The Factors that Affect GPA of Undergraduate Students

ABSTRACT

The purpose of this case study is to show the correlation between GPA and amount of

hours worked. Using a t-test it was found that we did not reject the hypothesis of µ = µ. The

regression equation for GPA vs. number of hours worked was f(x) = 8.899132 + 0.555767x. The

coefficient of determination was found to be 0.000602, and the correlation coefficient was

0.024546. As a second variable, the number of hours of sleep was tested as a factor that affects

GPA. For GPA vs. number of hours of sleep the regression equation was y = 3.357+ -0.00665x.

The coefficient of determination was 0.021732, and the correlation coefficient was -0.147416.

INTRODUCTION

This case study is to show the relationship between students’ GPA and the number of

hours they work per week. It is thought that a student that works more will have a lower GPA

because the student has less time to study. This means the mean GPA of the group of students

that are unemployed should have a higher GPA than the students that are employed. Additional

factors that could affect GPA, such as sleep, the number of hours of sleep each student got were

also recorded. This hypothesis has been discussed in the article “Term-Time Employment and

the Academic Performance of Undergraduates”. In this article they state that their survey data of

students through the years 2004 to 2008 “finds that an increase in work hours has negative

effects on GPA” (Wenz). This was our original thought process as well and the motivation

behind collecting and analyzing the data collected.

We collected our data by creating a survey online and inviting our friends using social

media to take the survey. We collected 50 survey entries. Our survey was conducted using the

website Kwiksurveys.com and consisted of 16 questions (see attached survey). From these

questions we were able to choose useful numbers in which we thought best correlated with GPA.

To analyze our collected data, we used Excel to organize our data and compute statistics and

graphs.

To show if there was a difference between the means of employed peoples’ GPA and

unemployed peoples’ GPA we used a 2-tailed non-pooled t-test. We first conducted a hypothesis

test with our Ho: µ1 = µ2. Our null hypothesis was that the mean GPA of unemployed students

was equal to the mean number of employed students. Our alternative hypothesis stated that the

mean GPA of unemployed students did not equal the mean GPA of employed students, Ha: µ≠ µ.

We did this at the 95% confidence interval. The test was performed at the 5% significance level,

α = 0.05. The null hypothesis was that the means would be equal. This would mean there is no

significant difference at the 95% confidence interval between the GPA of people who work and

the people who don’t work.

To figure this out we first needed to find out what the means of both employed peoples’

GPA and unemployed peoples’ GPA. The means were actually quite close. Employed people

had a mean GPA of 3.08, and unemployed people had a mean GPA of 3.016. This already goes

against our thought that unemployed people would have a higher GPA.

The critical values, t were calculated to indicate any discrepancy between the two means

was present. To solve for t you have to subtract the mean of one minus the mean of the other and

divide it by the square root of the standard deviation squared of one data set over the number of

variables plus the standard deviation squared of the other data set over the number of variables.

From this calculation, t value was obtained, which equaled to 0.530. Since this is a 2-tailed test

0.530 had to be between the two significant values. Since alpha is 0.05 and it was a two tailed

test, the alpha value was then split to be α = 0.025. We found the t value to be ±2.011. The value

0.530 falls in the region where we cannot reject the null. This means there is no significant

difference between the mean GPA of employed people and the GPA of unemployed people. This

appears contradictory for we expected the GPA of unemployed people to be higher since they

would be able to study more.

From this result, we’ve considered other additional factors that could affect the findings

we made and realized just because people have more time to study doesn’t mean they are using

it. We also realized this could be due to the fact that working people are more motivated due to

the fact they realize what it is like to work and most likely not be getting paid a wage they are

happy with.

The purpose of a 2-tailed hypothesis test is to show that the true mean of a population is

between two numbers given by the t value within a certain confidence level. The higher the

confidence level, the closer together the two numbers that the true mean needs to be between. If

the number does not fall between the numbers the null hypothesis is rejected. If it lies within the

range the null is not rejected, like what happened with our experiment. This is how we were able

to show that the two means are equal at the 95% confidence interval.

Our hypothesis test shows that the mean GPA of students who are employed is equal to

the mean GPA of students who are unemployed. To further prove this, we conducted a

regression analysis. By conducting this analysis we would be able to recognize if there was any

correlation between GPA and the number of hours worked per week by a student. If there is no

correlation seen between these two variables, then our hypothesis test would prove to be correct.

