Are exam results gender related?
Exam Results
Aim
Are exam results gender related?
Introduction
This coursework aims to determine if exam results are independent of gender. There may be many reasons as to why one sex may perform better in exams than the other. However why one sex may perform better than another is not an aim of this coursework, only to determine which sex, if any. Should one sex be found achieve better grades in exams, various possible reasons will be discussed. However, the question "Why?" is only a secondary aim of this coursework; as simply put, this is Maths, not Sociology.
Hypothesis
There have been many articles in the media that girls do better in exams than boys and my own experience agrees with this statement. I think girls are generally better at all subjects, and excel at subjects such as art, languages and English, but perform slightly less well in subjects such as science, maths and IT. Also, I generally feel that girls do better at essay writing subjects than boys.
Having made a hypothesis, I must remain working through this investigation. I must not try to make the data fit the hypothesis, but make calculations using the data. Once calculations have been made, then I may think about to what extent the data matches the hypothesis, if at all.
Also I must keep in mind not to try to disprove the hypothesis, as I am a boy and I "would like" to find that boys do better than girls.
What Data Will Be Analysed
In order to answer the question, "Are exam results gender related?" data concerning exam results must be analysed. This data are the results from Beauchamp College students taking exams in June 1999. This data was chosen because it was easily available. It must be noted, that as this is data from one school, the conclusions drawn in this coursework does not necessarily apply to students in other schools, in other parts of the country. This will be re-iterated later in this coursework.
Data for all subjects available will be "grouped" to determine if exam results are gender related overall. Also, data for specific subjects will be analysed in order to discover if one sex does better than the other.
The GCSE subjects that will be analysed and discussed are English, Science, French, Maths and I.T. Also, all A Level subjects will be "grouped" and analysed. I think this gives an appropriate spread of subjects with different skills involved, so the data could be used to comment on whether girls and boys have different skills.
I would have liked to analyse all the subjects at both GCSE and A Level, to find if any relationships change, but nearly all of the selected GCSE subjects, there are not enough students to analyse to results using the method that is to be used for the other subjects.
How The Data Will Be Analysed
Results for the individual subjects, as well as the total results, will be split into two categories for ease of analysis. The categories will be "Pass" and "Fail". For both GCSE and A Level, grades A* to C are passes, and D to U are fails. Students that did not sit the examinations will not be included as fails (or passes), as it is not known what they could have achieved.
The chi-squared test will be used to determine whether passing an exam is independent of gender. This entails first creating a model where exam results are independent of gender. Then calculating the X2 statistic, to determine how well the modelled results compare with the observed values. Finally the chi-squared test is used to analyse the X2 statistic and determine if the model fits the data, in this case this means if exam results are independent of gender.
The chi-squared test does not give an indication of which sex does better in exams, if any. It only finds if results are independent of gender. If the chi-squared test shows that the two are not independent then to determine which sex performs better a simple calculation using percentages will be used. Whichever sex had a high percentage of passes will be the one that does better. This is a basic method of determining which sex performs better, but it should be sufficient once the chi-squared test has revealed that exam results and gender are not independent.
The results of the various chi-squared tests for each subject will then be discussed.
Details Of The Chi-Squared Test
The following describes how the chi-squared test will be performed for each subject.
Creating a model
A model must be created and it's results compared with the observed values for the chi-squared test. This model will assume that exam results are independent of gender. To create this model a contingency table such as the one below will be created, but including the data that will be tested.
Pass
Fail
Male
Female
...
This is a preview of the whole essay
The results of the various chi-squared tests for each subject will then be discussed.
Details Of The Chi-Squared Test
The following describes how the chi-squared test will be performed for each subject.
Creating a model
A model must be created and it's results compared with the observed values for the chi-squared test. This model will assume that exam results are independent of gender. To create this model a contingency table such as the one below will be created, but including the data that will be tested.
Pass
Fail
Male
Female
To method for calculating the expected values (assuming result and gender are independent) for each cell in the table, is best shown in using an example - the expected value for the male/pass cell. The formula is as follows:
The above formulae are changed for the other cells in the contingency table, replacing "boys" with "girls" and "passes" with "fails" where appropriate.
Calculating the X2 statistic
The X2 statistic is a measure of how well the model fits the data. The formula is as follows:
The smaller this value is, the better the model fits the data. In this case, the model is that exam results are independent of gender. If the value is large, the model is not a good fit, and so exam results are not independent of gender - one sex does better than the other.
Chi-Squared Test
The chi-squared test is a test whether a model fits the data. It uses the X2 statistic. If the X2 statistic is below a certain value, the difference between the modelled results and the observed value is not significant; therefore the model is a good fit. This certain value is called the critical value, and varies according to the degrees of freedom, and to how sure you wish to be that the model fits the data.
