Analysis of GRE Scores of MEMS Applicants
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Contents
Introduction[edit]
MEMS program is an international master program in economics and management science that is being run at the Humboldt University zu Berlin. The fact that many of its courses are in English makes it attractive for students who do not master the German language properly and therefore many students all over the world apply to the program. In order to apply for it, students need, besides the classical forms as diplomas, transcripts, etc, 2 tests: GRE and TOEFL. GRE is a test that comprises of 3 parts: quantitative (mathematic problems), analytical (language skills) and verbal (essay). Using the results of this test, I will try to verify the assumptions that I refer to in the opening part, that are based on my personal experience of a senior MEMS student.
Dataset[edit]
The dataset ranges over 11 years, from 1998 (when the MEMS program started) until 2008. It has 1677 observations, out of which 418 were eliminated due to missing values (some of the applicants did not send their scores). 10 variables were taken into consideration.
 Application Year
 Decision Acceptance
 Immatriculated
 Gender
 Continent
 Average Grades (AG)
 Average Grades Quantitative (AGQ)
 GRE Quantitative (GRE Q)
 GRE Analytical (GRE A)
 GRE Verbal (GRE V)
Assumptions[edit]
The assumptions on which the analysis was performed were the following:
 Males score more than females in the quantitative tests
 Asians score better than the other students in quantitative tests
 Asians score less in analytical and verbal tests than students from Europe, N America and Oceania
 N Americans score more than the other students in analytical and verbal test
 Africans score the least in all fields.
General Statistics[edit]
Application Year  Decision Acceptance  Immatriculated  Gender  Continent  

N Valid  1259  1259  1259  1259  1259 
N Valid  0  0  0  0  0 
AG  AGQ  GRE Q  GRE A  GRE V  
N Valid  1259  1259  1259  1259  1259 
N Valid  0  0  0  0  0 
As one can see from the table, there are 1259 observations with no missing values for all the fields, except AG (average grade) and AGQ (average grade quantitative). However, the missing values from the 2 fields will not affect the analysis since the two variables will not be used.
Figure 1:
From the 1259 applicants, 677 were accepted in the program and 582 were rejected, showing that there is a high rate of acceptance rate.
Figure 2:
Out of the 677 accepted students, 429 decided to join the program and 200 did not.
A filter was applied for the 2008 intake, since at the date this project was made there was no information about the students who decided to join the program or not.
In any case, the immatriculation rate is higher than 64%, which means that two out of three accepted students accept to join the program
Figure 3:
Looking at the number of applications per year, one may see that in the begining there were few applicantions, probably due to the fact that the master was unknown, but there was however a positive trend peaking in 2003 with 218 applications. Since than, the number of applications decreased reaching 77 (excluding the number of application from the partner universities).
Figure 4:
Looking at the gender statistics, one may observe that there are more males that apply to MEMS than females, and to be more precise there are 54% male applicants and 46% female applicants.
The above issue may be explained by the quantitative aspect of the master which is usually more appealing to boys rather than girls.
However, the difference is minor and probably not significant.
Figure 5:
By checking the continent statistics of the applicants one may see that most of the applications come from Europe, which is normal since the master is in Berlin.
Following Europe, the next continent that provides also a high number of candidates is Asia, which may also not come as a surprise taking into account the million of people living on this continent.
Far away from Europe and Asia come North America, South America, Africa and Oceania.
Figure 6:
Figure 6 shows the continent origin of the accepted students. Comparing it with figure 5 one may see that the order as well as the the proportions do not change.
Descriptive Statistics[edit]
This chapter provides comparative information about the mean, standard deviation and outliers of applicant as well as accepted stdents according to the gender and continent. Checking the equivalence of means will be the focus of chapter 4.
Descriptive Statistics by Gender[edit]
Figure 7:
Looking at the GRE tests, AQ (average grades) and AGQ (average grades quantitative) of the applicants sorted by gender, one may see that they are, more ore less, the same.
Figure 8:
Performing the same analysis for the accepted students sorted by gender, we get, of course, better averages for males and females.
However, even if the average changed, the averages moved in the same direction and show very close results.
The next 6 box plots will show the GRE quantitative, analytical and verbal of applicants and accepted students sorted by gender. This graphical method will show that the means of males and females are, without making the appropriate tests, the same.
Figure 9: GRE quantitative of male and female applicants.
Figure 10: GRE quantitative of male and female accepted students.
Figure 11: GRE analytical of male and female applicants.
Figure 12: GRE analytical of male and female accepted students.
Figure 13: GRE verbal of male and female applicants.
