Appendix
Essentials of Statistics for
Psychological Science
OUTLINE OF RESOURCES
I. GRAPHIC REPRESENTATIONS AND DESCRIPTIVE STATISTICS
Classroom Exercise A.1: Yo Yo Yo: Listen Up
Classroom Exercise A.2: Smile! You’re a Scientist!
Classroom Exercise A.3: Collecting Class Data
Classroom Exericse A.4: I’ve Got a Golden Ticket
Multimedia Suggestions
PsychSim 5 Tutorials:
Descriptive Statistics
Chapter Objectives as a frequency distribution. Frequency distribu-
tions come in a variety of shapes, but one particular
After studying this chapter, students should be able to: distribution deserves special mention. The normal
1. Interpret a frequency distribution, and distribution is a symmetrical, bell-shaped curve
describe the basic characteristics of a normal that is of particular value to psychologists (and stat-
distribution. isticians) because we know the exact area underneath
2. Define three measures of central tendency different portions of the curve.
and explain how these measures differ when Another way to summarize data is through num-
distributions are skewed. bers known as descriptive statistics. The two most
useful descriptive statistics are measures of central
3. Define two measures of variability, the range
tendency, which tell us the value that best summa-
and the standard deviation.
rizes the entire data set, and measures of variability,
4. Describe how a correlation coefficient measures which provide information about how the data are
both the direction and strength of this distributed around that measure of central tendency.
relationship. The measure of central tendency that is used most
often is the mean, which is the average value of all
of the measurements. However, there are times when
I. Graphic Representations and we can’t use the mean because the data set contains a
Descriptive Statistics few extreme scores that result in the mean presenting
an inaccurate measure of central tendency. In that
After collecting data, psychologists need some way situation, the middle score of the distribution, the
to summarize the information collected. One way is median, is a more appropriate measure of central
to create a graphic representation of the data, known tendency. Then there are situations when our data
389
,390 • APPENDIX • Essentials of Statistics for Psychological Science
cannot be averaged because the data are categorical, Plot your normal curve
e.g., measures of political affiliation or gender, and Sixty-eight percent of scores
numbers don’t convey any useful information about Fall within just one
the variable’s properties. In this situation, the mode, Standard deviation mark
the most common score, is the appropriate measure Above and below the mean
of central tendency. Common measures of variability Ninety-six percent of scores
are the range, the difference between the highest Fall within just two
and lowest scores in a distribution, and the stan- Standard deviations
dard deviation, which is the average difference Above and below the mean
between each score in a distribution and the mean Ninety-nine percent of scores
of that distribution. Fall right within three
Standard deviations
Above and below the mean
Classroom Exercise A.1
Sources:
Yo Yo Yo: Listen Up VanVoorhis, C. R. W. (2002). Stat jingles: To sing or not to
sing. Teaching of Psychology, 29, 249–250.
Depending on how much detail you devote to statis- http://www.youtube.com/watch?v=RIbOztw_W-U
tics in the methodology section of your course, you http://www.youtube.com/watch?v=IcuAh7
might consider sharing with your students some fun Nafrc&feature=related
ways of keeping track of basic statistical concepts. http://www.contextualpsychology.org/
There are, inexplicably, several videos circulating statistics_rap_video_yo
on the Internet that incorporate statistics into musi- http://psychblog.wordpress.com/2006/07/11/stats-rap-yo/
cal productions. Most of these go by some variation http://video.google.com/videoplay?doc
of “stat rap,” “statz rap,” or “statistics rap,” but even id=489221653835413043
a cursory search of YouTube or GoogleVideo should http://youtube.com/watch?v=JS9GmU5hr5w
yield a bounty of possibilities. The URLs for several of
the more popular ones are given below.
