Below is a table of the data I am going to use. Firstly there is only the boys data. Secondly I have put on the girls data and finally I have put both the boys and the girls data.
Boys
Girls
Boys and Girls
Spearmans Co-efficient of Rank Correlation
Below is a table showing my preparation for Spearmans Co-efficient of Rank Correlation. Below each table I have written my conclusion for the following. Firstly there is only the boys data. Secondly I have put the girls data and finally I have put both the boys and the girls data.
Boys
Spearmans Co-efficient of Rank Correlation = 1 – (6 258.5) (25(625 – 1))
= 1 – (1551) (15600)
= 1 – 0.0994230769230769230769230769230769
= 0.900576923076923076923076923077
= 0.9 (1DP)
Using Spearmans Co-efficient of Rank Correlation I have found out that there is a very strong correlation between boys average GCSE results and boys average SAT results. The strongest positive correlation is 1 and the results above show that in this case the correlation is 0.9, which is very strong. The correlation is also stronger than both the girls and the combined boys and girls average results. For example if the SAT result was 5 the GCSE mark would be very likely to be 5 or close to 5.
Girls
Using Spearmans Co-efficient of Rank Correlation I have found out that there is a fairly strong correlation between girls average GCSE results and girls average SAT results. The strongest positive correlation is 1 and the results above show that in this case the correlation is 0.69, which is fairly strong. The correlation is smaller than both the boys and the combined boys and girls average results.
Boys and girls
Spearmans Co-efficient of Rank Correlation = 1 – (6 3491.82) (55(3025 – 1))
= 1 – (20950.92) (166320)
= 1 – 0.125967532467532467532467532467532
= 0.874032467532467532467532467533
= 0.87 (2DP)
Using Spearmans Co-efficient of Rank Correlation I have found out that there is a very strong correlation between boys and girls average GCSE results and boys and girls average SAT results. The strongest positive correlation is 1 and the results above show that in this case the correlation is 0.87, which is very strong. The correlation is also stronger than the girls average results and smaller than the boys average results.
Plotting the mean point
On my three scatter graphs I am going to plot the mean point for a line of best fit to go through. Below I will write what I did for each of the three graphs.
Boys
SAT mean point = (Sum of SAT average)
(Amount of boys)
= 132.68
25
= 5. 3072
= 5.3 (1.D.P)
GCSE mean point = (Sum of GCSE average)
(Amount of boys)
= 118.6
25
= 4.744
= 4.74 (2.D.P)
Girls
SAT mean point = (Sum of SAT average)
(Amount of girls)
= 177.98
30
= 5.932666667
= 5.93 (2.D.P)
GCSE mean point = (Sum of GCSE average)
(Amount of boys)
= 176.76
30
= 5.85866667
= 5.86 (2.D.P)
Boys and Girls
SAT mean point = (Sum of SAT average)
(Amount of boys and boys)
= 310.66
55
= 5. 648363636
= 5.65 (2.D.P)
GCSE mean point = (Sum of GCSE average)
(Amount of boys and girls)
= 294.36
55
= 5.352
= 5.35 (2.D.P)
Analysis
From my three scatter graphs I notice that their is a strong positive correlation between average SAT and GCSE results. The results from the graphs also back this point up.
Boys Analysis
Looking at my scatter graph for the boys average SAT and GCSE results I can see that there is a large range of 4.33 (7 – 2.67) for SAT results and 5.92 (7.2 – 1.28) for GCSE results. Using the line y = 0.72x + 1.95 I can predict approximately what people who got a level in their SAT’s would get in their GCSE’s. Of course you can do this the other way around (GCSE to SAT). For example if a boy got an average grade of 6 in their GCSE they would be likely to have got around 6.2 as their SAT average. Boys appear to get a higher level in their SAT’s.
Girls Analysis
Looking at my scatter graph for the girls average SAT and GCSE results I can see that there is a small range of 3.66 (7.33 – 3.67) for SAT results and 3.49 (7.59 – 3.49) for GCSE results. Using the line y = 0.57x + 1.75 I can predict approximately what people who got a level in their SAT’s would get in their GCSE’s. Of course you can do this the other way around (GCSE to SAT). For example if a girl got an average grade of 6 in their GCSE they would be likely to have got around 6.5 as their SAT average. Girls appear to get a higher level in their SAT’s.