This data was taken from 50 randomly selected college students, all attending different

universities. The GPA range from 2.0 to 3.95 and the number of hours worked per week range

from 0 to 40. This regression analysis analyzed the following data:

Table I: GPA and Number of Hours Worked

GPA Hours Work

2 20

2.4 10

2.5 10

2.5 3

2.5 20

2.56 8

2.56 0

2.56 12

2.58 0

2.6 10

2.6 0

2.7 15

2.89 15

2.9 20

2.9 20

2.9 25

2.9 0

2.9 0

2.9 0

2.9 0

2.9 15

3 15

3 0

3 15

3 20

GPA Hours Work

3.01 5.5

3.1 6

3.1 20

3.17 27

3.2 0

3.2 0

3.2 20

3.24 20

3.27 10

3.29 20

3.3 10

3.32 6.5

3.33 12.5

3.4 10

3.4 0

3.43 40

3.5 5

3.5 0

3.5 0

3.6 0

3.67 9

3.69 5

3.7 20

3.8 25

3.95 5.5

By placing this data into a scatter plot, it was easy to distinguish a weak relationship between the

number of hours worked per week and GPA. A scatter plot for the data collected is shown as:

Figure 1: GPA and Number of Hours Worked

Obtaining a regression equation for the data in the scatter plot does not seem reasonable.

This is because all of the data is spread out across the scatter plot. Also, if a line of best fit were

to be drawn, most of the points would not hit the line, which is an indication that there is little to

no correlation between GPA and number of hours worked per week. The data presents as:

SUMMARY OUTPUT for figure 1

Regression Statistics

Multiple R 0.024546 R Square 0.000602 Adjusted R Square -0.02022 Standard Error 9.493849 Observations 50

ANOVA df SS MS F Significance F

Regression 1 2.608213 2.608213 0.028937 0.865639 Residual 48 4326.392 90.13316

Total 49 4329

0 1 2 3 4 5

0

5

10

15

20

25

30

35

40

45

GPA

Nu

mb

er

of

ho

urs

wo

rke

d p

er

we

ek

Number of Hours worked and GPA

Number of Hours working

Linear (Number of Hours working)

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 8.899132 10.08839 0.882116 0.38211 -11.3849 29.18321

X Variable 1 0.555767 3.267106 0.17011 0.865639 -6.01319 7.124724

The regression equation for the data is f(x) = 8.899132 + 0.555767x. From this regression line

you get a slope of 0.555767. The slope is slightly increasing, inferring that GPA increases as the

number of hours worked increases. The best fit line only passes through a few of these points on

the scatter plot, showing that it would not be a good idea to make an inference about GPA and

number of hours worked from this regression equation. It does not represent enough of the data

correctly.

The coefficient of determination is 0.000602. This shows how much variation in the GPA

is explained by the number of hours worked. This number shows how much of the actual data

was explained by the regression line. If only 0.06% of the data was explained by the regression

line, this indicates that a lot of the data goes unexplained by the line of best fit. The correlation

coefficient is 0.024546. This was found by taking the square root of r2 (0.000602). The

correlation coefficient shows that since 0.024546 is not close to 1 (which would be a perfect line

of fit) that it would not be a good idea to say that GPA depends on number of hours worked. This

is an example of an extremely weak positive linear correlation. Because it is so close to zero, it

could possibly be considered linearly uncorrelated. This implies that the regression equation is

not useful for making predictions.

An outlier is an observation that lies outside the overall pattern of the data. In this data

set, the outlier seen is at (3.43, 40). This data points lies far from the regression line, relative to

the other data points. This point is determined an outlier because it is located outside of where all

the other points seemed to be condensed. This point was not near the regression line and had

great affect upon the regression equation. The removal of this point changes the line of

regression. A potential influential observation would also be seen at (3.43, 40). This point’s

removal would change the coefficient of determination to 0.001429. This would also change the

regression equation to f(x) = 12.35081 – 0.77004x. Both the slope and y-intercept change

considerably.

By removing the outlier listed above, a new scatter plot would present as:

Figure 2: Number of Hours Worked and GPA (removal of outlier)

Obtaining a regression equation for the data in the scatter plot would once again be

unreasonable. This is because the data is still all so far away from the regression line. Most of the

points do not hit the line of best fit, indicating that there is little to no correlation between GPA

and number of hours worked per week.

