1. Exploring the Effects of Exercise on Academic Success
Spencer Nelson, Brandon Yu, Kandace Mok, Katherine Delk
Introduction
In this study, we are using survey sampling techniques on UCLA students to investigate the relationship
between the amount of time spent exercising and academic performance. We are using the variables GPA,
gender, major type, time spent studying in hours per week, and the number of days per week spent exercising.
Ultimately, we would like to see if a student’s GPA is positively affected by devoting time to exercise. Our
goal is to use this project to encourage students to maintain a healthy lifestyle, and potentially demonstrate
the link between exercising and success in the classroom.
Many studies have proven that there are positive by-products from exercising outside of an individual’s
physical health. One such study, conducted by researchers at Purdue University [1], indicated that exercise
leads to reduced stress levels and this, in turn, makes students more awake (thus, allowing them to study
more). An article by the New York times [2] also noted that the more committed a student is to studying,
the more likely they are to be committed to exercising as well - we want to see if commitment to exercise
promotes a strong work ethic in the classroom. Additional research has proven that children who exercise
often are more attentive, have better time management, and have superior memory and problem solving
skills, which lead to higher scores on tests. We would like to test if a similar relationship exists within our
sample of college students, and see how strong that relationship is. Additionally, we want to see if certain
groups, such as different majors, have an influence on academic achievement.
Let’s quickly define a few parameters. By “exercise,” we mean activity requiring physical effort, carried out
especially to improve health or fitness; examples include weight-lifting, yoga, and sports. We have recorded
this variable in days spent exercising per week. Next, we define “academic achievement” strictly as GPA.
We hypothesize that we will observe evidence that increased frequency of exercise has a positive effect on a
college student’s GPA; in addition, we hypothesize that upon subsetting our data by different categories,
such as major type, this trend will still hold. In addition, we also believe that GPA will also be observed to
be strongly dependent on other factors, such as hours spent studying per week.
In total, we collected 151 responses via surveys conducted through a Google form sent out to peers.
Data Analysis
Please See Appendix for Enlarged Graphics for Data Analysis and Modeling Sections
Firstly, we would like to get a quick glimpse at our data in preparation for our modeling. For example, let’s
take a look at the distributions of our various variables of interest. We would like to investigate whether we
can realize any type of relationship between certain categories. Some interesting questions we would like to
answer include: are high levels of exercise tied to high GPAS; are high frequencies of studying tied to high
GPAs; are there differences in the distribution of GPA as you move across different majors?
Humanities Quan Science
Major Type Distribution (1a)
0102030405060
25
64 62
Low Medium High
Exercise Level Distribution (1b)
Frequency
020406080
80
42
29
<= 2 days/wk
3,4 days/wk
>= 5 days/wk
Low Medium High
GPA Level Distribution (1c)
Frequency
010203040506070
25
53
73
< 3.19
3.2−3.59
> 3.59
Low Medium High
Study Frequency Distribution (1d)
Frequency
0102030405060
33
53
65
<10 hr/wk
10−20 hr/wk
>20 hr/wk
1