Boys and Girls Analysis
Looking at my scatter graph for the boys and girls average SAT and GCSE results I can see that there is a large range of 4.66 (7.33 – 2.67) for SAT results and 6 (7.59 – 1.29) for GCSE results. Using the line y = 1.16x - 1.22 I can predict approximately what people who got a level in their SAT’s would get in their GCSE’s. Of course you can do this the other way around (GCSE to SAT). For example if a person got an average grade of 6 in their GCSE they would be likely to have got around 6.3 as their SAT average. Boys and girls appear to get a higher level in their SAT’s.
For boys, girls and boys and girls I used Spearmans Co-efficient of Rank Correlation which also showed that there is a strong correlation between average SAT and GCSE results.
Summary
I believe that I have proved my first hypothesis right. The original hypothesis was “People’s average SAT and average GCSE results will have a strong positive correlation between them.” I believe that I have proved this with the use of scatter graphs and Spearmans Co-efficient of Rank Correlation.
Hypothesis Two
As I stated in my plan I am going to use stratified sampling of 55 people. There are 167 pieces of data but I have not included the 167th piece of data because it does not specify whether it is for a boy or girl, it also does not contain any data apart from the total score and the average score of the GCSE results. This means that I have counted 75 boys and 92 girls with the 167th piece of data not being used I have now got 91 girls. Below I will show how I have done this.
Boys
(75 166) 55 = 24.8493975903614457831325301204819
24.8 (1DP)
I have decided to round up the 24.8 to 25 so that I have a whole amount of people to use.
To find out how large or small the difference is I have done 75 25 this equals 3. This means that the person that I am going to pick is every third person. To find out whether I start on the first, second or third person I rolled a dice. This dice landed on a three so the first piece of data is the third.
Girls
(91 166) 60 = 32.891566265060240963855421686747
32.9 (1DP)
I have decided to round up the 32.9 to 33 so that I have a whole amount of people to use.
To find out how large or small the difference is I have done 91 33 this equals 2.75. Again I have rounded this up. This time it has been rounded to 3. This means that the person that I am going to pick is every third person. To find out whether I start on the first, second or third person I rolled a dice. This dice landed on a three so the first piece of data is the third.
Below is a table of the data I am going to use. Firstly there is only the boys data. Secondly I have put on the girls data and finally I have put both the boys and the girls data. For my second hypothesis I am only going to use GCSE results.
Boys
Girls
Boys and Girls
Cumulative Frequency Tables
Boys
Girls
Histogram Preparation Table
Boys
Frequency of one = 4%
Girls
Frequency of one = 3.3%
Pie Chart Preparation Table
Boys
Frequency of one = 14.4 degrees
Girls
Frequency of one = 12 degrees
Boys and Girls
Frequency of one = 7
Analysis
Histogram
Using my percentage histogram I can see that a higher percentage of girls got higher average grades than the boys. For example at an average grade of between one and two there was 0% girls and 4% of boys. Looking at the higher levels 37% of girls got between six and seven and only 16% of boys attained this average level. Looking at my histogram I can see that the peak in the amount of boys is 28% at between three and four. The girls peak is 40% at five and six. Also looking at my graph I can see that there is a larger spread of data for the boys compared to the girls. This means that not only did the girls appear to get higher grades than the boys but they also consistently got higher grades.
Cumulative Frequency
Looking at my cumulative frequency graph for boys I can see that there is a large spread of data. There appears to be a very slight negative skew on the box plot. Looking at the box plot I can see that the four-quarters of data is very evenly distributed. Looking at the cumulative frequency graph for girls I can see that it is very different to the boys. Firstly the spread of data is very small but the spread seems to be at a higher point on the graph compared to the boys. Looking at the box plots I can see that there is a slightly positive skew and that the highest quarter of people seem to get higher grades (6.5 – 8).
Pie Chart
Again looking at my three pie charts I can see that a higher amount of girls attained higher average levels compared to the boys. This tends to agree with what my other graphs have also proven. Looking at the boys and girls pie chart I can see that the majority of boys and girls got a average level between five and seven.