 28
3.4 Data Transformations 29
4 Data Management in SPSS 33
4.1
Computing a New Variable 33
4.2 Selecting Cases 34
4.3
Recoding Variables into Same or Different Variables 36
4.4 Sort Cases 37
4.5 Transposing Data 38
5 Inferential Tests on Correlations, Counts, and Means 41
5.1 Computing z‐Scores in SPSS 41
5.2 Correlation Coefficients 44
5.3
A Measure of Reliability: Cohen’s Kappa 52
5.4 Binomial Tests 52
5.5
Chi‐square Goodness‐of‐fit Test 54
Contents

5.6 One‐sample t‐Test for a Mean 57
5.7 Two‐sample t‐Test for Means 59
6 Power Analysis and Estimating Sample Size 63
6.1
Example Using G*Power: Estimating Required Sample Size for
Detecting Population Correlation 64
6.2
Power for Chi‐square Goodness of Fit 66
6.3 Power for Independent‐samples t‐Test 66
6.4 Power for Paired‐samples t‐Test 67
7 Analysis of Variance: Fixed and Random Effects 69
7.1
Performing the ANOVA in SPSS 70
7.2 The F‐Test for ANOVA 73
7.3 Effect Size 74
7.4
Contrasts and Post Hoc Tests on Teacher 75
7.5
Alternative Post Hoc Tests and Comparisons 78
7.6
Random Effects ANOVA 80
7.7
Fixed Effects Factorial ANOVA and Interactions 82
7.8
What Would the Absence of an Interaction Look Like? 86
7.9
Simple Main Effects 86
7.10
Analysis of Covariance (ANCOVA) 88
7.11
Power for Analysis of Variance 90
8 Repeated Measures ANOVA 91
8.1
One‐way Repeated Measures 91
8.2
Two‐way Repeated Measures: One Between and One Within Factor 99
9 Simple and Multiple Linear Regression 103
9.1
Example of Simple Linear Regression 103
9.2
Interpreting a Simple Linear Regression: Overview of Output 105
9.3
Multiple Regression Analysis 107
9.4 Scatterplot Matrix 111
9.5
Running the Multiple Regression 112
9.6
Approaches to Model Building in Regression 118
9.7
Forward, Backward, and Stepwise Regression 120
9.8
Interactions in Multiple Regression 121
9.9
Residuals and Residual Plots: Evaluating Assumptions 123
9.10
Homoscedasticity Assumption and Patterns of Residuals 125
9.11
Detecting Multivariate Outliers and Influential Observations 126
9.12 Mediation Analysis 127
9.13 Power for Regression 129
10 Logistic Regression 131
10.1
Example of Logistic Regression 132
10.2
Multiple Logistic Regression 138
10.3
Power for Logistic Regression 139

11 Multivariate Analysis of Variance (MANOVA) and Discriminant Analysis 141
11.1 Example of MANOVA 142
11.2 Effect Sizes 146
11.3
Box’s M Test 147
11.4
Discriminant Function Analysis 148
11.5
Equality of Covariance Matrices Assumption 152
11.6
MANOVA and Discriminant Analysis on Three Populations 153
11.7 Classification Statistics 159
11.8 Visualizing Results 161
11.9
Power Analysis for MANOVA 162
12 Principal Components Analysis 163
12.1 Example of PCA 163
12.2
Pearson’s 1901 Data 164
12.3 Component Scores 166
12.4
Visualizing Principal Components 167
12.5
PCA of Correlation Matrix 170
13 Exploratory Factor Analysis 175
13.1
The Common Factor Analysis Model 175
13.2
The Problem with Exploratory Factor Analysis 176
13.3
Factor Analysis of the PCA Data 176
13.4
What Do We Conclude from the Factor Analysis? 179
13.5 Scree Plot 180
13.6
Rotating the Factor Solution 181
13.7
Is There Sufficient Correlation to Do the Factor Analysis? 182
13.8
Reproducing the Correlation Matrix 183
13.9 Cluster Analysis 184
13.10
How to Validate Clusters? 187
13.11
Hierarchical Cluster Analysis 188
14 Nonparametric Tests 191
14.1
Independent‐samples: Mann–Whitney U 192
14.2
Multiple Independent‐samples: Kruskal–Wallis Test 193
14.3
Repeated Measures Data: The Wilcoxon Signed‐rank
Test and Friedman Test 194
14.4
The Sign Test 196
Closing Remarks and Next Steps 199
References 201
Index 203

The goals of this book are to present a very concise, easy‐to‐use introductory primer of a host of
computational tools useful for making sense out of data, whether that data come from the social,
behavioral, or natural sciences, and to get you started doing data analysis fast. The emphasis on the
book is data analysis and drawing conclusions from empirical observations. The emphasis of the
book is not on theory. Formulas are given where needed in many places, but the focus of the book is
on concepts rather than on mathematical abstraction. We emphasize computational tools used in
the discovery of empirical patterns and feature a variety of popular statistical analyses and data
management tasks that you can immediately apply as needed to your own research. The book features
analysesanddemonstrationsusingSPSS.Mostofthedatasetsanalyzedareverysmallandconvenient,
so entering them into SPSS should be easy. If desired, however, one can also download them from
www.datapsyc.com. Many of the data sets were also first used in a more theoretical text written by
the same author (see Denis, 2016), which should be consulted for a more in‐depth treatment of the
topics presented in this book. Additional references for readings are also given throughout the book.

Target Audience and Level
This is a “how‐to” book and will be of use to undergraduate and graduate students along with
researchers and professionals who require a quick go‐to source, to help them perform essential
statistical analyses and data management tasks. The book only assumes minimal prior knowledge of
statistics, providing you with the tools you need right now to help you understand and interpret your
data analyses. A prior introductory course in statistics at the undergraduate level would be helpful,
but is not required for this book. Instructors may choose to use the book either as a primary text for
an undergraduate or graduate course or as a supplement to a more technical text, referring to this
book primarily for the “how to’s” of data analysis in SPSS. The book can also be used for self‐study. It
is suitable for use as a general reference in all social and natural science fields and may also be of
interest to those in business who use SPSS for decision‐making. References to further reading are
provided where appropriate should the reader wish to follow up on these topics or expand one’s
knowledge base as it pertains to theory and further applications. An early chapter reviews essential
statistical and research principles usually covered in an introductory statistics course, which should
be sufficient for understanding the rest of the book and interpreting analyses. Mini brief sample
write‐ups are also provided for select analyses in places to give the reader a starting point to writing
up his/her own results for his/her thesis, dissertation, or publication. The book is meant to be an
Preface

easy, user‐friendly introduction to a wealth of statistical methods while simultaneously demonstrat-
ing their implementation in SPSS. Please contact me at daniel.denis@umontana.edu or email@data-
psyc.com with any comments or corrections.

Glossary of Icons and Special Features
When you see this symbol, it means a brief sample write‐up has been provided for the
accompanying output. These brief write‐ups can be used as starting points to writing up
your own results for your thesis/dissertation or even publication.
When you see this symbol, it means a special note, hint, or reminder has been provided or
signifies extra insight into something not thoroughly discussed in the text.
When you see this symbol, it means a special WARNING has been issued that if not fol-
lowed may result in a serious error.
Acknowledgments
Thanks go out to Wiley for publishing this book, especially to Jon Gurstelle for presenting the idea to
Wiley and securing the contract for the book and to Mindy Okura‐Marszycki for taking over the
project after Jon left. Thank you Kathleen Pagliaro for keeping in touch about this project and the
former book. Thanks goes out to everyone (far too many to mention) who have influenced me in one
way or another in my views and philosophy about statistics and science, including undergraduate and
graduate students whom I have had the pleasure of teaching (and learning from) in my courses taught
at the University of Montana.
This book is dedicated to all military veterans of the United States of America, past, present, and
future, who teach us that all problems are relative.

1
The purpose of statistical modeling is to both describe sample data and make inferences about that
sample data to the population from which the data was drawn. We compute statistics on samples
(e.g. sample mean) and use such statistics as estimators of population parameters (e.g. population
mean). When we use the sample statistic to estimate a parameter in the population, we are engaged
in the process of inference, which is why such statistics are referred to as inferential statistics, as
opposed to descriptive statistics where we are typically simply describing something about a sample
or population. All of this usually occurs in an experimental design (e.g. where we have a control vs.
treatment group) or nonexperimental design (where we exercise little or no control over variables).
As an example of an experimental design, suppose you wanted to learn whether a pill was effective
in reducing symptoms from a headache. You could sample 100 individuals with headaches, give them
a pill, and compare their reduction in symptoms to 100 people suffering from a headache but not
receiving the pill. If the group receiving the pill showed a decrease in symptomology compared with
the nontreated group, it may indicate that your pill is effective. However, to estimate whether the
effect observed in the sample data is generalizable and inferable to the population from which the
data were drawn, a statistical test could be performed to indicate whether it is plausible that such a
difference between groups could have occurred simply by chance. If it were found that the difference
was unlikely due to chance, then we may indeed conclude a difference in the population from which
the data were drawn. The probability of data occurring under some assumption of (typically) equality
is the infamous p‐value, usually set at 0.05. If the probability of such data is relatively low (e.g. less
than 0.05) under the null hypothesis of no difference, we reject the null and infer the statistical alter‑
native hypothesis of a difference in population means.
Much of statistical modeling follows a similar logic to that featured above – sample some data,
apply a model to the data, and then estimate how good the model fits and whether there is inferential
evidence to suggest an effect in the population from which the data were drawn. The actual model you
will fit to your data usually depends on the type of data you are working with. For instance, if you have
collected sample means and wish to test differences between means, then t‐test and ANOVA tech‑
niques are appropriate. On the other hand, if you have collected data in which you would like to see
if there is a linear relationship between continuous variables, then correlation and regression are
usually appropriate. If you have collected data on numerous dependent variables and believe these
variables, taken together as a set, represent some kind of composite variable, and wish to determine
mean differences on this composite dependent variable, then a multivariate analysis of variance
(MANOVA) technique may be useful. If you wish to predict group membership into two or more
1
Review of Essential Statistical Principles
Big Picture on Statistical Modeling and Inference