2. *A note on how “major types” were divided in Fig.1a
We define Humanities as creative-thinking majors, including writing, political science, and linguistics.
Quantitative majors include statistics, mathematics, economics, and engineering. Science majors relate to
subjects tied with the life sciences, including biology, chemistry, and psychology.
Low Exerc. Med Exerc. High Exerc.
GPA and Exercise Levels (Fig. 2a)
010203040
Low GPA (<3.19)
Med GPA (3.2−3.59)
High GPA (>3.6)
Female Male
GPA and Gender (Fig. 2b)
0102030
Humanities Quantitative Science
GPA and Major Type (Fig. 2c)
05152535
(2a)/(2b)/(2c) Here we have created barplots to observe how GPA varies across different categories in
preparation for our chi-squared test of independence. Notice how in Fig. 2a, the distribution of GPA does
not seem to vary very much across different levels of exercise. Similarly in Fig. 2b, we can see that the
distributions of GPA among males and females are not radically different. However, in Fig. 2c, the distribution
of GPA seems to change as you move across different major types. In particular, under Humanities, medium
GPA levels makes the largest chunk, while for Science majors, high GPAs is the most prevalent.
Humanities Quantitative Science
01234567
Major Type and Frequency of Exercise (Fig.3a)
DaysSpentExercisingperWeek
Humanities Quantitative Science
0103050
Major Type and Hours Studied (Fig.3b)
HoursSpentStudyingperWeek
0 1 2 3 4 5 6 7
0103050
Study Hours v. Exercise Days (Fig. 3c)
Days Spent Exercising per Week
HoursSpentStudyingperWeek
(3a)/(3b) We would like to investigate if there are particular differences among our different majors which
may be accounting for the varying distributions of GPA. In Fig. 3a, we notice that the distribution of the
number of days per week spent exercising is quite similar across our three majors; in fact, Quantitative majors
and Science majors have identical distributions. However, by contrast in Fig. 3b, we can see the distribution
of the number of hours per week spent studying varies much more; in particular, Humanities majors seem to
be spending less time studying, whereas Quantitative majors have the highest median in hours spent studying
per week. We will investigate the independence of Major Type and Study Levels in the next section.
Modeling
Now, we would like to perform chi-squared tests to observe if there exists independence between our categories
of interest. For example, we will start by observing if there is any independence between GPA Levels and
Exercise Levels. We noted in our Data Analysis section that we noticed that, upon visual inspection, there
did not seem to be much variation in the distribution of GPA Levels across different Exercise Levels (see Fig.
2a). Thus, we suspect that GPA Levels and Exercise Levels are independent of one another; or in other words,
that we cannot predict GPA from Exercise Levels. More formally, we construct our hypothesis as follows.
Ho : GPA Levels and Exercise Levels are independent of one another.
Ha : GPA Levels and Exercise Levels are NOT independent of one another.
Running a chi-squared test of independence yields the following results:
2

3. 0 5 10 15 20
0.000.10
Chi−Square Density Graph: df = 4
<−−− p = 0.8248
χ2
= 1.5105
−3 −2 −1 0 1 2 3
0.00.20.4
Standard Normal
dnorm(x,0,1)
Low Ex (<= 2)
Med Ex
High Ex (>=5)
Rejection region
At a significance level of α = 0.05, we fail to reject the null hypothesis; there is convincing evidence that
knowing a student’s Exercise Level will not help us predict his or her GPA Level, and that these two variables
are independent. Notice how our standardized residuals, which can be thought of as z-values under a standard
normal curve, stay between our rejection regions.
GPA v. Major Test
Observing Fig.2c from our Data Analysis section, we notice that we do NOT have similar GPA distributions
across our different major types. In particular, medium GPA seems to make up a large proportion of the
observation in Humanities students compared to students studying Quantitative and Science topics. We
would like to test if this implies the two variables are not independent. We set up our hypotheses similarly:
Ho : GPA Levels and Major Types are independent of one another.
Ha : GPA Levels and Major Types are NOT independent of one another.
Running a chi-squared test of independence yields the following results:
0 5 10 15 20
0.000.10
Chi−Square Density Graph
χ2
= 10.589
p = 0.03159
−3 −2 −1 0 1 2 3
0.00.20.4
Standard Normal
dnorm(x,0,1)
−2.344
1.995
2.856
Humanities
Quantitative
Science
Rejection region
Because our p-value is under the significance level of α = 0.05, we can safely reject the null hypothesis; there is
convincing evidence that GPA Levels and Major Type are NOT independent. We are able to pinpoint which
categories are statistically significant. Notice how we have a highly negative residual for Science students
under our Medium GPA category of -2.34 and a highly positive residual for Science students for our High
GPA category of 1.99. This is an indication that our sample data underestimated the expected number of
Science students in the high GPA category, which was counterbalanced by overestimating the number of
Science students in the medium GPA category. Similarly, for Humanities students, our stray residual of 2.86
demonstrates our sample data underestimated the expected number of Humanities students in the Medium
GPA category, which was counterbalanced by overestimates in the Low GPA and High GPA categories.
Different Habits among Students of Different Majors?
3