Carmen VanVoorhis, of the University of
Classroom Exercise A.2
Wisconsin–La Crosse, also suggests using jingles as a Smile! You’re a Scientist!
way of understanding statistics. A few of these, sung
to the melody of well-known songs, are reproduced Angela Lipsitz, of Northern Kentucky University,
here. suggests a simple and engaging way to introduce stu-
Whatever avenue you take—cool and classic, or dents to the concepts of operationalization, hypoth-
fierce and funky—have a laugh with your students esis testing, and data collection.
while helping them learn. ■ Ask your students to bring to class a yearbook from
a coeducational high school or college. This is usu-
I’m a Standard Deviation
ally a simple request; most students have their own
(to the tune of “I’m a Yankee Doodle Dandy”)
yearbooks readily available or can borrow one from
I’m a standard deviation a roommate or friend.
A standard deviation am I ■ Tell them that they will be testing a hypothesis
I estimate the average distance from the mean regarding gender differences in smiling. Explain
Across a group of scores. that the null hypothesis would be that women and
I’m a standard deviation men do not differ in their rates of smiling, whereas
A standard deviation am I a two-tailed alternative hypothesis would be that
Subtract the mean from each score there is some difference. A more focused one-tailed
Square and add them up test would specifically predict that women smile
Divide by n, take the square root more than men (which, as you probably know,
And that’s a standard deviation has been documented in several studies over the
years). You can decide how much detail you’d like
Plot, Plot, Plot Your Curve to devote to discussing null and alternative hypoth-
(to the tune of “Row, Row, Row Your Boat”) eses, but students should grasp the basics of posing
Plot, plot, plot your curve a research question of this type.
Plot your normal curve ■ Next, discuss with your students which opera-
The mean is equal to the median and mode tional definition of “smiling” should be adopted.
, APPENDIX • Essentials of Statistics for Psychological Science • 391
Teeth showing? Smirk? Visible uplift of zygomati- you have collected the data, you can easily export to
cus major? Anything more pleasant-looking than SPSS or Excel or create tables and figures for export-
a neutral expression? Should there be agreement ing to PowerPoint.
across multiple raters? Students should come to You should ask questions that aren’t too personal
appreciate the necessity and (sometimes) com- and yet allow you to demonstrate the appropriateness
plexity of formulating appropriate operational of different measures of central tendency. Categorical
definitions of variables. data such as gender or handedness highlight the util-
■ Ask your students to work in pairs to code the pho- ity of the mode, and asking students to identify their
tos in the yearbooks they’ve provided. Working in favorite number clearly demonstrates the advantage
pairs allows the chance for inter-rater agreement, of the median over the mode, as you can pretty much
and also makes the exercise more enjoyable. count on one or two students identifying some huge
number as their favorite, e.g., 2.86 ⫻ 1020, which results
■ Students should code two classes (e.g., juniors and
in most everyone else’s choices falling below the mean.
seniors) from each of two yearbooks if possible,
For data that is most appropriate to demonstrate the
although this will be dictated by time constraints.
use of the mean, you could ask questions ranging from
For example, if the class size in the yearbook is
height to family size. A particularly good method is to
large, ask students to code only one group (e.g.,
ask students how much time they spend watching TV
just sophomores), or to code every fourth face, or
each week. Not only are these data best summarized
to code just one yearbook. You can decide how
by the mean, but you can reliably count on differences
much data to generate to fit your needs.
between male and female students should you want to
■ Students should record the number (and percent) of discuss inferential statistics. Depending on the amount
women and men smiling, the total class size, their of time you want to devote to this activity and how
own sex, the year the yearbook was published, the large the class is, you can treat the class as a popula-
class rank analyzed, and any other data you’d like to tion and draw a random sample from the larger group
include. Again, let your desire for elaboration guide of responses and see how well the sample matches the
your choices: a simple count of women versus men population. You could also explore/demonstrate why
smiling or not might suffice, or you might like the we don’t need to sample every subject in the popula-
ability to compute correlations with rater sex, year tion to get a fairly accurate picture of the population
of publication, class standing, and so on. comparing the means and standard deviations of
Lipsitz reports that about 90% of women smile in samples of 20, 40, and 80 students with the population
yearbook photos, compared to about 67% of men. means and standard deviations.