1 Review of Essential Statistical Principles
2
categories based on a set of predictors, then discriminant analysis or logistic regression would be
an option. If you wished to take many variables and reduce them down to fewer dimensions, then
principal components analysis or factor analysis may be your technique of choice. Finally, if you
are interested in hypothesizing networks of variables and their interrelationships, then path analysis
and structural equation modeling may be your model of choice (not covered in this book). There
are numerous other possibilities as well, but overall, you should heed the following principle in guid‑
ing your choice of statistical analysis:
1.1
Variables and Types of Data
Recall that variables are typically of two kinds – dependent or response variables and independent
or predictor variables. The terms “dependent” and “independent” are most common in ANOVA‐
type models, while “response” and “predictor” are more common in regression‐type models, though
their usage is not uniform to any particular methodology. The classic function statement Y = f(X) tells
the story – input a value for X (independent variable), and observe the effect on Y (dependent vari‑
able). In an independent‐samples t‐test, for instance, X is a variable with two levels, while the depend‑
ent variable is a continuous variable. In a classic one‐way ANOVA, X has multiple levels. In a simple
linear regression, X is usually a continuous variable, and we use the variable to make predictions of
another continuous variable Y. Most of statistical modeling is simply observing an outcome based on
something you are inputting into an estimated (estimated based on the sample data) equation.
Data come in many different forms. Though there are rather precise theoretical distinctions
between different forms of data, for applied purposes, we can summarize the discussion into the fol‑
lowing types for now: (i) continuous and (ii) discrete. Variables measured on a continuous scale can,
in theory, achieve any numerical value on the given scale. For instance, length is typically considered
to be a continuous variable, since we can measure length to any specified numerical degree. That is,
the distance between 5 and 10in. on a scale contains an infinite number of measurement possibilities
(e.g. 6.1852, 8.341364, etc.). The scale is continuous because it assumes an infinite number of possi‑
bilities between any two points on the scale and has no “breaks” in that continuum. On the other
hand, if a scale is discrete, it means that between any two values on the scale, only a select number of
possibilities can exist. As an example, the number of coins in my pocket is a discrete variable, since I
cannot have 1.5 coins. I can have 1 coin, 2 coins, 3 coins, etc., but between those values do not exist
an infinite number of possibilities. Sometimes data is also categorical, which means values of the
variable are mutually exclusive categories, such as A or B or C or “boy” or “girl.” Other times, data
come in the form of counts, where instead of measuring something like IQ, we are only counting the
number of occurrences of some behavior (e.g. number of times I blink in a minute). Depending on
the type of data you have, different statistical methods will apply. As we survey what SPSS has to
offer, we identify variables as continuous, discrete, or categorical as we discuss the given method.
However, do not get too caught up with definitions here; there is always a bit of a “fuzziness” in
The type of statistical model or method you select often depends on the types of data you have
and your purpose for wanting to build a model. There usually is not one and only one method
that is possible for a given set of data. The method of choice will be dictated often by the ration-
aleofyourresearch.Youmustknowyourvariablesverywellalongwiththegoalsofyourresearch
to diligently select a statistical model.

1.2 Significance Tests and Hypothesis Testing 3
learning about the nature of the variables you have. For example, if I count the number of raindrops
in a rainstorm, we would be hard pressed to call this “count data.” We would instead just accept it as
continuous data and treat it as such. Many times you have to compromise a bit between data types to
best answer a research question. Surely, the average number of people per household does not make
sense, yet census reports often give us such figures on “count” data. Always remember however that
the software does not recognize the nature of your variables or how they are measured. You have to
be certain of this information going in; know your variables very well, so that you can be sure
SPSS is treating them as you had planned.
Scales of measurement are also distinguished between nominal, ordinal, interval, and ratio. A
nominal scale is not really measurement in the first place, since it is simply assigning labels to objects
we are studying. The classic example is that of numbers on football jerseys. That one player has the
number 10 and another the number 15 does not mean anything other than labels to distinguish
between two players. If differences between numbers do represent magnitudes, but that differences
between the magnitudes are unknown or imprecise, then we have measurement at the ordinal level.
For example, that a runner finished first and another second constitutes measurement at the ordinal
level. Nothing is said of the time difference between the first and second runner, only that there is a
“ranking” of the runners. If differences between numbers on a scale represent equal lengths, but that
an absolute zero point still cannot be defined, then we have measurement at the interval level. A classic
example of this is temperature in degrees Fahrenheit – the difference between 10 and 20° represents
the same amount of temperature distance as that between 20 and 30; however zero on the scale does
not represent an “absence” of temperature. When we can ascribe an absolute zero point in addition
to inferring the properties of the interval scale, then we have measurement at the ratio scale. The
number of coins in my pocket is an example of ratio measurement, since zero on the scale represents
a complete absence of coins. The number of car accidents in a year is another variable measurable on
a ratio scale, since it is possible, however unlikely, that there were no accidents in a given year.
The first step in choosing a statistical model is knowing what kind of data you have, whether they
are continuous, discrete, or categorical and with some attention also devoted to whether the data are
nominal, ordinal, interval, or ratio. Making these decisions can be a lot trickier than it sounds, and
you may need to consult with someone for advice on this before selecting a model. Other times, it is
very easy to determine what kind of data you have. But if you are not sure, check with a statistical
consultant to help confirm the nature of your variables, because making an error at this initial stage
of analysis can have serious consequences and jeopardize your data analyses entirely.
1.2 Significance Tests and Hypothesis Testing
In classical statistics, a hypothesis test is about the value of a parameter we are wishing to estimate
with our sample data. Consider our previous example of the two‐group problem regarding trying to
establish whether taking a pill is effective in reducing headache symptoms. If there were no differ‑
ence between the group receiving the treatment and the group not receiving the treatment, then we
would expect the parameter difference to equal 0. We state this as our null hypothesis:
Null hypothesis: The mean difference in the population is equal to 0.
The alternative hypothesis is that the mean difference is not equal to 0. Now, if our sample means
come out to be 50.0 for the control group and 50.0 for the treated group, then it is obvious that we do

4
not have evidence to reject the null, since the difference of 50.0 – 50.0=0 aligns directly with expecta-
tion under the null. On the other hand, if the means were 48.0 vs. 52.0, could we reject the null? Yes,
there is definitely a sample difference between groups, but do we have evidence for a population

difference? It is difficult to say without asking the following question:
What is the probability of observing a difference such as 48.0 vs. 52.0
under the null hypothesis of no difference?
When we evaluate a null hypothesis, it is the parameter we are interested in, not the sample statis‑
tic. The fact that we observed a difference of 4 (i.e. 52.0–48.0) in our sample does not by itself indicate
that in the population, the parameter is unequal to 0. To be able to reject the null hypothesis, we
need to conduct a significance test on the mean difference of 48.0 vs. 52.0, which involves comput‑
ing (in this particular case) what is known as a standard error of the difference in means to estimate
how likely such differences occur in theoretical repeated sampling. When we do this, we are compar‑
ing an observed difference to a difference we would expect simply due to random variation. Virtually
all test statistics follow the same logic. That is, we compare what we have observed in our sample(s)
to variation we would expect under a null hypothesis or, crudely, what we would expect under simply
“chance.” Virtually all test statistics have the following form:
Test statistic=observed/expected
If the observed difference is large relative to the expected difference, then we garner evidence that
such a difference is not simply due to chance and may represent an actual difference in the popula‑
tion from which the data were drawn.
As mentioned previously, significance tests are not only performed on mean differences, however.
Whenever we wish to estimate a parameter, whatever the kind, we can perform a significance test on
it. Hence, when we perform t‐tests, ANOVAs, regressions, etc., we are continually computing sample
statistics and conducting tests of significance about parameters of interest. Whenever you see such
output as “Sig.” in SPSS with a probability value underneath it, it means a significance test has been
performed on that statistic, which, as mentioned already, contains the p‐value. When we reject the
null at, say, p 0.05, however, we do so with a risk of either a type I or type II error. We review these
next, along with significance levels.
1.3
Significance Levels and Type I and Type II Errors
Whenever we conduct a significance test on a parameter and decide to reject the null hypothesis, we
do not know for certain that the null is false. We are rather hedging our bet that it is false. For
instance, even if the mean difference in the sample is large, though it probably means there is a dif‑
ference in the corresponding population parameters, we cannot be certain of this and thus risk falsely
rejecting the null hypothesis. How much risk are we willing to tolerate for a given significance test?
Historically, a probability level of 0.05 is used in most settings, though the setting of this level should
depend individually on the given research context. The infamous “p 0.05” means that the probabil-
ity of the observed data under the null hypothesis is less than 5%, which implies that if such data are
so unlikely under the null, that perhaps the null hypothesis is actually false, and that the data are
more probable under a competing hypothesis, such as the statistical alternative hypothesis. The
point to make here is that whenever we reject a null and conclude something about the population

1.4 Sample Size and Power 5
parameters, we could be making a false rejection of the null hypothesis. Rejecting a null hypothesis
when in fact the null is not false is known as a type I error, and we usually try to limit the probability
of making a type I error to 5% or less in most research contexts. On the other hand, we risk another
type of error, known as a type II error. These occur when we fail to reject a null hypothesis that in
actuality is false. More practically, this means that there may actually be a difference or effect in the
population but that we failed to detect it. In this book, by default, we usually set the significance level
at 0.05 for most tests. If the p‐value for a given significance test dips below 0.05, then we will typically
call the result “statistically significant.” It needs to be emphasized however that a statistically signifi‑
cant result does not necessarily imply a strong practical effect in the population.
For reasons discussed elsewhere (see Denis (2016) Chapter 3 for a thorough discussion), one can
potentially obtain a statistically significant finding (i.e. p 0.05) even if, to use our example about the
headache treatment, the difference in means is rather small. Hence, throughout the book, when we
note that a statistically significant finding has occurred, we often couple this with a measure of effect
size, which is an indicator of just how much mean difference (or other effect) is actually present. The
exact measure of effect size is different depending on the statistical method, so we explain how to
interpret the given effect size in each setting as we come across it.
1.4
Sample Size and Power
Power is reviewed in Chapter 6, but an introductory note about it and how it relates to sample size
is in order. Crudely, statistical power of a test is the probability of detecting an effect if there is an
effect to be detected. A microscope analogy works well here – there may be a virus strain present
under the microscope, but if the microscope is not powerful enough to detect it, you will not see it.
It still exists, but you just do not have the eyes for it. In research, an effect could exist in the popula‑
tion, but if you do not have a powerful test to detect it, you will not spot it. Statistically, power is the
probability of rejecting a null hypothesis given that it is false. What makes a test powerful? The
determinants of power are discussed in Chapter 6, but for now, consider only the relation between
effect size and sample size as it relates to power. All else equal, if the effect is small that you are trying
to detect, you will need a larger sample size to detect it to obtain sufficient power. On the other hand,
if the effect is large that you are trying to detect, you can get away with a small sample size in detect‑
ing it and achieve the same degree of power. So long as there is at least some effect in the population,
then by increasing sample size indefinitely, you assure yourself of gaining as much power as you like.
That is, increasing sample size all but guarantees a rejection of a null hypothesis! So, how big do
you want your samples? As a rule, larger samples are better than smaller ones, but at some point,
collecting more subjects increases power only minimally, and the expense associated with increasing
sample size is no longer worth it. Some techniques are inherently large sample techniques and require
relatively large sample sizes. How large? For factor analysis, for instance, samples upward of 300–500
are often recommended, but the exact guidelines depend on things like sizes of communalities and
other factors (see Denis (2016) for details). Other techniques require lesser‐sized samples (e.g. t‐tests
and nonparametric tests). If in doubt, however, collecting larger samples than not is preferred, and
you need never have to worry about having “too much” power. Remember, you are only collecting
smaller samples because you cannot get a collection of the entire population, so theoretically and
pragmatically speaking, larger samples are typically better than smaller ones across the board of
statistical methodologies.