The data with the removal of the outlier, (3.43, 40) would present as:

0 1 2 3 4 5

0

5

10

15

20

25

30

GPA

Nu

mb

er

of

ho

urs

wo

rke

d p

er

we

ek

Number of Hours worked and GPA

Number of Hours working

Linear (Number of Hours working)

SUMMARY OUTPUT for figure 2

Regression Statistics

Multiple R 0.037797 R Square 0.001429 Adjusted R Square -0.01982 Standard Error 8.557783

Observations 49

ANOVA

df SS MS F Significance F

Regression 1 4.924384 4.924384 0.06724 0.796531

Residual 47 3442.076 73.23565

Total 48 3447

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 12.35081 9.147798 1.35014 0.183437 -6.05218 30.75381

X Variable 1 -0.77004 2.969591 -0.25931 0.796531 -6.74408 5.20401

The new regression equation for this data is f(x) = 12.35081 – 0.77004x. From this

regression line you get a slope of -0.77004. This slope is slightly decreasing, inferring that as

number of hours worked decreases, the GPA increases. The best fit line still only passes through

a few points on the scatter plot, inferring that it is still not a good idea to make an inference about

the data using this new regression equation. The coefficient of determination is 0.001429. This

shows how much variation in the GPA is explained by the number of hours worked per week. If

only 0.14 % of the data is explained by the regression line, this indicates that a lot of the data

goes unexplained by the line of best fit. The correlation coefficient is -0.037797. This was found

by taking the square root of r2 (0.001429). The correlation coefficient shows that since -0.037797

is not close to 1 (a perfect line of fit), that it would not be a good idea to say that GPA depends

on the number of hours worked per week. This would be an example of a weak negative linear

correlation. Because it is so close to zero it could possibly be considered linearly uncorrelated.

This implies that the regression equation is not useful in making predictions.

By removing the outlier at (3.43, 40), the regression equation went from a positive to a

negative slope. This dramatically changes the analysis of the data from inferring that GPA

increases as the number of hours worked increases, to inferring that as GPA increases, the

number of hours worked decreases. These two regression equations are completely different and

infer different ideas about the relationship between GPA and number of hours worked. Although

both regression equations differed considerably, the amount of data explained by these lines was

so small that neither equation could be used dependably. Both equations had extremely small

coefficients of determination, suggesting that both regression lines were not strong enough to

make inferences about the data.

The regression analysis of the data further explained our results for the hypothesis test of

whether or not the mean GPA of unemployed students was equal to the mean GPA of employed

students. By taking the actual number of hours worked by employed and unemployed students

we were able to conclude that there is no relationship between the GPA and number of hours

worked, and that GPA does not depend on the number of hours worked. By finding this, it

proved that the result of the hypothesis test was reasonable in showing that the mean GPA of

unemployed students was equal to the mean GPA of employed students, proving that both mean

GPAs are equal because there is no affect upon GPA by number of hours worked (employment).

The original data of the GPA scores and hours worked per week can be further tested by

a residual analysis. The first assumption of regression inferences is that a plot of the residuals

against the values of GPA should fall roughly in a horizontal band centered and symmetric

around the x-axis. The second assumption is that a normal probability plot of the residuals should

be roughly linear. If these assumptions are not met, the validity of the assumptions for regression

inferences of the data is lost. The residual plot of the data for GPA presents as:

Figure 3: Residual Plot GPA Data

This residual plot shows that the residuals fall roughly in a horizontal band that is

centered and symmetric about the x-axis. This meets the first assumption of a residual plot.

Figure 4: Normal Probability Plot for Residuals

The normal probability plot shows that the plot for the residuals is roughly linear. This

meets the second assumption of a normal probability plot. Interpreting these plots, we can

conclude that there are no obvious violations of the assumptions for regression inferences for the

variables GPA and number of hours worked per week.

-20

0

20

40

0 1 2 3 4 5

Re

sid

ual

s

GPA

GPA Residual Plot

0

20

40

60

0 20 40 60 80 100 120

Nu

mb

er

of

ho

urs

wo

rke

d

Residual

Normal Probability Plot

Table II: GPA and Number of Hours of Sleep

Table II: The data was obtained for a sample Undergraduate students at the University of Rhode

Island.