Analyze the data with your students and see what
results. Be sure to tie the discussion in to the meth- Classroom Exercise A.4
odological themes they are learning about.
“I’ve Got a Golden Ticket…”
Source:
You and your students no doubt fondly remem-
Lipsitz, A. (2000). Research methods with a smile: A gender
difference exercise that teaches methodology. Teaching ber Roald Dahl’s children’s classic, Charlie and the
of Psychology, 27, 111–113. Chocolate Factory. Chances are good that some of you
read the book, but chances are better that more of
you saw the original movie (the 1971 version…the
Classroom Exercise A.3 good version…although Dahl apparently detested
it) or its more recent remake, starring Johnny Depp
Collecting Class Data
as Willie Wonka. In fact, speaking of chances, you’ll
You can help students better grasp statistical con- recall that a central theme of the story is that Charlie
cepts by collecting data from them and then sum- faces astronomical odds of gaining admittance to
marizing the results for presentation in class. Not Wonka’s factory. Only five golden tickets are distrib-
only is this easy to do, but since their “own” data uted worldwide…and Charlie, pure of heart, has the
are being summarized, students are more interested. good fortune to get one.
This data collection can easily be done with a variety You can capitalize on these themes of random-
of free survey tools, such as SurveyMonkey, or per- ness, fortune, luck, and candy, with a simple demon-
haps even within your course management system. stration of sampling.
One of the simplest tools to use is Qualtrics (http:// ■ Gather as many minipackages of M&M’s plain
www.qualtrics.com/). Not only is survey creation chocolate candies as you’ll need for the number of
incredibly straightforward and intuitive, but once students in your class. The 1.69-ounce size works
, 392 • APPENDIX • Essentials of Statistics for Psychological Science
best (the kind that typically comes in a big pack- Discussion:
age; shop Costco or Sam’s Club for a good deal). Armed with this information, you can discuss not
■ Distribute one package to each student, and ask only the concept of the sampling distribution and its
students to open the package and to sort the use in inferential statistics, but also the law of large
M&M’s by color. numbers and the importance of sampling in general.
■ They should then count the number of each color
Ask your students to consider what would happen
and record that information; after that, they’re if you selected larger samples each time (i.e., if you
free to eat the M&M’s (producing a nice positive had collected 30 samples of 20 students each, rather
mood manipulation, by the way!). than 30 samples of 5 students each). Also ask them
how much faith they would put in any one student’s
■ As a group, determine one color to focus on (blue?
report on the number of green M&M’s. Would that be
green?), and with the aid of a table of random
indicative of the whole population? What would be a
numbers, select 5 or 10 students to report the
methodological improvement over that, to get better
number of those M&M’s they had. For example,
knowledge of the distribution of M&M’s? If the chanc-
from the table of random numbers (or visit www.
es of finding five golden tickets in the entire human
random.org), students 4, 12, 18, 20, and 45 might
population seem astronomical, could your students at
be selected, who each report that they had 3, 1, 3,
least estimate the reasonable odds of finding a green
2, and 2 green M&M’s in their possession.
M&M in a package selected at random?
■ Ask all students to calculate the mean of this
group, then repeat the process 20 or 30 more Source:
times (as class time will allow). Dyck, J. L., & Gee, N. R. (1998). A sweet way to teach stu-
■ The means of each of the 30 or so samples that dents about the sampling distribution of the mean.
result can be plotted on the blackboard, and Teaching of Psychology, 25, 192–195.
should reveal the shape of a normal distribution.
■ Now, ask students to calculate the mean of the
Multimedia Suggestions
means, or the average of all the sample means.
