Statistical Analysis for Psychology
PSY 348
Lead Instructor: Greg Jensen
Co-Instructor: Sabrina Schroerlucke
TAs: Kati Wolcott & Gavin Leonard
Class Requirements
• Readings. There is assigned reading associated with most classes,
sometimes from the textbook, sometimes from elsewhere.
o Textbook: Statistics for Psychology, 7th Edition (“Aron et al.”).
6th
o Additional readings are available electronically on Moodle.
• Class Participation. Everything in this class is cumulative, and
applied exercises are built into the class period. As such, I expect
you to come to class.
o You get four free unexcused absences.
o For each subsequent unexcused absence, your final overall
grade in the course will be lowered by 13%.
o There are many valid reasons to miss class (such as flu-like
symptoms, or dealing with emergencies), and absences are
more likely to be excused if you contact me in advance.
Class Requirements
• Exams. There are four exams (three midterms and a final) that
each reflect 25% of your overall grade.
o However! You may retake any section on the midterms up to
two additional times.
• Each section of an exam is graded on a three-point scale:
o M (“mastered”)
o G (“room to grow”)
o N (“needs work”)
• If you earned a G or an N on a section of the exam, you may retake
just that section to improve it.
o The exams are not intended to be an ordeal. They are
opportunities for you to express what you have mastered
Class Requirements
• Homework. Is strictly optional! You don’t need to do it!
o Homework is also graded on the M/G/N scale, to let you know
how you’re doing.
o However! If you wish to retake any section of an exam, you
need to turn in the relevant homework first.
o Additionally, you will be asked to revise homework answers
that score less than an M before being allowed to retake the
relevant exam section.
• Different people get tripped up on different statistical concepts.
o We ask that you use the opportunities presented to get as
much practice as you can for topics that are initially confusing.
Class Philosophy
• It’s a guarantee that you will find some of the ideas we discuss this
semester confusing at first.
o The course is designed to avoid punishing that confusion.
o We believe that everyone one of you can master these
concepts, and we are here to support your doing so.
o As such, we want to take the pressure off for the exams and
the homework, so you can focus on your own understanding.
• In turn, we ask that you take advantage of the opportunities for
support that the course provides.
o Come to class regularly and ask questions.
o Come to office hours to discuss the material further.
o Do the homework for its own sake, even if you believe you
already understand the material.
Math Notation: Finding X
𝑋 = 0,4,9,3,1,0,1
𝑋 is a set of observations.
𝑋3 = 9
𝑋𝑁 = 1
𝑋𝑖 is a single observation, the 𝑖 th in the data
𝑁 is the number of elements, so 𝑋𝑁 is the last observation
Math Notation: Finding X
𝑋𝐷𝑜𝑢𝑔 = AB positive
XDoug is the observation associated with Doug.
𝑋 can consists of a set of non-numbers.
As a rule, subscripts are labels and should be read as the
sentence, “The element in [blank] associated with [label].”
Try to read every equation as a sentence.
Math Notation: Greek
𝜇 = Greek letter 𝑚𝑢
𝜇 is used to denote the “arithmetic mean” (i.e. the average)
𝜇height = Mean of height
𝜇height describes something about a variable, making it a statistic.
Usually, in this class, a Greek letter will refer to some statistic.
Math Notation: Summation
𝑋 = Sum of 𝑋
Σ is an upper-case Greek sigma.
We will use it to as a summation operator.
𝑋 = 𝑋1 + 𝑋2 + ⋯ + 𝑋𝑁
𝑋 = 0 + 4 + 9 + 3 + 1 + 0 + 1 = 18
Math Notation: Summation
Σ is an “item-wise” operator.
𝑋 + 𝑌 = 𝑋1 + 𝑌1 + 𝑋2 + 𝑌2 + ⋯ + 𝑋𝑁 + 𝑌𝑁
𝑋 ∙ 𝑌 = 𝑋1 ∙ 𝑌1 + 𝑋2 ∙ 𝑌2 + ⋯ + 𝑋𝑁 ∙ 𝑌𝑁
𝑋 + 1 = 𝑋1 + 1 + 𝑋2 + 1 + ⋯ + 𝑋𝑁 + 1 = 𝑁 +
𝑋
Math Notation: Summation
Summation notation can be ambiguous.
𝑋+1 ≠
𝑋+1
𝑋 + 1 = 𝑋1 + 𝑋2 + ⋯ 𝑋𝑁 + 1
𝑋 + 1 = 𝑋1 + 1 + 𝑋2 + 1 + ⋯ + 𝑋𝑁 + 1
Math Notation: Order of Operations
P arentheses
E xponents
Multiplication
D ivision
A ddition
S ubstraction
5∙4
3
4
+ 5 ∙ 3 − 6 5 = 8403.8
Math Notation: Rounding
• Provide correct answers to two decimal places.
o Percentages, in particular, should be correct to
two decimal places, which means proportions
should be correct to four decimal places.
o Therefore, don’t round 1 3 to 0.3, or you’ll
introduce error into your calculations.
• Try to do hand calculations to at least one extra
decimal places throughout, and round at the end.
o Using spreadsheet software protects against
rounding errors, and saves a lot of time generally.
Practicing Math
• We’ll be doing some computation by hand.