Figure 14: GRE verbal of male and female accepted students.
As one may observe, the number of outliers decreses dramatically as we move from the applicants to accepted statistics.
Descriptive Statistics by Continent[edit]
Figure 15:
By looking at the GRE statistics of the applicants grouped by continents, one may observe that there is the following order:
 for GRE Quantitative
 Asia (743), Europe (727), Oceania (723), S America (695), N America (667), Africa (588)
 For GRE analytical
 Oceania (4.62), Asia (4.41), N America (4.30), Europe (4.14), S America (3.98), Africa (3.78)
 For GRE Verbal
 Asia (477), N America (472), Oceania (470), S America (415), Africa (398), Europe (392)
Figure 16:
By looking at the GRE statistics of the applicants grouped by continents, one may observe that there is the following order:
 for GRE Quantitative
 Asia (771), Oceania (765), Europe (753), S America (738), N America (729), Africa (690)
 For GRE analytical
 Oceania (4.83), Asia (4.82), N America (4.77), Africa (4.60), S America (4.48), Africa (4.42)
 For GRE Verbal
 Asia (524), Oceania (521), N America (507), S America (473), Africa (428), Europe (417)
Comparing the continent statistics of the applicants and accepted, one may see that there were only small changes in the order and namely:
 for GRE quantitative: Oceania switched places with Europe
 for GRE analytical: Africa switched places with Europe
 for GRE verbal: Oceania switched places with N America
The graphical representation of the continent statistics gives us a quite clear image of the standings as well as the number of outliers. However, the number of outliers decreses dramatically as we move from the applicants to accepted statistics.
Figure 17: GRE Quantitative of Applicants by Continent
Figure 18: GRE Quantitative Accepted by Continent
Figure 19: GRE Analytical Applicants by Continent
Figure 20: GRE Analytical Accepted by Continent
Figure 21: GRE Verbal Applicants by Continent
Figure 22:
GRE Verbal Accepted by Continent.
We may conclude by saying that, even though there are differences in means, as one may clearly see from the graphical representation above, one still needs to check statistically the equality between them, which will be the focus of next chapter.
GRE Score Analysis[edit]
Chapter 4 is the main chapter of this analysis and will clearly show the differencies between the compared groups.
Testing the equality between the means is performed by different test, depending on their distribution.
First of all, we will check if the data is normally distributed. If this is case, then we may performed the ttest for comparing means. In the case of not normally distributed data, then the nonparametric KruskallWallis test will be performed for the variables. If the test will not show significant results, than the MannWhitney U test will be performed for 2 by 2 varibles. Finally, by looking at the pvalues we may conclude which group scored better than the other and see if our conclusions match the assumptions.
KolmogorovSmirnov Test on Normality[edit]
Figure 23. KolmogorovSmirnov Test on Normality Applicants
The p value of the KolmogorovSmirnov test shows a value of 0.00 for all 3 GRE test which is smaller that the significance level of 0.05, meaning that the H_{0} hypothesis of normality is rejected.
According to the methodology used, the next step, in order to compare the means of the applicants, will be the KruskallWallis test.
Figure 24. KolmogorovSmirnov Test on Normality Accepted
The p value of the KolmogorovSmirnov test shows a value of 0.00 for all 3 GRE test which is smaller that the significance level of 0.05, meaning that the H_{0} hypothesis of normality is rejected.
According to the methodology used, the next step, in order to compare the means of the accepted, will be the KruskallWallis test.
KruskallWallis Test[edit]
Figure 25. KruskallWallis Test Applicants by Gender
By performing the KruskallWallis Test on applicants sorted by gender, we will accept the H_{0} hypothesis of equality in means between the males and females, since the obtained p value is higher than the significance level of 0.05 or 5%.
Figure 26. KruskallWallis Test Accepted by Gender
By performing the KruskallWallis Test on accepted students sorted by gender, we will accept the H_{0} hypothesis of equality in means between the males and females, since the obtained p value is higher than the significance level of 0.05 or 5%.
Figure 27. KruskallWallis Test Applicants by Continent
By performing the KruskallWallis Test on applicants sorted by continents, we will reject the H_{0} hypothesis of equality in means between the Europeans, Asians, North Americans, South Americans, Oceanians and Africans, since the obtained p value is lower than the significance level of 0.05 or 5%.
Figure 28. KruskallWallis Test Accepted by Continent
By performing the KruskallWallis Test on accepted students sorted by continents, we will reject the H_{0} hypothesis of equality in means between the Europeans, Asians, North Americans, South Americans, Oceanians and Africans, since the obtained p value is lower than the significance level of 0.05 or 5%.