This value should be very similar to the popula-
tion mean, or the average of all of the individual PsychSim 5 Tutorials
green M&M’s in every student’s possession. Descriptive Statistics
Essentials of Statistics for
Psychological Science
OUTLINE OF RESOURCES
I. GRAPHIC REPRESENTATIONS AND DESCRIPTIVE STATISTICS
Classroom Exercise A.1: Yo Yo Yo: Listen Up
Classroom Exercise A.2: Smile! You’re a Scientist!
Classroom Exercise A.3: Collecting Class Data
Classroom Exericse A.4: I’ve Got a Golden Ticket
Multimedia Suggestions
PsychSim 5 Tutorials:
Descriptive Statistics
Chapter Objectives as a frequency distribution. Frequency distribu-
tions come in a variety of shapes, but one particular
After studying this chapter, students should be able to: distribution deserves special mention. The normal
1. Interpret a frequency distribution, and distribution is a symmetrical, bell-shaped curve
describe the basic characteristics of a normal that is of particular value to psychologists (and stat-
distribution. isticians) because we know the exact area underneath
2. Define three measures of central tendency different portions of the curve.
and explain how these measures differ when Another way to summarize data is through num-
distributions are skewed. bers known as descriptive statistics. The two most
useful descriptive statistics are measures of central
3. Define two measures of variability, the range
tendency, which tell us the value that best summa-
and the standard deviation.
rizes the entire data set, and measures of variability,
4. Describe how a correlation coefficient measures which provide information about how the data are
both the direction and strength of this distributed around that measure of central tendency.
relationship. The measure of central tendency that is used most
often is the mean, which is the average value of all
of the measurements. However, there are times when
I. Graphic Representations and we can’t use the mean because the data set contains a
Descriptive Statistics few extreme scores that result in the mean presenting
an inaccurate measure of central tendency. In that
After collecting data, psychologists need some way situation, the middle score of the distribution, the
to summarize the information collected. One way is median, is a more appropriate measure of central
to create a graphic representation of the data, known tendency. Then there are situations when our data
389
,390 • APPENDIX • Essentials of Statistics for Psychological Science
cannot be averaged because the data are categorical, Plot your normal curve
e.g., measures of political affiliation or gender, and Sixty-eight percent of scores
numbers don’t convey any useful information about Fall within just one
the variable’s properties. In this situation, the mode, Standard deviation mark
the most common score, is the appropriate measure Above and below the mean
of central tendency. Common measures of variability Ninety-six percent of scores
are the range, the difference between the highest Fall within just two
and lowest scores in a distribution, and the stan- Standard deviations
dard deviation, which is the average difference Above and below the mean
between each score in a distribution and the mean Ninety-nine percent of scores
of that distribution. Fall right within three
Standard deviations
Above and below the mean
Classroom Exercise A.1
Sources:
Yo Yo Yo: Listen Up VanVoorhis, C. R. W. (2002). Stat jingles: To sing or not to
sing. Teaching of Psychology, 29, 249–250.
Depending on how much detail you devote to statis- http://www.youtube.com/watch?v=RIbOztw_W-U
tics in the methodology section of your course, you http://www.youtube.com/watch?v=IcuAh7
might consider sharing with your students some fun Nafrc&feature=related
ways of keeping track of basic statistical concepts. http://www.contextualpsychology.org/
There are, inexplicably, several videos circulating statistics_rap_video_yo
on the Internet that incorporate statistics into musi- http://psychblog.wordpress.com/2006/07/11/stats-rap-yo/
cal productions. Most of these go by some variation http://video.google.com/videoplay?doc
of “stat rap,” “statz rap,” or “statistics rap,” but even id=489221653835413043
a cursory search of YouTube or GoogleVideo should http://youtube.com/watch?v=JS9GmU5hr5w
yield a bounty of possibilities. The URLs for several of
the more popular ones are given below.
