Analysis of GRE Scores of MEMS Applicants

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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]

Table 1. General Statistics
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. Frequency Decision Acceptance

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. Frequency Immatriculated

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. Frequency Application by Year

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. Frequency Applicants by Gender

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. Frequency Applicants by Continent

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. Frequency Accepted by Continent

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. Statistics Gender Applicants

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. Statistics Gender Accepted

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 Applicants by Gender

Figure 9: GRE quantitative of male and female applicants.

Figure 10. GRE Quantitative Accepted by Gender

Figure 10: GRE quantitative of male and female accepted students.

Figure 11. GRE Analytical Applicants by Gender

Figure 11: GRE analytical of male and female applicants.

Figure 12. GRE Analytical Accepted by Gender

Figure 12: GRE analytical of male and female accepted students.

Figure 13. GRE Verbal Applicants by Gender

Figure 13: GRE verbal of male and female applicants.

Figure 14. GRE Verbal Accepted by Gender

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. General Statistics Applicants by Continent

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. General Statistics Accepted by Continent

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 Applicants by Continent

Figure 17: GRE Quantitative of Applicants by Continent

Figure 18. GRE Quantitative Accepted by Continent

Figure 18: GRE Quantitative Accepted by Continent

Figure 19. GRE Analytical Applicants by Continent

Figure 19: GRE Analytical Applicants by Continent

Figure 20. GRE Analytical Accepted by Continent

Figure 20: GRE Analytical Accepted by Continent

Figure 21. GRE Verbal Applicants by Continent

Figure 21: GRE Verbal Applicants by Continent

Figure 22. GRE Verbal Accepted 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 t-test for comparing means. In the case of not normally distributed data, then the non-parametric Kruskall-Wallis test will be performed for the variables. If the test will not show significant results, than the Mann-Whitney U test will be performed for 2 by 2 varibles. Finally, by looking at the p-values we may conclude which group scored better than the other and see if our conclusions match the assumptions.

Kolmogorov-Smirnov Test on Normality[edit]

Figure 23. Kolmogorov-Smirnov Test on Normality Applicants

Figure 23. Kolmogorov-Smirnov Test on Normality Applicants


The p value of the Kolmogorov-Smirnov 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 H0 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 Kruskall-Wallis test.


Figure 24. Kolmogorov-Smirnov Test on Normality Accepted

Figure 24. Kolmogorov-Smirnov Test on Normality Accepted


The p value of the Kolmogorov-Smirnov 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 H0 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 Kruskall-Wallis test.

Figure 25. Kruskall-Wallis Test Applicants by Gender

Kruskall-Wallis Test[edit]

Figure 25. Kruskall-Wallis Test Applicants by Gender


By performing the Kruskall-Wallis Test on applicants sorted by gender, we will accept the H0 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. Kruskall-Wallis Test Accepted by Gender

Figure 26. Kruskall-Wallis Test Accepted by Gender


By performing the Kruskall-Wallis Test on accepted students sorted by gender, we will accept the H0 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. Kruskall-Wallis Test Applicants by Continent

Figure 27. Kruskall-Wallis Test Applicants by Continent


By performing the Kruskall-Wallis Test on applicants sorted by continents, we will reject the H0 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. Kruskall-Wallis Test Accepted by Continent

Figure 28. Kruskall-Wallis Test Accepted by Continent


By performing the Kruskall-Wallis Test on accepted students sorted by continents, we will reject the H0 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 H0 hypothesis of equality in means of the Kruskall-Wallis 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 Mann-Whitney U test.

Mann-Whitney 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 H0 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. Mann-Whitney Test on GRE Quantitative Applicants by Continent


Figure 29. Mann-Whitney Test on GRE Quantitative Applicants by Continent

Performing the Mann-Whitney 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. Mann-Whitney Test on GRE Quantitative Accepted by Continent

Figure 30. Mann-Whitney Test on GRE Quantitative Accepted by Continent

Performing the Mann-Whitney 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. Mann-Whitney Test on GRE Analytical Applicants by Continent

Figure 31. Mann-Whitney Test on GRE Analytical Applicants by Continent

Performing the Mann-Whitney 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. Mann-Whitney Test on GRE Analytical Accepted by Continent

Figure 32. Mann-Whitney Test on GRE Analytical Accepted by Continent

Performing the Mann-Whitney 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. Mann-Whitney Test on GRE Verbal Applicants by Continent

Figure 33. Mann-Whitney Test on GRE Verbal Applicants by Continent

Performing the Mann-Whitney 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. Mann-Whitney Test on GRE Verbal Accepted by Continent

Figure 34. Mann-Whitney Test on GRE Verbal Accepted by Continent

Performing the Mann-Whitney 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, Humboldt-Universität zu Berlin
  • www.wikipedia.com
  • MEMS database

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