Since the H_{0} hypothesis of equality in means of the KruskallWallis test for applicants and accepted students for the grouping variable continent was rejected, the next step to make will be a 2 by 2 comparison between the variables, comparison that is performed by the MannWhitney U test.
MannWhitney U Test[edit]
The results of the Mann Whitney U test are displayed in the the following tables.
Notice that in case of rejection of the H_{0} hypothesis of equal means, we have a significance level lwer than 0.05 which is showed in red for a better understanding of the table.
Figure 29. MannWhitney Test on GRE Quantitative Applicants by Continent
Performing the MannWhitney test on GRE quantitative for the applicants, we get the following results:
 Asians score significantly more than Europeans, Africans, N Americans and S Americans
 Asians score statistically the same as Ocenians
 Europeans score significamtly more than Africans, N Americans and S Americans
 Europeans score the same as Oceanians
 N americans, S Americans and Oceanians score statistically the same, but significantly higher than Africans
Figure 30. MannWhitney Test on GRE Quantitative Accepted by Continent
Performing the MannWhitney test on GRE quantitative for the accepted students, we get the following results:
 Asians score significantly more than Europeans, Africans, N Americans and S Americans
 Asians score statistically the same as Ocenians
 Europeans score significamtly more than Africans, N Americans and S Americans
 Europeans score the same as Oceanians
 N americans, S Americans and Oceanians score statistically the same, but significantly higher than Africans
Figure 31. MannWhitney Test on GRE Analytical Applicants by Continent
Performing the MannWhitney test on GRE analytical for the applicants, we get the following results:
 Asians score significantly more than Europeans, Africans and S Americans
 Asians score statistically the same as Ocenians and N Americans
 N Americans score significantly more than S Americans and Africans
 Europeans And Ocenians score significamtly more than Africans
 Europeans score the same as Oceanians, N Americans and S Americans
 N Americans score the same as Oceanians and Asians
 Africans score the same as S Americans
Figure 32. MannWhitney Test on GRE Analytical Accepted by Continent
Performing the MannWhitney test on GRE analytical for the accepted students, we get the following results:
 Europeans score significantly less than N Americans and Asians
 All other tests show equality in means for all other groups
Figure 33. MannWhitney Test on GRE Verbal Applicants by Continent
Performing the MannWhitney test on GRE verbal for the applicants, we get the following results:
 Asians score significantly more than Europeans, Africans and S Americans
 Asians score statistically the same as Ocenians and N Americans
 N Americans score significantly more than Europeans, S Americans and Africans
 Europeans score less than Asian, N Americans, S Americans at 5% confidence level and less than Oceanian at 10% confidence level
 Europeans score the same as Africans
Figure 34. MannWhitney Test on GRE Verbal Accepted by Continent
Performing the MannWhitney test on GRE verbal for the accepted students, we get the following results:
 Asians score significantly more than Europeans, Africans and S Americans
 Asians score statistically the same as Ocenians and N Americans
 N Americans score significantly more than Europeans, Africans at 5% confidence level and more than the S Americans at 10 percent confidence level
 Europeans score less than Asian, N Americans, S Americans and Oceanian
 Europeans score the same as Africans
Conclusions[edit]
Refering to assumptions which were stated in the beginning of the project we may conclude that:
 males do not score more than females in quantitative areas
 Asians score more than Europeans, Africans, N Americans and S Americans
 Asians score the same as Oceanians in quantitative
 Asians do not score less than students from Europe, N America and Oceania in analytical, they actually score more than Europeans
 N Americans only score more than Europeans (10% interval level), Africans and S Americans in analytical for the application stage and only more than European in the acceptance stage
 N Americans only score more than Europeans Africans and S Americans(10% interval level for the acceptance stage), in verbal
 Africans score the least only in quantitative; for verbal and analytical parts their score are competitive
However, these results are interpretable and show the competitive level of the students that apply for the program and not of the overall continents. One needs a much wider sample of observations to come up with better results.
References[edit]
 Good P., HardinJ., Common Errors in Statistics, 2nd Edition, John Wiley & Sons, Inc., Publication, 2006
 Härdle, W., Simar, L.: "Applied Multivariate Statistical Analysis", Springer, 2003
 Kanji, G.: 100 Statistical Tests: Third Edition, Sage Publications, 2006
 Klinke, S.: "Applied Quantitative Methods  Lecture Notes", Institut für Statistik und Ökonometrie, HumboldtUniversität zu Berlin
 www.wikipedia.com
 MEMS database