Carmen VanVoorhis, of the University of
Classroom Exercise A.2
Wisconsin–La Crosse, also suggests using jingles as a Smile! You’re a Scientist!
way of understanding statistics. A few of these, sung
to the melody of well-known songs, are reproduced Angela Lipsitz, of Northern Kentucky University,
here. suggests a simple and engaging way to introduce stu-
Whatever avenue you take—cool and classic, or dents to the concepts of operationalization, hypoth-
fierce and funky—have a laugh with your students esis testing, and data collection.
while helping them learn. ■ Ask your students to bring to class a yearbook from
a coeducational high school or college. This is usu-
I’m a Standard Deviation
ally a simple request; most students have their own
(to the tune of “I’m a Yankee Doodle Dandy”)
yearbooks readily available or can borrow one from
I’m a standard deviation a roommate or friend.
A standard deviation am I ■ Tell them that they will be testing a hypothesis
I estimate the average distance from the mean regarding gender differences in smiling. Explain
Across a group of scores. that the null hypothesis would be that women and
I’m a standard deviation men do not differ in their rates of smiling, whereas
A standard deviation am I a two-tailed alternative hypothesis would be that
Subtract the mean from each score there is some difference. A more focused one-tailed
Square and add them up test would specifically predict that women smile
Divide by n, take the square root more than men (which, as you probably know,
And that’s a standard deviation has been documented in several studies over the
years). You can decide how much detail you’d like
Plot, Plot, Plot Your Curve to devote to discussing null and alternative hypoth-
(to the tune of “Row, Row, Row Your Boat”) eses, but students should grasp the basics of posing
Plot, plot, plot your curve a research question of this type.
Plot your normal curve ■ Next, discuss with your students which opera-
The mean is equal to the median and mode tional definition of “smiling” should be adopted.
, APPENDIX • Essentials of Statistics for Psychological Science • 391
Teeth showing? Smirk? Visible uplift of zygomati- you have collected the data, you can easily export to
cus major? Anything more pleasant-looking than SPSS or Excel or create tables and figures for export-
a neutral expression? Should there be agreement ing to PowerPoint.
across multiple raters? Students should come to You should ask questions that aren’t too personal
appreciate the necessity and (sometimes) com- and yet allow you to demonstrate the appropriateness
plexity of formulating appropriate operational of different measures of central tendency. Categorical
definitions of variables. data such as gender or handedness highlight the util-
■ Ask your students to work in pairs to code the pho- ity of the mode, and asking students to identify their
tos in the yearbooks they’ve provided. Working in favorite number clearly demonstrates the advantage
pairs allows the chance for inter-rater agreement, of the median over the mode, as you can pretty much
and also makes the exercise more enjoyable. count on one or two students identifying some huge
number as their favorite, e.g., 2.86 ⫻ 1020, which results
■ Students should code two classes (e.g., juniors and
in most everyone else’s choices falling below the mean.
seniors) from each of two yearbooks if possible,
For data that is most appropriate to demonstrate the
although this will be dictated by time constraints.
use of the mean, you could ask questions ranging from
For example, if the class size in the yearbook is
height to family size. A particularly good method is to
large, ask students to code only one group (e.g.,
ask students how much time they spend watching TV
just sophomores), or to code every fourth face, or
each week. Not only are these data best summarized
to code just one yearbook. You can decide how
by the mean, but you can reliably count on differences
much data to generate to fit your needs.
between male and female students should you want to
■ Students should record the number (and percent) of discuss inferential statistics. Depending on the amount
women and men smiling, the total class size, their of time you want to devote to this activity and how
own sex, the year the yearbook was published, the large the class is, you can treat the class as a popula-
class rank analyzed, and any other data you’d like to tion and draw a random sample from the larger group
include. Again, let your desire for elaboration guide of responses and see how well the sample matches the
your choices: a simple count of women versus men population. You could also explore/demonstrate why
smiling or not might suffice, or you might like the we don’t need to sample every subject in the popula-
ability to compute correlations with rater sex, year tion to get a fairly accurate picture of the population
of publication, class standing, and so on. comparing the means and standard deviations of
Lipsitz reports that about 90% of women smile in samples of 20, 40, and 80 students with the population
yearbook photos, compared to about 67% of men. means and standard deviations.