Previous attempt to determine whether the Number of Hours of Working has an effect on

students’ GPA resulted in a strongly weak correlation, more close to no correlation. To explore

additional factors that are associated to the GPA of students, the relationship between GPA and

the number of hours of sleep was examined. First, a scatterplot was developed in order to

visualize any apparent relationship between GPA and the number of hour of sleep:

GPA Hours of Sleep GPA Hours of Sleep

2 45 3.01 49

2.4 75 3.09 35

2.5 42 3.1 45

2.5 50 3.1 49

2.5 56 3.17 32

2.56 30 3.2 50

2.56 56 3.2 36

2.56 35 3.2 35

2.58 56 3.24 45

2.6 40 3.27 42

2.6 42 3.3 50

2.7 49 3.32 42

2.89 49 3.33 54

2.9 42 3.4 50

2.9 40 3.4 56

2.9 32 3.45 49

2.9 55 3.5 35

2.9 50 3.5 40

2.9 35 3.5 60

2.9 56 3.6 49

2.9 50 3.67 42

3 35 3.69 35

3 33 3.7 35

3 50 3.8 35

3 49 3.95 56

Figure 5: Scatterplot for the GPA and the average number of hours of sleep students get over

the span of 7 days data from Table II.

The scatterplot is consists of data from Table II with the horizontal axis used number of

hours of sleep and the vertical axis used for GPA. Although the GPA – number of hours sleep

data points do not fall exactly on a line, they appear to scatter about a line, so a regression

equation is obtained to further examine the relationship between the two variables.

Figure 6: Regression line and data points for GPA – Number of Hours of Sleep data

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50 60 70 80

GP

A

Number of Hours of Sleep (Hours)

GPA vs. Number of Hours of Sleep

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 20 40 60 80

GP

A

Number of Hours of Sleep (Hours)

GPA vs. Number of Hours of Sleep

GPA

Regression Equation

SUMMARY OUTPUT for Figure 6

Regression Statistics

Multiple R -0.147416

R Square 0.021732

Adjusted R

Square 0.001351

Standard Error 0.413929

Observations 50

ANOVA

df SS MS F Significance F

Regression 1 0.182695 0.182695 1.066288 0.30696

Residual 48 8.224193 0.171337

Total 49 8.406888

Coefficients

Standard

Error t Stat P-value Lower 95% Upper 95%

Intercept 3.356989 0.296543 11.3204 3.75E-15 2.760749 3.953229

X Variable 1 -0.00665 0.006437 -1.03261 0.30696 -0.01959 0.006296

Table III: Summary of Regression Statistics; obtained from Microsoft Excel using Data Analysis.

From the summary of regression statistics table, the linear equation can be obtained: y =

3.357+ -0.00665x; where the y-intercept is 3.357 and the slope is -0.00665. The line slopes

downward, where the y-values decrease as x increase because the slope is negative. This

indicates that the GPA decreases as the number of hour of sleep increases, which is no surprise.

Although the data points are slightly scattered, they are scattered about a line, so it would be

appropriate to determine a regression line, which indicates that it is acceptable to determine the

coefficient of determination.

The coefficient of determination, represented r2, is a descriptive measure of the utility of

the regression equation for making predictions. It can be calculated using the proportion of

variation in the observed values of the response variable (SSR) by the total regression (SST).

From the summary of regression statistics table, the values of SSR and SST can be obtained,

which are 0.182695 and 8.406888, respectively, resulting in the coefficient of determination

value of 0.021732. The coefficient of determination value will always lie between 0 and 1, where

a value near 0 indicates that the regression equation is not very useful for making predictions,

while a value near 1 indicates that the regression is very useful for making predictions. Given

that the coefficient of determination value of 0.021732, which is extremely close to 0, suggests

that the regression is rather not very useful in making assumptions.

Another method used to measure the correlation between two quantitative variables is the

linear correlation coefficient, r. This value measures the strength of the linear relationship

between two variables. From the regression statistics table, the value of the linear correlation

coefficient can be determined, which is -0.147416. From this value of the linear correlation, we

can conclude a number of properties about the data. First, the negative r value reflects the slope

of the scatterplot, which in this case is negative. Second, the sign of r also suggests the type of

linear relationship. This r value is negative (-), suggesting the variables are negatively linearly

correlated, meaning that y will decrease linearly as x increases. Third, the magnitude of the r

value indicates the strength of the linear relationship between the two variables; for this

scatterplot it is -0.147416, which is not near ±1, but rather 0. Therefore this shows most weak

linear relationship between the variables. Furthermore, this can strongly conclude that the

variable x is a poor linear predictor of the variable y. Lastly but not least, the sign of r and the

sign of the slope of the regression line is identical. The sign of both r and the slope of the

regression line are negative. This identical sign implies that the regression equation and the

linear correlation are useful for making predictions. This results in an extreme weak negative

linear correlation with an r value of -0.147416, which can be concluded that the two variables are

linearly uncorrelated.