The degrees of freedom for a contingency table is calculated from this formula:
As all the contingency tables in this coursework will have 2 columns and 2 rows, the degrees of freedom will always be 1.
The X2 distribution can be approximated with a family of distributions called the ?2 (chi-squared) distributions, each for a certain number of degrees of freedom.
Here is a approximate sketch of the ?2 density function, for ?=1 (1 degree of freedom):
This graph should help explain how the chi-squared test works.
If the X2 value is greater than 3.84 (the critical value at 5%), the difference between the observed values and the expected values is said to be significant, and so the model is not a good fit. This is because such a high value would only happen in every 5% of the cases by chance alone if the model is true. This shows that it is improbable that the model is correct. There is a 95% chance that the model does not fit the data.
If the X2 value is less than 3.84, the difference is not significant. This means that the model is a good fit.
If you want to be surer of significance, you can choose a smaller percentage, which gives you a higher critical value. If the X2 value is still larger than this value, it is even more improbable that the model is true. If the percentage is 1%, and the X2 value is larger than the critical value, such a high value would only happen in 1% of the cases by chance alone if the model were true, so there is a 99% probability that the model is not true, and so should be rejected as false.
It must be remembered that there are two conditions for the chi-squared test. There must be a total frequency of 50 or greater, and in each cell, the expected value must be 5 or greater. This is the reason why I could not have studied many of the selected subjects at A Level.
Calculations
Here is a description of how a chi-squared test was performed for the total passes and fails for all subjects at GCSE Level.
Contingency table for total GCSE passes/fails:
Pass
Fail
Male
194
932
Female
775
582
Calculating the expected frequencies, assuming that pass/fail are independent of gender:
Pass
Fail
Male
p(male) x p(pass) x no. students
p(male) x p(fail) x no. students
Female
p(female) x p(pass) x no. students
p(female) x p(fail) x no. students
Pass
Fail
Male
408.007
717.993
Female
560.993
796.007
Calculating X2 value.
The critical value for 1 degree of freedom at 5% significance level is 3.84. 183.19 is greater than the critical value, therefore the difference is significant.
The critical value at 1% is 6.63. 183.19 is greater than this value, so the difference is very significant.
The critical value at 0.01% is 10.83, which is lot less than the X2 value 183.19. This means that the difference is extremely significant. This means that if the model were correct, such a high X2 value would only happen in less than 0.01% of the cases. This means that it is highly improbable that the model is true, and so should be rejected. The evidence suggests that exam results, overall, are not independent of gender.
To determine which sex does better, a simple percentage calculation can be carried out. About 75% of girls pass, which the boy pass rate is about 56%. This, along with the chi-squared test shows that overall for GCSE, girls do better at exams than boys. Possible reasons for this will follow later in this coursework.
Other Calculations
In addition to performing a chi-squared test for independence for all GCSE subjects together, they were done separately also, for both A Level and GCSE. The results of these tests are outlined in the following table. The X2 statistic is the value comparing the observed frequencies of pass and fail, against expected frequencies if exam results are independent of gender. The actual frequencies used in these calculations can be found in Appendix A.
If the X2 value is significant at 5%, it is said to be significant. If the X2 value is significant at 0.1% it is extremely significant.
If the X2 statistic is significant, then it is improbable that the model is correct, as such a high value would only occur in a small amount of cases if the model was correct (the amount is about the percentage level of significance). This means that the model of independence is not correct, and exam performance is dependent of gender. The percentage value gives an indication as to which gender performs better.
The percentage values for each gender passing are not included in subjects whose X2 value was not significant, as any difference they show is not significant.
Subject
X2
Significance Level
% Boy Pass
% Girl Pass
All GCSE
83.1901
Extremely
56.16%
75.31%
Science
5.62236
Extremely
61.68%
73.95%
French
26.89082
Extremely
55.13%
80.49%
Maths
5.587027
Significant
50.00%
60.07%
English
32.7049
Extremely
58.23%
81.67%
IT
.39244
Not
N/A
N/A
All A Level
0.807663
Not
N/A
N/A
Analysis Of Calculated Values
All Subjects
At GCSE level, the X2 value is extremely significant, and the percentage of girls that pass is higher than the percentage of boys that pass. This shows that girls do much better than boys, generally, in GCSE exams.
There may be many reasons for this. The most basic one is that girls simply have a higher innate ability. Girls may simply be better.
Girls may be more mature than boys at that age, as it is a common fact that girls mature faster than boys, and so girls would try harder and/or be better. Relating to my personal experience, it was usually boys who messed around the most in GCSE lessons. This reason is strengthened by the fact that at A Level, exam success does appear to be independent from gender. This could be because boys have matured, and so "caught up" with the girls. However, this could be because the boys that wouldn't have done well in A Levels simply did not do them, or possibly that the immature boys did not do A Levels.
