Data Description: The Marks of Mathematics and Statistics of First-Year Students - Research Paper Example

Data The Marks of Mathematics and Statistics of First Year in the past have been given along with the gender. The number of students enrolled in the ten years of study has been at the maximum of 50 and always above 30. The trend is increasing enrollment of female students. The number of Female students in the year 1991 is 34. The number of female students in year 2000 is 50. The increase has taken place gradually over the period of 10 years under the study. The vice versa has happened in case of male students. The number of male students enrolling for the course has gradually come down from 50 to 40. This is evident from the table and chart in the appendix 1. The proportion of male students to the total students has fallen from 60% in year 1991 to 44% in year 2000. This is evident from the table and chart in the appendix.1 The average marks taken by both Male and Female students over a period of ten years under study is 52. The average marks of Female students in 10 years have never been below 50. But for Male students the average marks fluctuated below and above 50 in the 10 years. That explains the deviation. The standard deviation in 10 years data is 16 for Male students and 14 for Female students. The range of marks taken by Male students is between 0 and 100 and by Female students is between 11 and 98 in ten years. This is consistent with standard deviation. This is evident from table and chart in the appendix 2 And standard deviations of marks taken in different years say the Female students scored similar marks in the initial 2 to 4 years period of the study and their marks started varying a lot in the later years. The standard deviation was 6, 9 and 11 during first four years of study and later the standard deviation was well above 11. For the male students the standard deviation is high from the beginning indicating the marks scored by Male students varied at large. For the year 2001, both Male and Female students have similar and high standard deviation above 16. This is evident from table and chart in the appendix 2 Over a period of time, the number and proportion of female students enrolled have gone up, but the marks they score have varied among themselves to a large extents post 1995. Relation Metrics The correlation coefficient is a measure of nature and strength of relationship. This can be used to measure the nature and strength of relationship that Male students and Female students share. The correlation coefficient is negative (-0.69) in case of enrollment of male and female students meaning the number of students enroll is inversely related between male and female students. The more the number of male students, the less the number of female students enrolled for the course in the past. But, the marks correlation and coefficient is positive (0.55) meaning both male and female students marks are positively related. The scores travel in same direction for both male and female students. This is evident from charts in the appendix 3. Tchebycheff's Inequality For any probability distribution & any value of h>0 we choose For example, set h=2, then Tchebycheff's Inequality gives Tested the above inequality with the scores data provided for 10 years for both male and female separately. The marks scored by Male students and Female Students are considered and the scores outside average and 2 standard deviations are identified and their proportions are well less than 0.25. The is evident from tables in Appendix4 The analysis of the outcome is higher proportion of females students have outlier values as compared to Male students. The same can be tested for different sigma levels. Bayes' Rule The Bayes' rule is used to find the probability of a gender given a student with more than or equal to 75% marks is chosen. i.e., A student with more than or equal to 75% marks has been chosen and the probability that student is male or female is found with the help of Bayes' rule. P(A) = P(Male scoring at least 75%) P(B) = P(Female scoring at least 75%) P(E) = P(Choosing a student with at least 75%) Total number of students in 10 years is 883. Total number of Male students in 10 years is 465 Total number of Female students in 10 years is 418 P(A) = P(Male scoring at least 75%) = 0.07957 (37/465) P(B) = P(Female scoring at least 75%) =0.05502 (23/418) P(E) = P(Choosing a student with at least 75%)=0.0680 (60/883) P(A n E)=0.04190 (37/883) P(B n E)=0.02604 (23/883) P(E/A) = P(Male student Chosen scores at least 75%) P(E/B) = P(Female student Chosen scores at least 75%) P(E/A)= P(A n E) / P(A) = 0.04190 / 0.07957 = 0.526614 P(E/B)= P(B n E) / P(B) = 0.02604 / 0.05502 = 0.473386 P(A/E) = P(Student chosen for at least 75% marks happen to be a Male student) P(B/E) = P(Student chosen for at least 75% marks happen to be a Female student) P(A/E) = (P(E/A) P(A)) / P(E) = (0.526614 * 0.07957) / 0.0680 = 0.616216 P(B/E) = (P(E/B) P(B)) / P(E) = (0.473386 * 0.05502) / 0.0680 = 0.383025 From the above calculations it must be evident that a students chosen scoring at least 75% has more chances of being male student and less chances of being Female students. So, in the past ten years the bright student most probably has been a Male student. Central limit theorem As n tends to infinity any distribution would follow normal distribution. That is the essence of Central limit theorem. The distribution of marks by Male students and Female students over a period of 10 years is distributed identically and independently. The aggregate of the marks distribution for Male and Female separately and all together would follow normal distribution. These are evident from the following charts. Appendix 1 Year Absolute Frequency Total Relative Frequency Male Female Male Female 1991 50 34 84 60% 40% 1992 48 35 83 58% 42% 1993 47 40 87 54% 46% 1994 46 41 87 53% 47% 1995 50 40 90 56% 44% 1996 44 42 86 51% 49% 1997 43 46 89 48% 52% 1998 49 43 92 53% 47% 1999 48 47 95 51% 49% 2000 40 50 90 44% 56% Appendix 2 Year Male Female Average STDEV Minimum Maximum Average STDEV Minimum Maximum 1991 48.60 11.21 28 79 49.82 6.41 38 66 1992 47.96 11.70 22 75 51.00 8.54 37 73 1993 52.81 13.26 18 79 52.45 10.88 29 75 1994 48.74 12.94 24 75 52.98 10.95 29 79 1995 53.98 17.57 27 100 53.18 16.46 21 91 1996 46.02 16.68 8 85 49.93 14.24 12 81 1997 56.91 16.96 3 95 52.87 14.61 22 98 1998 51.14 19.82 0 100 55.79 16.36 22 95 1999 53.23 17.94 10 84 52.40 14.99 17 80 2000 57.03 16.42 20 100 53.46 16.95 11 93 Appendix 3 Appendix 4 Year # in limits (male) Outlier % Minimum Marks Maximum Marks 1991 49 1 0.22% 28 79 1992 45 3 0.67% 22 75 1993 46 1 0.22% 18 79 1994 45 1 0.22% 24 75 1995 46 4 0.90% 27 100 1996 42 2 0.45% 8 85 1997 41 2 0.45% 3 95 1998 47 2 0.45% 0 100 1999 47 1 0.22% 10 84 2000 38 2 0.45% 20 100 Year # in limits (Female) Outlier % Minimum Marks Maximum Marks 1991 33 1 0.25% 38 66 1992 34 1 0.25% 37 73 1993 38 2 0.50% 29 75 1994 38 3 0.75% 29 79 1995 39 1 0.25% 21 91 1996 40 2 0.50% 12 81 1997 44 2 0.50% 22 98 1998 41 2 0.50% 22 95 1999 45 2 0.50% 17 80 2000 48 2 0.50% 11 93 Read More