4. Upon our results which show that GPA and Major are not independent, we would like to investigate if there
are certain habitual differences among students of different majors. In particular, we would like to test if
there exists independence between a student’s major against two factors: his or her level of exercise and how
frequently he or she studies per week. Let’s investigate exercise as our first variable of interest. Again, we set
up the hypotheses:
Ho : Major Types and Exercise Levels are independent of one another.
Ha : Major Types and Exercise Levels are NOT independent of one another.
Running a chi-squared test of independence yields the following results:
0 5 10 15 20
0.000.10
Chi−Square Density Graph: df = 4
χ2
= 6.559
p−value = 0.195
−3 −2 −1 0 1 2 3
0.00.20.4
Standard Normal
dnorm(x,0,1)
Humanities
Quantitative
Science
Rejection region
At a p-value of 0.195, we fail to reject the null hypothesis; there is, in fact, convincing evidence demonstrating
that Major Types and Exercise Levels are independent of one another. This confirms our first chi-squared
test, which showed that a student’s GPA and his or her exercise level were not dependent on one another.
Notice again how our standardized residuals stay outside of the critical regions of our standard normal graph.
If not exercise level, we suspect that there must be another factor influencing the differences in GPA among
different major types. We will now focus our attention on analyzing if there exists independence between a
student’s particular major and how frequently he or she studies per week.
Major Type v. Study Levels
Here, we will be investigating if there is variation in the frequency of a student’s studying based on his or her
major. Again, we set up our hypotheses similarly:
Ho : Major Types and Study Levels are independent of one another.
Ha : Major Types and Study Levels are NOT independent of one another.
Running a chi-squared test of independence produces the following results:
0 5 10 15 20
0.000.10
Chi−Square Density Graph: df = 4
χ2
= 13.123
p = 0.01069
−3 −2 −1 0 1 2 3
0.00.20.4
Standard Normal
dnorm(x,0,1)
−2.548
−1.987 2.478
2.93
Humanities
Quantitative
Science
Rejection region
4

5. At a p-value of 0.01, we reject the null hypothesis; there is convincing evidence that Major Levels and Study
Levels are NOT independent. Thus, we have shown that there does, indeed, exists differences in study habits
among students of different majors. We can observe our residual summary to pinpoint which categories are
contributing to the test’s statistical significance. In particular, notice that in our Quantitative category, our
residual of 2.48 indicates we vastly underestimated the number of students who study frequently, and this
was counterbalanced by by an overestimation of Quantitative students who had low frequencies of studying,
as indicated by the negative residual of -1.99. In addition, the opposite trend occurred among Humanities
students, where our sample data overestimated the expected number of these students with high frequencies
of studying, indicated by the negative residual of -2.55; this was counterbalanced by the underestimation of
Humanities students with low levels of studying, as indicated by the highly positive residual of 2.93.
Conclusion
Overall, our findings in this study reject our initial hypothesis that we would observe differing distributions
of academic performance among students who exercised at different weekly frequencies. Rather, it appears
that GPA distribution varies when we categorize students based upon their field of study. Furthermore, upon
dividing students by major types, we find that it is likely differences in the number of weekly hours dedicated
to studying which accounts for this non-uniform GPA distribution.
Let us revisit some of our statistical results which led to the aforementioned conclusions. After running a
chi-squared test of independence between student’s GPA levels (low, medium, high) and their weekly exercise
frequency, we obtained a p-value of 0.82, a strong indication that the two categories are independent. In
other words, we do not expect to see substantially variable GPA distributions among students who exercise
at different rates. A similar analysis between GPA levels and Major Types yielded an extremely low p-value
of 0.03; again, this was a very good indicator that we expect the distribution of GPAs to change as we move
across different major types. Indeed, our residual analysis proved this to be true; for Science students, a
highly positive residual of 1.99 showed our sample data underestimated the expected number of students
in High GPA category, and this was compensated by overestimating the number of Science students in the
Medium GPA category - indicated by a highly negative residual of -2.34. Similarly, a residual of 2.86 indicated
our sample data underestimated the expected number of Humanities students in the Medium GPA category,
which was offset by an overestimation of Humanities students in the High GPA category.
We were interested in investigating potential reasons as to why GPA distribution varied across Major Types,
so we ran two separate chi-squared tests of independence: Major Type v. Exercise Levels and Major Type v.
Study Frequency. Unsurprisingly, our test between Major Type and Exercise Levels yielded a p-value of 0.195,
demonstrating that knowing a student’s major does not give us information about his or her frequency of
exercise. This adds a level of confirmation to our first chi-squared test which showed GPA Levels and Exercise
Levels were independent. However, running a chi-squared test between Major Type and Study Frequency
yielded a p-value of 0.01, exemplifying that the distribution of student’s study frequency should be expected
to be different among different majors. Indeed, residual analysis demonstrated that we underestimated the
number of Humanities students with Low study frequency and overestimated the number of Humanities
students with High frequency of study. This, indeed, aligns with the fact that the Humanities lacked a high
proportion of its students in the High GPA category, a strong indication that hours spent studying and GPA
are strongly dependent.
Let’s discuss the real-world implications of our results. We have found evidence against the claim that there
is dependency between GPA Level and Exercise Level. And this results does make sense; one would not
expect exercise alone to be a contributor to a high GPA. Some make the claim that students who exercise
more have higher GPAs, because students who exercise more also tend to be more active academically. For
our particular sample, however, as seen in Fig. 3c, whether a student exercises zero days per week or seven
days a week, the distribution of hours spent studying seems fairly uniform. We then conclude that exercise
alone has little effect on GPA, and that higher GPAs are largely a byproduct of simply longer hours dedicated
to studying; studies which claim a relationship exists between exercise and GPA likely derive their results
from samples containing students who BOTH study highly frequently AND exercise highly frequently.
5