• If doing math problems isn’t your thing, it gets easier
with practice. The most important thing is to show
the steps you did, in order.
• Performing a computation is like following a recipe:
do the steps in the order prescribed.
o Showing your work is like writing a recipe for
someone else to follow.
o If we ask you to show your work, we want to see
all the steps, whether you used a machine or not.
Machine-Assisted Analysis
• You may use software to do computation.
o In fact, many standard computations in modern
statistics require computers.
o To show your work clearly, you need to be able to
report the steps you took, even if you relied on
calculators or spreadsheets.
• You should trying to solve problems using software,
then transcribe the steps you took.
o Microsoft Excel, Google Sheets, and LibreOffice
are all valid tools for making the tedious parts of
math the computer’s problem.
A Warning About AI
• Large Language Models
such as ChatGPT are an
important exception. Do
not use them.
o LLMs are mainly good
at producing text that
has the right sort of
overall look.
o They are consistently
unreliable because that
overall look is achieved
with no understanding.
Why Are You Here?
• All science depends on measurement.
o The output of our measurements are data.
• All measurement has uncertainty.
• We have to understand and reduce the uncertainty in
our estimates.
• Statistics (the field) is the study of what we can say
about our measurements.
o Each expression of that understanding is achieved
by computing a relevant statistic.
What’s In A Datum?
• Anything we need to measure is a variable.
o “Age” is a variable. So is “religious affiliation.”
• Each variable is limited to certain values.
o “12 years” is a valid age. “Catholic” is not a valid
value for age, but it is for religious affiliation.
• The specific value we actually measure is a score.
o “12 years” could be an age but it’s not my score.
• Scores are only as reliable as our measurement
paradigm. They can be vague, or wrong.
Psychologists Measure Absurd Things
• “Happiness.”
• “Memory.”
• “Intelligence.”
• We need to measure these things in order to discover
what they are and how they work.
• How do you measure something you don’t
understand?
Levels of Measurement
• When we define a variable, we also define the sorts
of things we can do to the resulting scores.
o Psychologists have widely adopted a “ladder” of
variable types, as proposed by Stevens (1946).
o As you climb to the next “level” of the ladder, you
are allowed to do more to your scores.
o This ladder is specific to psychology; other
sciences mostly don’t use this framework.
• For this class, we will consider three levels: nominal,
rank-order, and equal-interval variables.
Nominal Data
• A nominal variable consist of discrete categories.
o These categories are assumed to never overlap;
they are always discrete.
• Categories have no ordering.
o “Vanilla > Chocolate” is not a valid statement.
• Categories can’t have arithmetic done to them.
o “Vanilla ÷ Chocolate” is not a valid statement.
• The main thing you can do with categories is count
how many scores of each kind you received.
Rank-Order Data
• A rank-order variable consists of values that have
some sort of ordering. This is also called an ordinal
variable.
o Gold vs. silver vs. bronze medals
o First-born, second-born, third-born
• We can compare ordinal values.
o “Gold Medal > Silver Medal” is valid.
• However, we can’t do arithmetic with them.
o “Gold Medal ÷ Silver Medal” is not valid.
• Ideal for things that are hard to measure precisely, or
for which sensible units don’t exist.
Equal-Interval Data
• When we collect data for an equal-interval variable,
the data have units that give the numbers meaning.
o Height, weight, distance, temperature…
• Equal-interval variables do support arithmetic!
o “10 > 5” is valid
o “10 ÷ 5” is valid (and equals 2)
• Many of the outcomes we’ll talk about in the class are
equal-interval variables (also called “scale” variables).
o Some are not equal-interval, which can be a
problem when the field pretends that they are!
Interval vs. Ordinal Data
• Many variables we might assume are at the scale level
are really only at the ordinal level.
oExample: IQ.
oThe gap between 40 and 50 doesn’t mean the
same thing as the gap between 140 and 150.
• When a variable is ordinal, the phenomena
underlying it are much more obscure.
oRegular meals might be enough to get a 3.2 GPA to
a 3.4, but won’t necessarily get a 3.7 get to a 3.9.
• The level of measurement puts important limits on
how rich our theories can become.
Measurement Is In Our Heads
• People often misinterpret “levels of measurement” as
being an objective property of data.
oNature itself doesn’t care about measurement.
oNearly all variables in nature are continuous.
• Level of measurement are a judgment we make about
the relationships between the values in our data.
oWe say the “wrong” level of measurement is used
when there is a disconnect between our
assumptions (and the math those assumptions
lead us to use) and the behavior of the world.
Measures vs. Statistics
• Statistics are extracted from measurements, but need
not be on the same level or of the same type.
oWhat is the average roll on an ordinary die?
oWhat is about “% who favor chocolate?”
• Typically, our statistics will be continuous interval
values, even when the data are not.
oHowever a statistic is only as informative of the
scale it is based on.
oFor example, a sprinter’s average ranking is less
informative than their average speed.
Practice!
Types of Variables
• Coin flip
• “Are you tall enough to ride?”
• Gender presentation
• % of folks tall enough to ride
• # of drinks per week
• Probability
• % body fat
• Credit rating
• 7-point satisfaction scale
• Semantic similarity
• Age
• Intensity of depression