It used to be that girls did worse than boys at school. This could be because of a change in values of girls. A study carried out in the 1970's showed that girls and women were more concerned about husbands, marriage and children than their career. In recent years, this emphasis may have moved to career, and so gaining qualifications is more important to girls and so they try harder at school.
Also, in England, a large proportion of industry has moved from the primary and secondary sectors (farming, mining and manufacturing) to tertiary (services and office work) in the last few decades. This could have also given girls more incentive to find work, and so they tried harder at school.
Also, work from women's movements may have increased equal opportunities for women, perhaps to an extent where women feel as though they have more opportunities than men. In many places you hear phrases such like "women don't have to marry a successful man, they can be successful themselves". This may cause girls to try harder as they feel they can be successful.
Girls may have the skills that are required in most subjects, and boys not.
Another way of thinking about it is not that girls do better, but that boys do worse. One reason already stated is that boys may be more immature at GCSE level, but "catch up" at A Level. This is implied in the results.
Another reason is that boys may not want qualifications, or they may not be good at school so they feel as though they don't want them, so they don't try.
Specific Subjects
As the results show, girls do better in most subjects. However, at maths the X2 value is significant at 5%, but not higher. Also, about 60% of girls pass maths, while only 50% of boys pass it. This means that although girls are better, they as better in the other subjects, such as English, French and Science.
This may be because boys are better at the type of thought, and have more of the skills maths required than in other subjects.
The X2 values are particularly high for English and French. Perhaps this is because girls have better language, communication and essay writing skills, and so they do better at these subjects.
At IT, the X2 value is not significant, showing the exam results are independent from gender. This could be because the people that take IT all have an approximately equal skill level in the subject, and so attain about the same grades.
Possibly, boys may have a higher skill in IT, but girls try harder to make up for lack of skill in the subject. This may not be a valid reason, as it is my personal, and quite biased opinion, that boys are "better at computers" than girls.
Short Summary
Girls achieve better grades in all subjects at GCSE Level, apart from IT where there is no difference. The performance of boys and girls are slightly similar in Maths, although girls are still better.
At A Level exam success is independent of gender. There is no difference between boys and girls.
Limitations
As was mentioned towards the start of this coursework, this data is from Beauchamp, as so does not apply definitely for other schools in Leicester and other areas of the country, or the country as a whole. In other areas situations may be different. The attitudes towards school may be different.
One of the possible reasons that girls do better which was listed earlier is that the tertiary (services and office) industry is larger than the secondary and primary industries (farming mining and manufacturing), so they have more of an incentive to find work as they might prefer to work in the tertiary sector. In some areas, the organisation of the local economy might be different.
All of the subjects were split into "pass" and "fail". This does not take into account the actual grade students achieved. Boys may have achieved more A's than Girls, but the found to get the same overall pass rate. This would affect my results.
The results may simply be "fluky". Other factors may have affected the results: illness, personal tragedies etc. However, it is highly improbably that enough of such events could have significantly affected the results and the conclusions drawn from them.
Possible Improvements
A system to take into account the actual grades achieved by both genders could improve the viability of any conclusions drawn from the results.
I carried out some research, and discovered that for 2x2 contingency tables with 1 degree of freedom, a correction for the X2 formula should have been carried out. This is called the Yates Correction. Here is an excerpt from a page from the Internet concerning this:
Yates Correction. The approximation of the Chi-square statistic in small 2 x 2 tables can be improved by reducing the absolute value of differences between expected and observed frequencies by 0.5 before squaring (Yates' correction). This correction, which makes the estimation more conservative, is usually applied when the table contains only small observed frequencies, so that some expected frequencies become less than 10 (for further discussion of this correction, see Conover, 1974; Everitt, 1977; Hays, 1988; Kendall & Stuart, 1979; and Mantel, 1974).
* http://tiger.ees.kyushu-u.ac.jp/~hu/Reference/Online/textbook/stbasic.html
This means that the formula for X2 should be:
This should especially be used in cases where the X2 value is close to the critical value. This could be used for, Maths GCSE, as its value was close to the critical value.
Possible Extensions
Data could be analysed, using the above refinements, for other schools, whole regions or whole countries to find if exam results are gender related. If data was available, analysing how students progress from GCSE to A Level, according to gender could be done. This could help find whether boys do mature at A Level, or some other reason is behind the fact that at A Level there is no distinction between the results of boys and the results of girls.
Appendix A
Actual frequencies used in X2 statistic calculations:
Subject
Boy Pass
Boy Fail
Girl Pass
Girl Fail
All GCSE
194
932
775
582
Science
264
64
352
24
French
86
70
65
40
Maths
36
36
64
09
English
45
04
205
46
IT
40
22
36
2
All A Level
311
50
302
65
2
MATHEMATICKS COURSEWORK.
OCTOBER 14, 2000