6
1.5 Model Assumptions
The majority of statistical tests in this book are based on a set of assumptions about the data that if
violated, comprise the validity of the inferences made. What this means is that if certain assumptions
about the data are not met, or questionable, it compromises the validity with which interpreting
p‑values and other inferential statistics can be made. Some authors also include such things as adequate
sample size as an assumption of many multivariate techniques, but we do not include such things
when discussing any assumptions, for the reason that large sample sizes for procedures such as factor
analysis we see more as a requirement of good data analysis than something assumed by the theoreti‑
cal model.
We must at this point distinguish between the platonic theoretical ideal and pragmatic reality. In
theory, many statistical tests assume data were drawn from normal populations, whether univari‑
ate, bivariate, or multivariate, depending on the given method. Further, multivariate methods usually
assume linear combinations of variables also arise from normal populations. But are data ever
drawn from truly normal populations? No! Never! We know this right off the start because perfect
normality is a theoretical ideal. In other words, the normal distribution does not “exist” in the real
world in a perfect sense; it exists only in formulae and theoretical perfection. So, you may ask, if nor‑
mality in real data is likely to never truly exist, why are so many inferential tests based on the assump‑
tion of normality? The answer to this usually comes down to convenience and desirable properties
when innovators devise inferential tests. That is, it is much easier to say, “Given the data are multi‑
variate normal, then this and that should be true.” Hence, assuming normality makes theoretical
statistics a bit easier and results are more tractable. However, when we are working with real data in
the real world, samples or populations while perhaps approximating this ideal, will never truly.
Hence, if we face reality up front and concede that we will never truly satisfy assumptions of a statisti‑
cal test, the quest then becomes that of not violating the assumptions to any significant degree such
that the test is no longer interpretable. That is, we need ways to make sure our data behave “reason‑
ably well” as to still apply the statistical test and draw inferential conclusions.
There is a second concern, however. Not only are assumptions likely to be violated in practice, but
it is also true that some assumptions are borderline unverifiable with real data because the data occur
in higher dimensions, and verifying higher‐dimensional structures is extremely difficult and is an
evolving field. Again, we return to normality. Verifying multivariate normality is very difficult, and
hence many times researchers will verify lower dimensions in the hope that if these are satisfied, they
can hopefully induce that higher‐dimensional assumptions are thus satisfied. If univariate and bivari‑
ate normality is satisfied, then we can be more certain that multivariate normality is likely satisfied.
However, there is no guarantee. Hence, pragmatically, much of assumption checking in statistical
modeling involves looking at lower dimensions as to make sure such data are reasonably behaved. As
concerns sampling distributions, often if sample size is sufficient, the central limit theorem will
assure us of sampling distribution normality, which crudely says that normality will be achieved as
sample size increases. For a discussion of sampling distributions, see Denis (2016).
A second assumption that is important in data analysis is that of homogeneity or homoscedastic-
ity of variances. This means different things depending on the model. In t‐tests and ANOVA, for
instance, the assumption implies that population variances of the dependent variable in each level of
the independent variable are the same. The way this assumption is verified is by looking at sample
data and checking to make sure sample variances are not too different from one another as to raise a
concern. In t‐tests and ANOVA, Levene’s test is sometimes used for this purpose, or one can also

1.5 Model Assumptions 7
use a rough rule of thumb that says if one sample variance is no more than four times another,
then the assumption can be at least tentatively justified. In regression models, the assumption of
homoscedasticity is usually in reference to the distribution of Y given the conditional value of the
predictor(s). Hence, for each value of X, we like to assume approximate equal dispersion of values
of Y. This assumption can be verified in regression through scatterplots (in the bivariate case) and
residual plots in the multivariable case.
A third assumption, perhaps the most important, is that of independence. The essence of this
assumption is that observations at the outset of the experiment are not probabilistically related. For
example, when recruiting a sample for a given study, if observations appearing in one group “know
each other” in some sense (e.g. friendships), then knowing something about one observation may tell
us something about another in a probabilistic sense. This violates independence. In regression analy‑
sis, independence is violated when errors are related with one another, which occurs quite frequently
in designs featuring time as an explanatory variable. Independence can be very difficult to verify in
practice, though residual plots are again helpful in this regard. Oftentimes, however, it is the very
structure of the study and the way data was collected that will help ensure this assumption is met.
When you recruited your sample data, did you violate independence in your recruitment
procedures?
The following is a final thought for now regarding assumptions, along with some recommenda‑
tions. While verifying assumptions is important and a worthwhile activity, one can easily get caught
up in spending too much time and effort seeking an ideal that will never be attainable. In consulting
on statistics for many years now, more than once I have seen some students and researchers obsess
and ruminate over a distribution that was not perfectly normal and try data transformation after data
transformation to try to “fix things.” I generally advise against such an approach, unless of course
there are serious violations in which case remedies are therefore needed. But keep in mind as well
that a violation of an assumption may not simply indicate a statistical issue; it may hint at a substan-
tive one. A highly skewed distribution, for instance, one that goes contrary to what you expected to
obtain, may signal a data collection issue, such as a bias in your data collection mechanism. Too often
researchers will try to fix the distribution without asking why it came out as “odd ball” as it did. As a
scientist, your job is not to appease statistical tests. Your job is to learn of natural phenomena
and use statistics as a tool in that venture. Hence, if you suspect an assumption is violated and are
not quite sure what to do about it, or if it requires any remedy at all, my advice is to check with a
statistical consultant about it to get some direction on it before you transform all your data and make
a mess of things! The bottom line too is that if you are interpreting p‐values so obsessively as to be
that concerned that a violation of an assumption might increase or decrease the p‐value by miniscule
amounts, you are probably overly focused on p‐values and need to start looking at the science (e.g.
effect size) of what you are doing. Yes, a violation of an assumption may alter your true type I error
rate, but if you are that focused on the exact level of your p‐value from a scientific perspective, that
is the problem, not the potential violation of the assumption. Having said all the above, I summarize
with four pieces of advice regarding how to proceed, in general, with regard to assumptions:
1) If you suspect a light or minor violation of one of your assumptions, determine a potential source
of the violation and if your data are in error. Correct errors if necessary. If no errors in data collec‑
tion were made, and if the assumption violation is generally light (after checking through plots
and residuals), you are probably safe to proceed and interpret results of inferential tests without
any adjustments to your data.

8
2) If you suspect a heavy or major violation of one of your assumptions, and it is “repairable,” (to the
contrary, if independence is violated during the process of data collection, it is very difficult or
impossible to repair), you may consider one of the many data transformations available, assum-
ing the violation was not due to the true nature of your distributions. For example, learning that
most of your subjects responded “zero” to the question of how many car accidents occurred to
them last month is not a data issue – do not try to transform such data to ease the positive skew!
Rather, the correct course of action is to choose a different statistical model and potentially reop‑
erationalize your variable from a continuous one to a binary or polytomous one.
3) If your violation, either minor or major, is not due to a substantive issue, and you are not sure
whether to transform or not transform data, you may choose to analyze your data with and then
without transformation, and compare results. Did the transformation influence the decision on
null hypotheses? If so, then you may assume that performing the transformation was worthwhile
and keep it as part of your data analyses. This does not imply that you should “fish” for statistical
significance through transformations. All it means is that if you are unsure of the effect of a viola‑
tion on your findings, there is nothing wrong with trying things out with the original data and
then transformed data to see how much influence the violation carries in your particular case.
4) A final option is to use a nonparametric test in place of a parametric one, and as in (3), compare
results in both cases. If normality is violated, for instance, there is nothing wrong with trying out
a nonparametric test to supplement your parametric one to see if the decision on the null changes.
Again, I am not recommending “fishing” for the test that will give you what you want to see (e.g.
p 0.05). What I am suggesting is that comparing results from parametric and nonparametric
tests can sometimes helps give you an inexact, but still useful, measure of the severity (in a very
crude way) of the assumption violation. Chapter 14 reviews select nonparametric tests.
Throughout the book, we do not verify each assumption for each analysis we conduct, as to save
on space and also because it detracts a bit from communicating how the given tests work. Further,
many of our analyses are on very small samples for convenience, and so verifying parametric assump‑
tions is unrealistic from the outset. However, for each test you conduct, you should be generally
aware that it comes with a package of assumptions, and explore those assumptions as part of your
data analyses, and if in doubt about one or more assumptions, consult with someone with more
expertise on the severity of any said violation and what kind of remedy may (or may not be) needed.
In general, get to know your data before conducting inferential analyses, and keep a close eye out
for moderate‐to‐severe assumption violations.
Many of the topics discussed in this brief introductory chapter are reviewed in textbooks such as
Howell (2002) and Kirk (2008).