Analyze the data with your students and see what
results. Be sure to tie the discussion in to the meth- Classroom Exercise A.4
odological themes they are learning about.
“I’ve Got a Golden Ticket…”
Source:
You and your students no doubt fondly remem-
Lipsitz, A. (2000). Research methods with a smile: A gender
difference exercise that teaches methodology. Teaching ber Roald Dahl’s children’s classic, Charlie and the
of Psychology, 27, 111–113. Chocolate Factory. Chances are good that some of you
read the book, but chances are better that more of
you saw the original movie (the 1971 version…the
Classroom Exercise A.3 good version…although Dahl apparently detested
it) or its more recent remake, starring Johnny Depp
Collecting Class Data
as Willie Wonka. In fact, speaking of chances, you’ll
You can help students better grasp statistical con- recall that a central theme of the story is that Charlie
cepts by collecting data from them and then sum- faces astronomical odds of gaining admittance to
marizing the results for presentation in class. Not Wonka’s factory. Only five golden tickets are distrib-
only is this easy to do, but since their “own” data uted worldwide…and Charlie, pure of heart, has the
are being summarized, students are more interested. good fortune to get one.
This data collection can easily be done with a variety You can capitalize on these themes of random-
of free survey tools, such as SurveyMonkey, or per- ness, fortune, luck, and candy, with a simple demon-
haps even within your course management system. stration of sampling.
One of the simplest tools to use is Qualtrics (http:// ■ Gather as many minipackages of M&M’s plain
www.qualtrics.com/). Not only is survey creation chocolate candies as you’ll need for the number of
incredibly straightforward and intuitive, but once students in your class. The 1.69-ounce size works
, 392 • APPENDIX • Essentials of Statistics for Psychological Science
best (the kind that typically comes in a big pack- Discussion:
age; shop Costco or Sam’s Club for a good deal). Armed with this information, you can discuss not
■ Distribute one package to each student, and ask only the concept of the sampling distribution and its
students to open the package and to sort the use in inferential statistics, but also the law of large
M&M’s by color. numbers and the importance of sampling in general.
■ They should then count the number of each color
Ask your students to consider what would happen
and record that information; after that, they’re if you selected larger samples each time (i.e., if you
free to eat the M&M’s (producing a nice positive had collected 30 samples of 20 students each, rather
mood manipulation, by the way!). than 30 samples of 5 students each). Also ask them
how much faith they would put in any one student’s
■ As a group, determine one color to focus on (blue?
report on the number of green M&M’s. Would that be
green?), and with the aid of a table of random
indicative of the whole population? What would be a
numbers, select 5 or 10 students to report the
methodological improvement over that, to get better
number of those M&M’s they had. For example,
knowledge of the distribution of M&M’s? If the chanc-
from the table of random numbers (or visit www.
es of finding five golden tickets in the entire human
random.org), students 4, 12, 18, 20, and 45 might
population seem astronomical, could your students at
be selected, who each report that they had 3, 1, 3,
least estimate the reasonable odds of finding a green
2, and 2 green M&M’s in their possession.
M&M in a package selected at random?
■ Ask all students to calculate the mean of this
group, then repeat the process 20 or 30 more Source:
times (as class time will allow). Dyck, J. L., & Gee, N. R. (1998). A sweet way to teach stu-
■ The means of each of the 30 or so samples that dents about the sampling distribution of the mean.
result can be plotted on the blackboard, and Teaching of Psychology, 25, 192–195.
should reveal the shape of a normal distribution.
■ Now, ask students to calculate the mean of the
Multimedia Suggestions
means, or the average of all the sample means.
This value should be very similar to the popula-
tion mean, or the average of all of the individual PsychSim 5 Tutorials
green M&M’s in every student’s possession. Descriptive Statistics