Examining both methods of Coefficient of Determination and Linear Correlation

Coefficient resulted in uncorrelated relationship between the GPA and the number of hours of

sleep. This leads to an assumption that there may be outliers or influential observations present in

the data. An outlier is any data point that lies far from the regression line. Influential observation

is any data point whose removal causes a significant change in the regression equation. Two

influential observations, (56, 3.95) and (45, 2) were removed and a new scatterplot was obtained.

Figure 7: New regression line and data points for GPA – Number of Hours of Sleep data with 2

influential observations, (56, 3.95) and (45, 2) removed.

SUMMARY OUTPUT for

figure 7

Regression Statistics

Multiple R -0.231041

R Square 0.05338 Adjusted R Square 0.032801 Standard Error 0.365501

Observations 48

0

0.5

1

1.5

2

2.5

3

3.5

4

0 20 40 60 80

GP

A

Number of Hours of Sleep

GPA vs. Number of Hours of Sleep

GPA

Regression Equation

ANOVA

df SS MS F Significance F

Regression 1 0.346528 0.346528 2.593951 0.114114 Residual 46 6.14517 0.133591

Total 47 6.491698

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 3.477697 0.264531 13.14663 3.45E-17 2.945223 4.010172

X Variable 1 -0.00929 0.005768 -1.61057 0.114114 -0.0209 0.002321

Table IV: Summary of Regression Statistics for new scatter plot with two influential observations

removed; obtained from Microsoft Excel using Data Analysis.

From the new summary of regression statistics table, the linear equation is obtained: y =

3.477 + -0.00929x; where the y-intercept is 3.477 and the slope is -0.00929. The line slopes still

negative, indicating that the GPA decreases as the number of hour of sleep increase. The

coefficient of determination for the new scatter plot is 0.05338. Although the r2 value has

increased significantly, it remains closer to 0, indicating that the regression is still not useful in

making assumptions. The linear correlation coefficient, r value for the new scatter plot is -

0.231041. This value has increased tremendously after having removed 2 influential

observations. The negative value reflects negative slope of the scatter plot which is identical for

the linear regression equation, also suggesting the negative linearly correlation relationship.

Although the magnitude of the r value is 0.231041 is still closer to 0 than ±1, it shows a better

weak correlation between two variables. From the linear correlation coefficient computation, we

can conclude there is a weak negative linear correlation with an r value of -0.231041.

CONCLUSION

The results were not what were initially expected. The mean GPA of both employed and

unemployed was not significantly different. This means working does not affect a student’s

GPA. Also, there was no correlation between GPA and the number of hours a student slept.

Possible reasons for this could be the students with jobs are more motivated to do well, or

perhaps the students all study for similar amounts of time even though some of them work during

the week.

References

Weiss, N.A. (2012). Introductory Statistics. 9th

Edition.

Wenz, M., & Yu, W. (2010). Term-Time Employment and the Academic Performance of

Undergraduates. Journal Of Education Finance, 35(4), 358-373.

SAMPLE SURVEY (Factors Affecting GPA)

1) Are you currently enrolled in college?

Yes

No

2) Are you a full-time or part-time student?

Full-time

Part-time

3) What is your current class status?

Freshman

Sophomore

Junior

Senior

4) What is your current field of study (major)?

Science

Math

English

History

Other

5) What is your current GPA? (Response in 0.00 format)

6) Do you live at home, on-campus or off-campus?

Home

On-campus

Off-campus

7) If you are an off-campus student, then how many hours per week do you spend driving to school? (If you

live on campus, then mark 0 as your response)

0

0-5

5-10

10-15

15 or more

8) Are you currently employed or unemployed?

Employed

Unemployed

9) On average, how many hours per week do you work? (Response in # hours format)

10) If you commute to work, how many hours per week do you spend driving to work? (If you are not

currently working, mark 0 as your answer)

0

0-5

5-10

10-15

15 or more

11) On average, how many hours of sleep do you get per week? (Response in 0.0 hours format)

12) How many hours per week do you spend doing school-related work (ex. homework, projects, paper, lab

reports, studying, etc)?

0-5

5-10

10-15

15 -20

20-25

25-30

30 or more

13) Are you currently involved in any extra-curricular activities? (Ex. sports, clubs, organizations, volunteer

work, etc)

Sports

Organizations

Clubs

Volunteer Work

None

Other

14) What is your family's average income?

0-20,000

20,000-40,000

40,000-60,000

60,000-80,000

80,000-100,000

100,000 or more

15) Are you financially independent?

Yes

No

16) What is your ethnicity?

Caucasian

African American

Native American

Asian

Hispanic

Other