9
In this second chapter, we provide a brief introduction to SPSS version 22.0 software. IBM SPSS

provides a host of online manuals that contain the complete capabilities of the software, and beyond
brief introductions such as this one should be consulted for specifics about its programming options.
These can be downloaded directly from IBM SPSS’s website. Whether you are using version 22.0 or an
earlier or later version, most of the features discussed in this book will be consistent from version to
version, so there is no cause for alarm if the version you are using is not the one featured in this book.
This is a book on using SPSS in general, not a specific version. Most software upgrades of SPSS ver-
sions are not that different from previous versions, though you are encouraged to keep up to date with
SPSS bulletins regarding upgrades or corrections (i.e. bugs) to the software. We survey only select
possibilities that SPSS has to offer in this chapter and the next, enough to get you started
performing
data analysis quickly on a host of models featured in this book. For further details on data manage-
ment in SPSS not covered in this chapter or the next, you are encouraged to consult Kulas (2008).
2.1
How to Communicate with SPSS
There are basically two ways a user can communicate with SPSS – through syntax commands
entered directly in the SPSS syntax window and through point‐and‐click commands via the graphi-
cal user interface (GUI). Conducting analyses via the GUI is sufficient for most essential tasks fea-
tured in this book. However, as you become more proficient with SPSS and may require advanced
computing commands for your specific analyses, manually entering syntax code may become neces-
sary or even preferable once you become more experienced at programming. In this introduction, we
feature analyses performed through both syntax commands and GUI. In reality, the GUI is simply a
reflection of the syntax operations that are taking place “behind the scenes” that SPSS has automated
through easy‐to‐access applications, similar to how selecting an app on your cell phone is a type of
fast shortcut to get you to where you want to go. The user should understand from the outset how-
ever that there are things one can do using syntax that cannot automatically be performed through
the GUI (just like on your phone, there is not an app for everything!), so it behooves one to learn at
least elementary programming skills at some point if one is going to work extensively in the field of
data analysis. In this book, we show as much as possible the window commands to obtaining output
and, in many places, feature the representative syntax should you ever need to adjust it to customize
your analysis for the given problem you are confronting. One word of advice – do not be
2
Introduction to SPSS

2 Introduction to SPSS
10
intimidated when you see syntax, since as mentioned, for the majority of analyses presented in this
book, you will not need to use it specifically. However, by seeing the corresponding syntax to the
window commands you are running, it will help “demystify” what SPSS is actually doing, and then
through trial and error (and SPSS’s documentation and manuals), the day may come where you are
adjusting syntax on your own for the purpose of customizing your analyses, such as one regularly
does in software packages such as R or SAS, where typing in commands and running code is the
habitual way of proceeding.
2.2 Data View vs. Variable View
When you open SPSS, you will find two choices for SPSS’s primary
window – Data View vs. Variable
View (both contrasted in Figure 2.1). The Data View is where you will manually enter data into SPSS,
whereas the Variable View is where you will do such things as enter the names of variables, adjust the
numerical width of variables, and provide labels for variables.
The case numbers in SPSS are listed along the left‐hand column. For
instance, in Figure 2.1, in the Data View (left), approximately 28 cases are
shown. In the Variable View, 30 cases are shown. Entering data into SPSS is
very easy. As an example, consider the following small hypothetical data set
(left) on verbal, quantitative, and analytical scores for a group of students
on a standardized “IQ test” (scores range from 0 to 100, where 0 indicates
virtually no ability and 100 indicates very much ability). The “group” variable
denotes whether students have studied “none” (0), “some” (1), or “much” (2).
Entering data into SPSS is no more complicated than what we have done
above, and barring a few adjustments, we could easily go ahead and start
conducting analyses on our data immediately. Before we do so, let us have
a quick look at a few of the features in the Variable View for these data and
how to adjust them.
Figure 2.1 SPSS Data View (left) vs. Variable View (right).

2.2 Data View vs. Variable View 11
Let us take a look at a few of the above column
headers in the Variable View:
Name – this is the name of the variable we have
entered.
Type – if you click on Type (in the cell), SPSS will
open the following window:
Verify for yourself that you are able to read the data correctly. The first person (case 1) in the data set
scored “56.00” on verbal, “56.00” on quant, and “59.00” on analytic and is in group “0,” the group that
studied “none.”The second person (case 2) in the data set scored “59.00” on verbal, “42.00” on quant,
and “54.00” on analytic and is also in group “0.”The 11th individual in the data set scored “66.00” on
verbal,“55.00”on quant, and“69.00”on analytic and is in group“1,”the group that studied“some”for
the evaluation.
Notice that under Variable Type are many options. We can specify the variable as numeric (default
choice) or comma or dot, along with specifying the width of the variable and the number of decimal
places we wish to carry for it (right‐hand side of window). We do not explore these options in this book
for the reason that for most analyses that you conduct using quantitative variables, the numeric varia-
ble type will be appropriate, and specifying the width and number of decimal places is often a matter
of taste or preference rather than one of necessity. Sometimes instead of numbers, data come in the
form of words, which makes the“string”option appropriate. For instance, suppose that instead of“0 vs.
1 vs. 2”we had actually entered“none,”“some,”or“much.”We would have selected“string”to represent
our variable (which I am calling“group_name”to differentiate it from“group”[see below]).

12
Having entered our data, we could begin conducting analyses immediately. However, sometimes
researchers wish to attach value labels to their data if they are using numbers to code categories.
This can easily be accomplished by selecting the Values tab. For example, we will do this for our
group variable:

There are a few other options available in Variable View such as Missing, Columns, and Measure,
but we leave them for now as they are not vital to getting started. If you wish, you can access the
Measure tab and record whether your variable is nominal, ordinal, or interval/ratio (known as scale
in SPSS), but so long as you know how you are treating your variables, you need not record this in
SPSS. For instance, if you have nominal data with categories 0 and 1, you do not need to tell SPSS the
variable is nominal; you can simply select statistical routines that require this variable to be nominal
and interpret it as such in your analyses.
2.3
Missing Data in SPSS: Think Twice Before Replacing Data!
Ideally, when you collect data for an experiment or study, you are able to collect measurements
from every participant, and your data file will be complete. However, often, missing data occurs.
For example, suppose our IQ data set, instead of appearing nice and complete, had a few missing
observations:
Whether we use words to categorize this variable or numbers makes little difference so
long as we are aware ourselves regarding what the variable is and how we are using the vari-
able. For instance, that we coded group from 0 to 2 is fine, so long as we know these
numbers represent categories rather than true measured quantities. Had we incorrectly analyzed
the data such that 0 to 2 is assumed to exist on a continuous scale rather than represent categories,
we risk ensuing analyses (e.g. such as analysis of variance) being performed incorrectly.

2.3 Missing Data in SPSS: Think Twice Before Replacing Data! 13
Any attempt to replace a missing data point, regard-
less of the approach used, is nonetheless an educated
“guess” at what that data point may have been had the
participant answered or it had not gone missing.
Presumably, the purpose of your scientific investigation
was to do
science, which means making measurements on
objects in nature. In conducting such a scientific investiga-
tion, the data is your only true link to what you are study-
ing. Replacing a missing value means you are prepared to
“guesstimate” what the observation is, which means it
is no longer a direct reflection of your measurement

process. In some cases, such as in repeated measures or
longitudinal designs, avoiding missing data is difficult
because participants may drop out of longitudinal studies
or simply stop showing up. However, that does not necessarily mean you should automatically replace
their values. Get curious about your missing data. For our IQ data, though we may be able to attribute
the missing observations for cases 8 and 13 as possibly “missing at random,” it may be harder to draw
this conclusion regarding case 18, since for that case, two points are missing. Why are they missing? Did
the participant misunderstand the task? Was the participant or object given the opportunity to respond?
These are the types of questions you should ask before contemplating and carrying out a missing data
routine in SPSS. Hence, before we survey methods for replacing missing data then, you should heed the
following principle:
Let us survey a couple approaches to replacing
missing data. We will demonstrate these proce-
dures for our quant variable. To access the feature:
TRANSFORM → REPLACE MISSING VALUES
We can see that for cases 8, 13, and 18, we have missing
data. SPSS offers many capabilities for replacing missing
data, but if they are to be used at all, they should be used
with extreme caution.
Never, ever, replace missing data as
an ordinary and usual process of data
analysis. Ask yourself first WHY the data
point might be missing and whether it is missing
“atrandom”orwasduetosomesystematicerroror
omission in your experiment. If it was due to some
systematic pattern or the participant misunder-
stood the instructions or was not given full oppor-
tunity to respond, that is a quite different scenario
than if the observation is missing at random due to
chance factors. If missing at random, replacing
missing data is, generally speaking, more appro-
priate than if there is a systematic pattern to the
missing data. Get curious about your missing data
instead of simply seeking to replace it.

14
In this first example, we will replace the missing observation with the series mean. Move quant over to New
Variable(s). SPSS will automatically rename the variable “quant_1,” but underneath that, be sure Series mean
is selected. The series mean is defined as the mean of all the other observations for that variable. The mean for
quant is 66.89 (verify this yourself via Descriptives). Hence, if SPSS is replacing the missing data correctly, the
new value imputed for cases 8 and 18 should be 66.89. Click on OK:
RMV /quant_1=SMEAN(quant).
Result Variables
Case Number of
Non-Missing Values
First
1
2
1 quant_1
Result
Variable
N of
Replaced
Missing
Values
N of Valid
Cases
Creating
Function
SMEAN
(quant)
30 30
Last
Replace Missing Values
●
● SPSS provides us with a brief report revealing that two
missing values were replaced (for cases 8 and 18, out
of 30 total cases in our data set).
●
● The Creating Function is the SMEAN for quant (which
means it is the“series mean”for the quant variable).
●
● In the Data View, SPSS shows us the new variable cre-
ated with the missing values replaced (I circled them
manually to show where they are).
Another option offered by SPSS is to replace with the mean of nearby points. For this option, under Method,
select Mean of nearby points, and click on Change to activate it in the New Variable(s) window (you will
notice that quant becomes MEAN[quant 2]). Finally, under Span of nearby points, we will use the number 2
(which is the default). This means SPSS will take the two valid observations above the given case and two
below it, and use that average as the replaced value. Had we chosen Span of nearby points=4, it would have
taken the mean of the four points above and four points below. This is what SPSS means by the mean of
“nearby points.”
●
● We can see that SPSS, for case 8, took the mean of
two cases above and two cases below the given
missing observation and replaced it with that
mean. That is, the number 47.25 was computed
by averaging 50.00 + 54.00 + 46.00 + 39.00, which
when that sum is divided by 4, we get 47.25.
●
● For case 18, SPSS took the mean of observations
74, 76, 82, and 74 and averaged them to equal
76.50, which is the imputed missing value.