6. Appendix
References
[1] A study by Purdue University students investigating the effects of exercise on academic success
http://www.purdue.edu/newsroom/releases/2013/Q2/college-students-working-out-at-campus-gyms-get-better-grades.
html
[2] A study by the New York Times investigating the positive effects of exercise on cognitive abilities and
mental health.
http://well.blogs.nytimes.com/2010/06/03/vigorous-exercise-linked-with-better-grades/
Below are enlarged graphics from our Data Analysis and Modeling Sections
Humanities Quan Science
Major Type Distribution (1a)
0102030405060
25
64 62
Low Medium High
Exercise Level Distribution (1b)
Frequency
020406080
80
42
29
<= 2 days/wk
3,4 days/wk
>= 5 days/wk
Low Medium High
GPA Level Distribution (1c)
010305070
25
53
73
< 3.19
3.2−3.59
> 3.59
Low Medium High
Study Frequency Distribution (1d)
Frequency
0102030405060
33
53
65
<10 hr/wk
10−20 hr/wk
>20 hr/wk
6

7. Low Exerc. Med Exerc. High Exerc.
GPA and Exercise Levels (Fig. 2a)010203040
Low GPA (<3.19)
Med GPA (3.2−3.59)
High GPA (>3.6)
Female Male
GPA and Gender (Fig. 2b)
0102030
Humanities Quantitative Science
GPA and Major Type (Fig. 2c)
05101520253035
7

8. Humanities Quantitative Science
01234567
Major Type and Frequency of Exercise (Fig.3a)
DaysSpentExercisingperWeek
Humanities Quantitative Science
0102030405060
Major Type and Hours Studied (Fig.3b)
HoursSpentStudyingperWeek
0 1 2 3 4 5 6 7
0102030405060
Study Hours v. Exercise Days (Fig. 3c)
Days Spent Exercising per Week
HoursSpentStudyingperWeek
0 5 10 15 20
0.000.050.100.15
Chi−Square Density Graph: df = 4
<−−− p = 0.8248
χ2
= 1.5105
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
Standard Normal
dnorm(x,0,1)
Low Ex (<= 2)
Med Ex
High Ex (>=5)
Rejection region
8

9. 0 5 10 15 20
0.000.050.100.15
Chi−Square Density Graph
χ2
= 10.589
p = 0.03159
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
Standard Normal
dnorm(x,0,1)
−2.344
1.995
2.856
Humanities
Quantitative
Science
Rejection region
0 5 10 15 20
0.000.050.100.15
Chi−Square Density Graph: df = 4
χ2
= 6.559
p−value = 0.195
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
Standard Normal
dnorm(x,0,1)
Humanities
Quantitative
Science
Rejection region
9

10. 0 5 10 15 20
0.000.050.100.15
Chi−Square Density Graph: df = 4
χ2
= 13.123
p = 0.01069
−3 −2 −1 0 1 2 3
0.00.10.20.30.4
Standard Normal
dnorm(x,0,1)
−2.548
−1.987 2.478
2.93
Humanities
Quantitative
Science
Rejection region
10