Replacing with the mean as we have done above is an easy way of doing it, though is often not the
most preferred (see Meyers et al. (2013), for a discussion). SPSS offers other alternatives, including
replacing with the median instead of the mean, as well as linear interpolation, and more sophisti-
cated methods such as maximum likelihood estimation (see Little and Rubin (2002) for details).
SPSS offers some useful applications for evaluating missing data patterns though Missing Value
Analysis and Multiple Imputation.
As an example of SPSS’s ability to identify patterns in missing data and replace these values using
imputation, we can perform the following (see Leech et al. (2015) for more details on this approach):
ANALYZE → MULTIPLE IMPUTATION → ANALYZE PATTERNS

Missing Value Patterns
Type
1
2
Pattern
verbal quant analytic
Variable
3
4
Nonmissing
Missing
The pattern analysis can help you identify whether there is any systematic
features to the missingness or whether you can assume it is random. SPSS
will allow us to replace the above missing values through the following:
MULTIPLE IMPUTATION → INPUT MISSING DATA VALUES

●
● Move over the variables of interest to the Variables in Model side.
●
● Adjust Imputations to 5 (you can experiment with greater values, but for demonstration, keep
it at 5).
The Missing Value
Patterns identifies
four patterns in the
data. The first row is
a pattern revealing
no missing data,
while the second
row reveals the

middle point (for
quant) as missing,
while two other pat-
terns are identified
as well, including
the final row, which
is the pattern of
missingness across
two variables.

16
●
● SPSS requires us to name a new file that will contain the upgraded data (that now includes filled
values). We named our data set “missing.” This will create a new file in our session called
“missing.”
●
● Under the Method tab, we will select Custom and Fully Conditional Specification (MCMC) as
the method of choice.
●
● We will set the Maximum Iterations at 10 (which is the default).
●
● Select Linear Regression as the Model type for scale variables.
●
● Under Output, check off Imputation model and Descriptive statistics for variables with
imputed values.
●
● Click OK.
SPSS gives us a summary report on the imputation results:
Imputation Results
Imputation Method
Imputation Sequence
Dependent Variables Imputed
Not Imputed (Too
Many Missing Values)
Not Imputed (No
Missing Values)
Fully Conditional Specification Method
Iterations
Fully Conditional Specification
quant, analytic
10
verbal
verbal, quant, analytic
Imputation Models
Model Missing
Values
Imputed
Values
Type Effects
quant Linear
Regression
Linear
Regression
analytic
verbal, analytic
verbal, quant
2
2
10
10
The above summary is of limited use. What is more useful is to look at the accompanying file that
was created, named “missing.” This file now contains six data sets, one being the original data and
five containing inputted values. For example, we contrast the original data and the first imputation
below:

We can see that the procedure replaced the missing data points for cases 8, 13, and 18. Recall

however that the imputations above are only one iteration. We asked SPSS to produce five iterations,
so if you scroll down the file, you will see the remaining iterations. SPSS also provides us with a

summary of the iterations in its output:
analytic
Data
Original Data
Imputed Values
Imputation N Mean Std. Deviation Minimum Maximum
28 70.8929 18.64352 29.0000 97.0000
2 79.0207 9.14000 72.5578 85.4837
2 80.2167 16.47851 68.5647 91.8688
2 79.9264 1.50806 78.8601 80.9928
2 81.5065 23.75582 64.7086 98.3044
2 67.5480 31.62846 45.1833 89.9127
30 71.4347 18.18633 29.0000 97.0000
30 71.5144 18.40024 29.0000 97.0000
30 71.4951 18.13673 29.0000 97.0000
30 71.6004 18.71685 29.0000 98.3044
30 70.6699 18.94268 29.0000 97.0000
1
2
3
4
5
Complete Data After
Imputation
1
2
3
4
5
Some procedures in SPSS will allow you to
immediately use the file with now the “com-
plete” data. For example, if we requested some
descriptives (from the “missing” file, not the
original file), we would have the following:
DESCRIPTIVES VARIABLES=verbal
analytic quant
/STATISTICS=MEAN STDDEV MIN MAX.
Descriptive Statistics
Imputation Number N Minimum
30
28
28
49.00
29.00
35.00
Maximum
98.00
97.00
98.00
Mean
72.8667
70.8929
66.8929
Std. Deviation
12.97407
18.64352
18.86863
27
30
30
30
49.00
29.00
35.00
98.00
97.00
98.00
72.8667
71.4347
66.9948
12.97407
18.18633
18.78684
30
30
30
30
49.00
29.00
35.00
98.00
97.00
98.00
72.8667
71.5144
66.2107
12.97407
18.40024
19.24780
30
30
30
30
49.00
29.00
35.00
98.00
97.00
98.00
72.8667
71.4951
66.9687
12.97407
18.13673
18.26461
30
30
30
30
49.00
29.00
35.00
98.00
98.30
98.00
72.8667
71.6004
67.2678
12.97407
18.71685
18.37864
30
30
30
30
49.00
29.00
35.00
98.00
97.00
98.00
72.8667
70.6699
66.0232
12.97407
18.94268
18.96753
30
30
30
30
72.8667
71.3429
66.6930
30
Original data verbal
analytic
quant
Valid N (listwise)
1 verbal
analytic
quant
Valid N (listwise)
2 verbal
analytic
quant
Valid N (listwise)
3 verbal
analytic
quant
Valid N (listwise)
4 verbal
analytic
quant
Valid N (listwise)
5 verbal
analytic
quant
Valid N (listwise)
Pooled verbal
analytic
quant
Valid N (listwise)
quant
Data
Original Data 28
Imputed Values
Imputation N Mean Std. Deviation Minimum Maximum
1
2
3
4
5
Complete Data After
Imputation
1
2
3
4
5
2
2
2
2
2
30
30
30
30
30
66.8929
68.4214
56.6600
68.0303
72.5174
53.8473
66.9948
66.2107
66.9687
67.2678
66.0232
18.86863
24.86718
30.58958
7.69329
11.12318
22.42527
18.78684
19.24780
18.26461
18.37864
18.96753
35.0000
50.8376
35.0299
62.5904
64.6521
37.9903
35.0000
35.0000
35.0000
35.0000
35.0000
98.0000
86.0051
78.2901
73.4703
80.3826
69.7044
98.0000
98.0000
98.0000
98.0000
98.0000
SPSS gives us first the original data on which
there are 30 complete cases for verbal, and 28
complete cases for analytic and quant, before the
imputation algorithm goes to work on replacing
the missing data. SPSS then created, as per our
request, five new data sets, each time imputing a
missing value for quant and analytic. We see
that N has increased to 30 for each data set, and
SPSS gives descriptive statistics for each data set.
The pooled means of all data sets for analytic
and quant are now 71.34 and 66.69, respectively,
which was computed by summing the means of
all the new data sets and dividing by 5.

18
Let us try an ANOVA on the new file:
ONEWAY quant BY group
/MISSING ANALYSIS.
ANOVA
quant
Imputation Number
Sum of
Squares
8087.967
1524.711
9612.679
2
25
4043.984 66.307 .000
60.988
27
Mean Square F Sig.
df
Original data Between Groups
Within Groups
Total
8368.807
1866.609
10235.416
2
27
4184.404 60.526 .000
69.134
29
1 Between Groups
Within Groups
Total
9025.806
1718.056
10743.862
2
27
4512.903 70.922 .000
63.632
29
2
3
Between Groups
Within Groups
Total
7834.881
1839.399
9674.280
2
27
3917.441 57.503 .000
68.126
29
Between Groups
Within Groups
Total
4 7768.562
2026.894
9795.456
2
27
3884.281 51.742 .000
75.070
29
Between Groups
Within Groups
Total
5 8861.112
1572.140
10433.251
2
27
4430.556 76.091 .000
58.227
29
Between Groups
Within Groups
Total
This is as far as we go with our brief discussion of
missing data. We close this section with reiterating the
warning – be very cautious about replacing missing
data. Statistically it may seem like a good thing to do for
a more complete data set, but scientifically it means you
are guessing (albeit in a somewhat sophisticated esti-
mated fashion) at what the values are that are missing. If
you do not replace missing data, then common methods
of handling cases with missing data include listwise and
pairwise deletion. Listwise deletion excludes cases with
missing data on any variables in the variable list, whereas
pairwise deletion excludes cases only on those variables for which the given analysis is being

conducted. For instance, if a correlation is run on two variables that do not have missing data, the

correlation will compute on all cases even though for other variables, missing data may exist (try a
few correlations on the IQ data set with missing data to see for yourself). For most of the procedures
in this book, especially multivariate ones, listwise deletion is usually preferred over pairwise deletion
(see Meyers et al. (2013) for further discussion).
SPSS gives us the ANOVA results for
each imputation, revealing that regard-
less of the imputation, each analysis
supports rejecting the null hypothesis.
We have evidence that there are mean
group differences on quant.
A one‐way analysis of variance
(ANOVA) was performed com-
paring students’ quantitative
performance, measured on a continuous
scale, based on how much they studied
(none, some, or much). Total sample size
was 30, with each group having 10 obser-
vations. Two cases (8 and 18) were missing
values on quant. SPSS’s Fully Conditional
Specification was used to impute values
for this variable, requesting five imputa-
tions.EachimputationresultedinANOVAs
that rejected the null hypothesis of equal
populationmeans(p0.001).Hence,there
is evidence to suggest that quant perfor-
manceisafunctionofhowmuchastudent
studies for the evaluation.

19
Due to SPSS’s high‐speed computing capabilities, a researcher can conduct a variety of exploratory
analyses to immediately get an impression of their data, as well as compute a number of basic sum-
mary statistics. SPSS offers many options for graphing data and generating a variety of plots. In this
chapter, we survey and demonstrate some of these exploratory analyses in SPSS. What we present
here is merely a glimpse at the capabilities of the software and show only the most essential functions
for helping you make quick and immediate sense of your data.
3.1 Frequencies and Descriptives
Before conducting formal inferential statistical analyses, it is always a good idea to get a feel for one’s
data by conducting so‐called exploratory data analyses. We may also be interested in conducting
exploratory analyses simply to confirm that our data has been entered correctly. Regardless of its
purpose, it is always a good idea to get very familiar with one’s data before analyzing it in any
significant way. Never simply enter data and conduct formal analyses without first exploring all of
your variables, ensuring assumptions of analyses are at least tentatively satisfied, and ensuring your
data were entered correctly.
3
Exploratory Data Analysis, Basic Statistics, and Visual Displays

3 Exploratory Data Analysis, Basic Statistics, and Visual Displays
20
SPSS offers a number of options for conducting a variety of data summary tasks. For example, sup-
pose we wanted to simply observe the frequencies of different scores on a given variable. We could
accomplish this using the Frequencies function:
As a demonstration, we will obtain frequency information for the variable verbal, along with a
number of other summary statistics. Select Statistics and then the options on the right:

ANALYZE → DESCRIPTIVE STATISTICS →
FREQUENCIES (this shows the sequence
of the GUI menu selection, as shown on
the left)

3.1 Frequencies and Descriptives 21
We have selected Quartiles under Percentile Values and Mean, Median, Mode, and Sum under
Central Tendency. We have also requested dispersion statistics Std. Deviation, Variance, Range,
Minimum, and Maximum and distribution statistics Skewness and Kurtosis. We click on Continue
and OK to see our output (below is the corresponding syntax for generating the above – remember,
you do not need to enter the syntax below; we are showing it only so you have it available to you
should you ever wish to work with syntax instead of GUI commands):
FREQUENCIES VARIABLES=verbal
/NTILES=4
/STATISTICS=STDDEV VARIANCE RANGE MINIMUM MAXIMUM MEAN MEDIAN MODE
SUM SKEWNESS SESKEW KURTOSIS SEKURT
/ORDER=ANALYSIS.
Valid
Missing
Statistics
N 30
0
72.8667
73.5000
56.00a
12.97407
168.326
–.048
–.693
.833
49.00
49.00
98.00
2186.00
62.7500
73.5000
84.2500
.427
verbal
Mean
Median
Mode
Std. Deviation
Variance
Skewness
Std. Error of Skewness
Std. Error of Kurtosis
Range
Minimum
Maximum
Sum
Percentiles 25
50
75
a. Multiple modes exist. The smallest
value is shown
Kurtosis
To the left are presented a number of useful summary and descrip-
tive statistics that help us get a feel for our verbal variable. Of note:
●
● There are a total of 30 cases (N =30), with no missing values (0).
●
● The Mean is equal to 72.87 and the Median 73.50. The mode
(most frequent occurring score) is equal to 56.00 (though multi-
ple modes exist for this variable).
●
● The Standard Deviation is the square root of the Variance, equal to
12.97. This gives an idea of how much dispersion is present in the
variable. For example, a standard deviation equal to 0 would mean
all values for verbal are the same. As the standard deviation is
greater than 0 (it cannot be negative), it indicates increasingly more
variability.
●
● The distribution is slightly negatively skewed since Skewness of
−0.048 is less than zero, indicating slight negative skew. The fact
that the mean is less than the median is also evident of a slightly
negatively skewed distribution. Skewness of 0 indicates no skew.
Positive values indicate positive skew.
●
● Kurtosis is equal to −0.693 suggesting that observations cluster
less around a central point and the distribution has relatively thin
tails compared with what we would expect in a normal distribu-
tion (SPSS 2017). These distributions are often referred to as
platykurtic.
●
● The range is equal to 49.00, computed as the highest score in the
data minus the lowest score (98.00 – 49.00=49.00).
●
● The sum of all the data is equal to 2186.00.
The scores at the 25th, 50th, and 75th percentiles are 62.75, 73.50,
and 84.25. Notice that the 50% percentile corresponds to the same
value as the median.

22
SPSS then provides us with the frequency information for verbal:
We can also obtain some basic descriptive statistics via Descriptives:
ANALYZE → DESCRIPTIVE STATISTICS → DESCRIPTIVES
Frequency
49.00
51.00
54.00
56.00
59.00
62.00
63.00
66.00
68.00
69.00
70.00
73.00
74.00
75.00
76.00
79.00
82.00
84.00
85.00
86.00
92.00
94.00
98.00
Total
1
1
1
2
1
1
1
1
2
1
1
2
2
1
1
2
1
1
2
2
1
1
1
30
3.3
3.3
3.3
6.7
3.3
3.3
3.3
3.3
6.7
3.3
3.3
6.7
6.7
3.3
3.3
6.7
3.3
3.3
6.7
6.7
3.3
3.3
3.3
100.0
3.3
3.3
3.3
6.7
3.3
3.3
3.3
3.3
6.7
3.3
3.3
6.7
6.7
3.3
3.3
6.7
3.3
3.3
6.7
6.7
3.3
3.3
3.3
100.0
3.3
6.7
10.0
16.7
20.0
23.3
26.7
30.0
36.7
40.0
43.3
50.0
56.7
60.0
63.3
70.0
73.3
76.7
83.3
90.0
93.3
96.7
100.0
Valid
Percent
verbal
Cumulative
Percent
Valid
Percent
We can see from the output that the value of 49.00
occurs a single time in the data set (Frequency=1) and
consists of 3.3% of cases. The value of 51.00 occurs a
single time as well and denotes 3.3% of cases.The cumu-
lative percent for these two values is 6.7%, which con-
sists of that value of 51.00 along with the value before it
of 49.00. Notice that the total cumulative percent adds
up to 100.0.
After moving verbal to the Variables window, select Options.
As we did with the Frequencies function, we select a variety of
summary statistics. Click on Continue then OK.

3.2 The Explore Function 23
Our output follows:
N
Statistic
Range Minimum
Statistic Statistic Statistic Statistic Statistic Statistic Statistic Statistic
Kurtosis
Skewness
Variance
Std. Deviation
Mean
Descriptive Statistics
Maximum
Std. Error Std. Error
49.00 49.00 98.00 72.8667 12.97407 168.326 –.048 .427 –.693 .833
30
30
verbal
Valid N (listwise)
3.2
The Explore Function
A very useful function in SPSS for obtaining descriptives as well as a host of summary plots is the
EXPLORE function:
ANALYZE → DESCRIPTIVE STATISTICS →
EXPLORE
Move verbal over to the Dependent List and
group to the Factor List. Since group is a

categorical (factor) variable, what this means
is that SPSS will provide us with summary sta-
tistics and plots for each level of the grouping
variable.
Under Statistics, select Descriptives, Outliers, and
Percentiles. Then under Plots, we will select, under
Boxplots, Factor levels together, then under Descriptive,
Stem‐and‐leaf and Histogram. We will also select
Normality plots with tests:

24
SPSS generates the following output:
verbal
group
Valid Missing Total
Cases
Percent Percent
100.0%
100.0%
100.0%
100.0%
100.0%
100.0%
0.0%
0.0%
0.0%
10
10
10
10
10
10
0
0
0
Percent N N
N
Case Processing Summary
.00
1.00
2.00
The Case Processing Summary above simply reveals the variable we are subjecting to analysis
(verbal) along with the numbers per level (0, 1, 2). We confirm that SPSS is reading our data file
correctly, as there are N =10 per group.
Statistic
group
verbal .00 Mean
95% confidence Interval
for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Mean
1.00
95% Confidence Interval
for Mean
5% Trimmed Mean
Median
Variance
Std. Deviation
Minimum
Maximum
Range
Interquartile Range
Skewness
Kurtosis
Mean
2.00
95% Confidence Interval
for Mean
Std. Error
2.44404
59.2000
53.6712
64.7288
58.9444
57.5000
59.733
7.72873
49.00
74.00
25.00
11.00
–0.25
.656 .687
1.334
1.70261
.687
1.334
2.13464
73.1000
69.2484
76.9516
72.8889
73.0000
28.989
5.38413
66.00
84.00
18.00
7.25
.818
.578
86.3000
81.4711
91.1289
86.2222
85.5000
45.567
6.75031
76.00
98.00
22.00
11.25
.306
–.371
.687
1.334
Lower Bound
Upper Bound
Lower Bound
Upper Bound
Lower Bound
Upper Bound
Descriptives In the Descriptives summary to the left, we can see
that SPSS provides statistics for verbal by group
level (0, 1, 2). For verbal =0.00, we note the
following:
●
● The arithmetic Mean is equal to 59.2, with a
standard error of 2.44 (we will discuss standard
errors in later chapters).
●
● The 95% Confidence Interval for the Mean has
limits of 53.67 and 64.73. That is, in 95% of sam-
ples drawn from this population, the true popu-
lation mean is expected to lie between this lower
and upper limit.
●
● The 5% Trimmed Mean is the adjusted mean by
deleting the upper and lower 5% of cases on the
tails of the distribution. If the trimmed mean is
very much different from the arithmetic mean, it
could indicate the presence of outliers.
●
● The Median, which represents the score that is
the middle point of the distribution, is equal to
57.5. This means that 1/2 of the distribution lay
below this value, while 1/2 of the distribution lay
above this value.
●
● The Variance of 59.73 is the average sum of
squared deviations from the arithmetic mean
and provides a measure of how much dispersion
(in squared units) exists for the variable. Variance
of 0 (zero) indicates no dispersion.
●
● The Standard Deviation of 7.73 is the square root
of the variance and is thus measured in the origi-
nal units of the variable (rather than in squared
units such as the variance).
●
● The Minimum and Maximum values of the data are also given, equal to 49.00 and 74.00, respectively.
●
● The Range of 25.00 is computed by subtracting the lowest score in the data from the highest
(i.e. 74.00 – 49.00=25.00).

group
.00 Highest
Case
Number Value
Extreme Values
Highest
Lowest
Lowest
Lowest
a. Only a partial list of cases with the value 73.00 are shown
in the table of upper extremes.
b. Only a partial list of cases with the value 73.00 are shown
in the table of lower extremes.
2.00
1.00
verbal
Highest
1
2
3
4
5
4
6
5
3
2
74.00
68.00
63.00
62.00
59.00
49.00
51.00
54.00
56.00
56.00
66.00
68.00
69.00
70.00
73.00b
84.00
79.00
75.00
74.00
73.00a
10
9
7
8
1
15
18
17
13
14
11
16
12
20
19
98.00
94.00
92.00
86.00
86.00
76.00
79.00
82.00
85.00
85.00
29
26
27
22
28
24
25
23
30
21
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
Tests of Normality
Shapiro-Wilk
Statistic
Statistic
group
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
verbal .00 .161
.162
.218 10
10
10
10
10
10
.200*
.200*
.197 .962
.948
.960 .789
.639
.809
1.00
2.00
df
df
Kolmogorov-Smirnova
Sig.
Sig.
●
● The Interquartile Range is computed as the third quartile (Q3) minus the first quartile (Q1) and hence is a
rough measure of how much variation exists on the inner part of the distribution (i.e. between Q1 and Q3).
●
● The Skewness index of 0.656 suggests a slight positive skew (skewness of 0 means no skew, and negative num-
bers indicate a negative skew).The Kurtosis index of −0.025 indicates a slight“platykurtic”tendency (crudely, a
bit flatter and thinner tails than a normal or“mesokurtic”distribution).
SPSS also reports Extreme Values that give the top 5
lowest and top 5 highest values in the data at each
level of the group variable. A few conclusions from this
table:
●
● In group=0, the highest value is 74.00, which is case
number 4 in the data set.
●
● In group=0, the lowest value is 49.00, which is case
number 10 in the data set.
●
● In group=1, the third highest value is 75.00, which is
case number 17 in the data set.
●
● In group=1, the third lowest value is 69.00, which is
case number 12 in the data set.
●
● In group=2, the fourth highest value is 86.00, which
is case number 22.
●
● In group=2, the fourth lowest value is 85.00, which
is case number 30.
SPSS reports Tests of Normality (left, at the bottom)
both the Kolmogorov–Smirnov and Shapiro–Wilk
tests. Crudely, these both test the null hypothesis that
the sample data arose from a normal population. We
wish to not reject the null hypothesis and hence desire
a p‐value greater than the typical 0.05. A few conclu-
sions we draw:
●
● For group=0, neither test rejects the null (p =0.200
and 0.789).
●
and 0.639).
●
and 0.809).
The distribution of verbal was evaluated for normality across groups of the independent variable.
Both the Kolmogorov–Smirnov and Shapiro–Wilk tests failed to reject the null hypothesis of a normal
population distribution, and so we have no reason to doubt the sample was not drawn from normal
populations in each group.

26
Below are histograms for verbal for each level of the group variable. Along with each plot is given
the mean, standard deviation, and N per group. Since our sample size per group is very small, it is
rather difficult to assess normality per cell (group), but at minimum, we do not notice any gross viola-
tion of normality. We can also see from the histograms that each level contains at least some variabil-
ity, which is important to have for statistical analyses (if you have a distribution that has virtually
almost no variability, then it restricts the kinds of statistical analyses you can do or whether analyses
can be done at all).
50.00
45.00
0
1
2
Frequency
3
for group = .00
Histogram
Mean = 59.20
Std. Dev. = 7.729
N = 10
55.00 60.00 65.00 70.00 75.00
verbal
Mean = 73.10
Std. Dev. = 5.384
N = 10
0
1
2
Frequency 3
for group = 1.00
Histogram
65.00 70.00 75.00 80.00 85.00
verbal
Mean = 86.30
Std. Dev. = 6.75
N = 10
0
1
2
Frequency
3
4
for group = 2.00
Histogram
75.00 85.00
80.00 90.00 95.00
verbal
The following are what are known as Stem‐and‐leaf Plots. These are plots that depict the distribu-
tion of scores similar to a histogram (turned sideways) but where one can see each number in each
distribution. They are a kind of “naked histogram” on its side. For these data, SPSS again plots them
by group number (0, 1, 2).
Frequency
Stem-and-Leaf Plots
verbal Stem–and–Leaf Plot for
group = .00
Stem width:
Each leaf:
10.00
1 case (s)
Stem Leaf
1.00
5.00
3.00
1.00
4
5
6
7
9
14669
238
4
.
.
.
.

group = 1.00
Stem width:
Each leaf:
10.00
1 case (s)
Frequency Stem Leaf
3.00
4.00
2.00
1.00
6
7
7
8
689
0334
59
4
.
.
.
.

group = 2.00
Stem width:
Each leaf:
10.00
1 case (s)
Frequency Stem Leaf
2.00
1.00
4.00
2.00
1.00
7
8
8
9
9
69
2
5566
24
8
.
.
.
.
Let us inspect the first plot (group=0) to explain how it is constructed. The first value in the data
for group=0 has a frequency of 1.00. The score is that of 49. How do we know it is 49? Because “4”
is the stem and “9” is the leaf. Notice that below the plot is given the stem width, which is 10.00.
What this means is that the stems correspond to “tens” in the digit placement. Recall that from

right to left before the decimal point, the digit positions are ones, tens, hundreds, thousands, etc.
SPSS also tells us that each leaf consists of a single case (1 case[s]), which means the “9” represents
a single case. Look down now at the next row; We see there are five values with stems of 5. What
are the values? They are 51, 54, 56, 56, and 59. The rest of the plots are read in a similar manner.
To confirm that you are reading the stem‐and‐leaf plots correctly, it is always a good idea to match
up some of the values with your raw data simply to make sure what you are reading is correct.
With more complicated plots, sometimes discerning what is the stem vs. what is the leaf can be a
bit tricky!
Below are what known as Q–Q Plots. As requested, SPSS also prints these out for each level
of the verbal variable. These plots essentially compare observed values of the variable with
expected values of the variable under the condition of normality. That is, if the distribution fol-
lows a normal distribution, then observed values should line up nicely with expected values.
That is, points should fall approximately on the line; otherwise distributions are not perfectly
normal. All of our distributions below look at least relatively normal (they are not perfect, but
not too bad).
40
–2
–1
0
Expected
Normal
Expected
Normal
Expected
Normal
1
2
3
–2
–1
0
1
2
–2
–1
0
1
2
3
50 60
Normal Q-Q Plot of verbal
for group = .00
for group = 1.00
for group = 2.00
70 80 80 80
85 85 90 95 100
70 75 70 75
65
Observed Value Observed Value Observed Value
To the left are what are called Box‐and‐
whisker Plots. For our data, they represent a
summary of each level of the grouping varia-
ble. If you are not already familiar with box-
plots, a detailed explanation is given in the
box below, “How to Read a Box‐and‐whisker
Plot.”As we move from group=0 to group=2,
the medians increase. That is, it would appear
that those who receive much training do bet-
ter (median wise) than those who receive
some vs. those who receive none.
40.00
.00 1.00 2.00
50.00
60.00
70.00
80.00
90.00
verbal
group
100.00

28
3.3
What Should I Do with Outliers? Delete or Keep Them?
In our review of boxplots, we mentioned that any point that falls below Q1 – 1.5×IQR or above
Q3+1.5×IQR may be considered an outlier. Criteria such as these are often used to identify extreme
observations, but you should know that what constitutes an outlier is rather subjective, and not quite
as simple as a boxplot (or other criteria) makes it sound. There are many competing criteria for defin-
ing outliers, the boxplot definition being only one of them. What you need to know is that it is a
mistake to compute an outlier by any statistical criteria whatever the kind and simply delete it from
your data. This would be dishonest data analysis and, even worse, dishonest science. What you
should do is consider the data point carefully and determine based on your substantive knowledge of
the area under study whether the data point could have reasonably been expected to have arisen
from the population you are studying. If the answer to this question is yes, then you would be wise to
keep the data point in your distribution. However, since it is an extreme observation, you may also
choose to perform the analysis with and without the outlier to compare its impact on your final model
results. On the other hand, if the extreme observation is a result of a miscalculation or a data error,
How to Read a Box‐and‐whisker Plot
Consider the plot below, with normal densities
given below the plot.
IQR
Q3
Q3 + 1.5 × IQR
Q1
Q1 – 1.5 × IQR
–4σ –3σ –2σ –1σ 0σ 1σ 2σ 3σ
2.698σ
–2.698σ 0.6745σ
–0.6745σ
24.65% 50% 24.65%
15.73%
68.27%
15.73%
4σ
–4σ –3σ –2σ –1σ 0σ 1σ 2σ 3σ 4σ
–4σ –3σ –2σ –1σ 0σ 1σ 2σ 3σ 4σ
Median
●
● The median in the plot is the point that divides the dis-
tribution into two equal halves. That is, 1/2 of observa-
tions will lay below the median, while 1/2 of
observations will lay above the median.
●
● Q1 and Q3 represent the 25th and 75% percentiles,
respectively. Note that the median is often referred to
as Q2 and corresponds to the 50th percentile.
●
● IQR corresponds to “Interquartile Range” and is com-
puted by Q3 – Q1. The semi‐interquartile range (not
shown) is computed by dividing this difference in half
(i.e. [Q3 − Q1]/2).
●
● On the leftmost of the plot is Q1 − 1.5×IQR. This corre-
sponds to the lowermost “inner fence.” Observations that
are smaller than this fence (i.e. beyond the fence, greater
negative values) may be considered to be candidates for
outliers.The area beyond the fence to the left corresponds
toaverysmallproportionofcasesinanormaldistribution.
●
● On the rightmost of the plot is Q3+1.5×IQR. This cor-
responds to the uppermost“inner fence.”Observations
that are larger than this fence (i.e. beyond the fence)
may be considered to be candidates for outliers. The
area beyond the fence to the right corresponds to a
very small proportion of cases in a normal distribution.
●
● The“whiskers”in the plot (i.e. the vertical lines from the
quartiles to the fences) will not typically extend as far
as they do in this current plot. Rather, they will extend
as far as there is a score in our data set on the inside of
the inner fence (which explains why some whiskers
can be very short). This helps give an idea as to how
compact is the distribution on each side.

then yes, by all means, delete it forever from your data, as in this case it is a “mistake” in your data,
and not an actual real data point. SPSS will thankfully not automatically delete outliers from any
statistical analyses, so it is up to you to run boxplots, histograms, and residual analyses (we will dis-
cuss these later) so as to attempt to spot unusual observations that depart from the rest. But again,
do not be reckless with them and simply wish them away. Get curious about your extreme scores, as
sometimes they contain clues to furthering the science you are conducting. For example, if I gave a
group of 25 individuals sleeping pills to study its effect on their sleep time, and one participant slept
well below the average of the rest, such that their sleep time could be considered an outlier, it may
suggest that for that person, the sleeping pill had an opposite effect to what was expected in that it
kept the person awake rather than induced sleep. Why was this person kept awake? Perhaps the drug
was interacting with something unique to that particular individual? If we looked at our data file
further, we might see that subject was much older than the rest of the subjects. Is there something
about age that interacts with the drug to create an opposite effect? As you see, outliers, if studied,
may lead to new hypotheses, which is why they may be very valuable at times to you as a scientist.
3.4 Data Transformations
Most statistical models make assumptions about the structure of data. For example, linear least‐
squares makes many assumptions, among which, for instance, are linearity and normality and inde-
pendence of errors (see Chapter 9). However, in practice, assumptions often fail to be met, and one
may choose to perform a mathematical transformation on one’s data so that it better conforms to
required assumptions. For instance, when sample data do not follow normal distributions to a large
extent, one option is to perform a transformation on the variable so that it better approximates nor-
mality. Such transformations often help “normalize” the distribution, so that the assumptions of such
tests as t‐tests and ANOVA are more easily satisfied. There are no hard and fast rules regarding when
and how to transform data in every case or situation, and often it is a matter of exploring the data and
trying out a variety of transformations to see if it helps. We only scratch the surface with regard to
transformations here and demonstrate how one can obtain some transformed values in SPSS and
their effect on distributions. For a thorough discussion, see Fox (2016).
The Logarithmic Transformation
The log of a number is the exponent to which we need to raise a number to get another number. For
example, the natural log of the number 10 is equal to
log .
e 10 2 302585093
Why? Because e2.302585093
=10, where e is a constant equal to approximately 2.7183. Notice that the
“base” of these logarithms is equal to e. This is why these logs are referred to as “natural” logarithms.
We can also compute common logarithms, those to base 10:
log10 10 1
But why does taking logarithms of a distribution help “normalize” it? A simple example will help
illustrate. Consider the following hypothetical data on a given variable:
2 4 10 15 20 30 100 1000

30
Though the distribution is extremely small, we nonetheless notice that lower scores are closer in
proximity than are larger scores. The ratio of 4 to 2 is equal to 2. The distance between 100 and 1000
is equal to 900 (the ratio is equal to 10). How would taking the natural log of these data influence
these distances? Let us compute the natural logs of each score:
0 69 1 39 2 30 2 71 2 99 3 40 4 61 6 91
. . . . . . . .
Notice that the ratio of 1.39–0.69 is equal to 2.01, which closely mirrors that of the original data.
However, look now at the ratio of 6.91–4.61, it is equal to 1.49, whereas in the original data, the ratio
was equal to 10. In other words, the log transformation made the extreme scores more “alike” the other
scores in the distribution. It pulled in extreme scores. We can also appreciate this idea through simply
looking at the distances between these points. Notice the distance between 100 and 1000 in the origi-
nal data is equal to 900, whereas the distance between 4.61 and 6.91 is equal to 2.3, very much less
than in the original data. This is why logarithms are potentially useful for skewed distributions.
Larger numbers get “pulled in” such that they become closer together. After a log transformation,
often the resulting distribution will resemble more closely that of a normal distribution, which makes
the data suitable for such tests as t‐tests and ANOVA.
The following is an example of data that was subjected to a log transformation. Notice how after
the transformation, the distribution is now approximately normalized:
0
(a) (b)
20 40 60 80
Enzyme Level Log of Enzyme Level
4
3
2
1
0
We can perform other transformations as well on data, including taking square roots and recipro-
cals (i.e. 1 divided by the value of the variable). Below we show how our small data set behaves under
each of these transformations:
TRANSFORM → COMPUTE VARIABLE

●
● Notice above we have named our Target Variable by the name of LOG_Y. For our example, we will
compute the natural log (LN), so under Functions and Special Variables, we select LN (be sure to
select Function Group=Arithmetic first). We then move Y, our original variable, under Numeric
Expression so it reads LN(Y).
●
● The output for the log transformation appears to the right of the window, along with other trans-
formations that we tried (square root (SQRT_Y) and reciprocal (RECIP_Y).
●
● To get the square root transformation, simply scroll down.
But when to do which transformation? Generally speaking, to correct negative skew in a distribu-
tion, one can try ascending the ladder of powers by first trying a square transformation. To reduce
positive skew, descending the ladder of powers is advised (e.g. start with a square root or a common
log transform). And as mentioned, often transformations to correct one feature of data (e.g. abnor-
mality or skewness) can help also simultaneously adjust other features (e.g. nonlinearity). The trick
is to try out several transformations to see which best suits the data you have at hand. You are allowed
to try out several transformations.
The following is a final word about transformations. While some data analysts take great care in
transforming data at the slight of abnormality or skewed distributions, generally, most parametric sta-
tistical analyses can be conducted without transforming data at all. Data will never be perfectly normal
or linear, anyway, so slight deviations from normality, etc, are usually not a problem. A safeguard against
this approach is to try the given analysis with the original variable, then again with the transformed
variable, and observe whether the transformation had any effect on significance tests and model results
overall. If it did not, then you are probably safe not performing any transformation. If, however, a
response variable is heavily skewed, it could be an indicator of requiring a different model than the one
that assumes normality, for instance. For some situations, a heavily skewed distribution, coupled with
the nature of your data, might hint a Poisson regression to be more appropriate than an ordinary least‐
squares regression, but these issues are beyond the scope of the current book, as for most of the proce-
dures surveyed in this book, we assume well‐behaved distributions. For analyses in which distributions
are very abnormal or “surprising,” it may indicate something very special about the nature of your data,
and you are best to consult with someone on how to treat the distribution, that is, whether to merely
transform it or to conduct an alternative statistical model altogether to the one you started out with. Do
not get in the habit of transforming every data set you see to appease statistical models.

33
Before we push forward with a variety of statistical analyses in the remainder of the book, it would
do well at this point to briefly demonstrate a few of the more common data management capacities
in SPSS. SPSS is excellent for performing simple to complex data management tasks, and often the
need for such data management skill pops up over the course of your analyses. We survey only a few
of these tasks in what follows. For details on more data tasks, either consult the SPSS manuals or
simply explore the GUI on your own to learn what is possible. Trial and error with data tasks is a
great way to learn what the software can do! You will not break the software! Give things a shot, and
see how it turns out, then try again! Getting what you want any software to do takes patience and trial
and error, and when it comes to data management, often you have to try something, see if it works,
and if it does not, try something else.
4.1 Computing a New Variable
Recall our data set on verbal, quantitative, and analytical scores. Suppose we wished to create a new
variable called IQ (i.e. intelligence) and defined it by summing the total of these scores. That is, we
wished to define IQ=verbal+quantitative+analytical. We could do so directly in SPSS syntax or via
the GUI:
4
Data Management in SPSS

4 Data Management in SPSS
34
We compute as follows:
●
● Under Target Variable, type in the name of the
new variable you wish to create. For our data,
that name is“IQ.”
●
● Under Numeric Expression, move over the vari-
ables you wish to sum. For our data, the expres-
sion we want is verbal + quant + analytic.
●
● We could also select Type Label under IQ to
make sure it is designated as a numeric variable,
as well as provide it with a label if we wanted.
We will call it“Intelligence Quotient”:
Once we are done with the creation of the variable, we verify that it has been computed in the Data View:
We confirm that a new variable has been
created
by the name of IQ. The IQ for the first case, for
example, is computed just as we requested, by
adding verbal + quant + analytic, which for the
first case is 56.00+56.00+59.00=171.00.
4.2 Selecting Cases
In this data management task, we wish to select particular cases of our data set, while excluding
others. Reasons for doing this include perhaps only wanting to analyze a subset of one’s data.
Once we select cases, ensuing data analyses will only take place on those particular cases. For
example, suppose you wished to conduct analyses only on females in your data and not males. If
females are coded “1” and males “0,” SPSS can select only cases for which the variable Gender = 1
is defined.
For our IQ data, suppose we wished to run analyses only on data from group=1 or 2, excluding
group=0. We could accomplish this as follows: DATA → SELECT CASES
TRANSFORM → COMPUTE VARIABLE

4.2 Selecting Cases 35
In the Select Cases window, notice that we bulleted If
condition is satisfied. When we open up this window, we
obtain the following window (click on IF):
Notice that we have typed in group=1 or group=2. The or
function means SPSS will select not only cases that are in
group 1 but also cases that are in group 2. It will exclude
cases in group=0.We now click Continue and OK and verify
in the Data View that only cases for group=1 or group=2
were selected (SPSS crosses out cases that are excluded and
shows a new “filter_$” column to reveal which cases have
been selected – see below (left)).
After you conduct an analysis with Select
Cases, be sure to deselect the option once
you are done, so your next analysis will be
performed on the entire data set. If you keep Select
Cases set at group=1 or group=2, for instance, then
all ensuing analyses will be done only on these two
groups, which may not be what you wanted! SPSS
does not keep tabs on your intentions; you have to be
sure to tell it exactly what you want! Computers, unlike
humans, always take things literally.

4 Data Management in SPSS
36
4.3
Recoding Variables into Same or Different Variables
Oftentimes in research we wish to recode a variable. For example, when using a Likert scale, some-
times items are reverse coded in order to prevent responders from simply answering each question
the same way and ignoring what the actual values or choices mean. These types of reverse‐coded
items are often part of a “lie detection” attempt by the investigator to see if his or her respondents are
answering honestly (or at minimum, whether they are being careless in responding and simply cir-
cling a particular number the whole way through the questionnaire). When it comes time to analyze
the data, however, we often wish to code it back into its original scores so that all values of variables
have the same direction of magnitude.
To demonstrate, we create a new variable on how much a responder likes pizza, where 1=not at all
and 5=extremely so. Here is our data:
Suppose now we wanted to
reverse the coding. To recode
these data into the same varia-
ble, we do the following:
TRANSFORM → RECODE INTO
SAME VARIABLES
To recode the variable, select Old and New
Values:
●
● Under Old Value enter 1. Under New Value
enter 5. Then, click Add.
●
● Repeat the above procedure for all values of
the variable.
●
● Notice in the Old → New window, we have
transformed all values 1 to 5, 2 to 4, 3 to 3, 4
to 2, and 5 to 1.
●
● Note as well that we did not really need to
add“3 to 3,”but since it makes it easier for us
to check our work, we decided to include it,
and it is a good practice that you do so as
well when recoding variables – it helps keep
your thinking organized.
●
● Click on Continue then Ok.
●
● We verify in our data set (Data View) that
the variable has indeed been recoded (not
shown).

4.4 Sort Cases 37
Sometimes we would like to recode the variable, but instead of recoding into the same variable,
recode it into a different variable (so that we can keep the original one intact):
TRANSFORM → RECODE INTO DIFFERENT VARIABLES
To recode into a different variable, move “pizza” over to the right‐hand side, then:
●
● Enter a name for the output variable. For our data, we will name it “pizza_recode” and label it
“pizza preference recoded.”
●
● Next, we click on Old and New Values and repeat the process we did for the Change into Same
Variable:
4.4 Sort Cases
Sometimes we want to sort cases by values of a variable. For instance, suppose we wished to sort
cases of pizza starting from the lowest values to the highest (i.e. 1–5 for our data set):
Next, click Continue.
●
● Finally, to finish up, select Change, and in the window
will appear the transformation we wanted to have:
pizza → pizza_recode.
●
● Click on OK and verify (in the Data View) that the vari-
able has been recoded into a different variable (not
shown), keeping the original variable intact.

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