The aim of my investigation is to use and apply my understanding of statistics and statistical techniques to investigate the two following hypothesis - There is a correlation between Key Stage 3 (KS3) and GCSE results.
INTRODUCTION
The aim of my investigation is to use and apply my understanding of statistics and statistical techniques to investigate the two following hypotheses:
SCHOOL A -Highgram School
Hypothesis 1: There is a correlation between KS3 & GCSE results.
RAW DATA (table 1)
GENDER
KS3 LEVEL
GCSE GRADE
GENDER
KS3 LEVEL
GCSE GRADE
BOY
3
U
BOY
7
A
BOY
3
U
GIRL
3
E
GIRL
3
G
GIRL
3
D
BOY
4
U
BOY
4
F
BOY
5
E
GIRL
5
E
GIRL
5
C
BOY
5
D
GIRL
5
E
GIRL
5
D
GIRL
5
E
BOY
5
D
GIRL
6
D
BOY
5
E
BOY
6
D
GIRL
5
D
BOY
6
D
GIRL
5
C
GIRL
6
C
BOY
5
D
BOY
6
D
BOY
5
D
GIRL
6
C
GIRL
5
C
GIRL
6
E
BOY
6
B
BOY
6
D
GIRL
6
C
GIRL
6
C
BOY
6
D
BOY
6
D
GIRL
6
D
GIRL
7
B
GIRL
6
A
BOY
7
A
GIRL
6
C
GIRL
3
F
GIRL
6
D
BOY
3
G
GIRL
6
C
GIRL
4
F
GIRL
7
A
BOY
4
U
GIRL
4
U
GIRL
5
E
BOY
5
E
GIRL
5
E
GIRL
6
B
GIRL
6
B
GIRL
6
C
BOY
6
B
BOY
6
E
GIRL
6
C
GIRL
6
C
GIRL
6
B
GIRL
6
B
With my acquired raw data (table 1) I will construct a frequency table (table 2) to condense the information and make it easier to read, understand and utilise.
FREQUENCY TABLE (table 2)
KS3
Level 3
Level 4
Level 5
Level 6
Level 7
TOTAL
Boys
3
3
7
9
2
24
Girls
4
2
0
8
2
36
TOTAL
7
5
7
27
4
60
GCSE
U
G
F
E
D
C
B
A
TOTAL
Boys
4
4
0
0
2
2
24
Girls
2
7
6
2
5
2
36
TOTAL
5
2
3
1
6
2
7
4
60
To make it easier to compare values and grades graphically I will use the Microsoft Excel package on my computer to construct bar charts and pie charts. Bar charts will make it easier to compare grades because they illustrate and from looking at them I will be able to compare frequencies. Yet, they can only be constructed using discrete data. Pie charts are useful as they can be used for both discrete and continuous data and from them I can compare ratios.
The charts that I will construct are:
i. Bar Chart showing KS3 results (chart 1, page 4)
ii. Bar Chart showing GCSE results (chart 2, page 5)
iii. Pie Chart showing KS3 results (chart 3, page 6)
iv. Pie Chart showing GCSE results (chart 4, page 7)
From viewing the bar charts I can see that there is a correlation between KS3 and GCSE results as the trend lines are the same shape. From looking at the pie charts it is difficult to see if there is a comparison a at KS3 ...
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The charts that I will construct are:
i. Bar Chart showing KS3 results (chart 1, page 4)
ii. Bar Chart showing GCSE results (chart 2, page 5)
iii. Pie Chart showing KS3 results (chart 3, page 6)
iv. Pie Chart showing GCSE results (chart 4, page 7)
From viewing the bar charts I can see that there is a correlation between KS3 and GCSE results as the trend lines are the same shape. From looking at the pie charts it is difficult to see if there is a comparison a at KS3 there are 5 levels whereas at GCSE there are 8 grades. Although, on both pie charts the percentages for level 7 and grade A correspond at 7%, the percentages for level 6 and grades C and D are close at 45% and 47% respectively. Also the percentages level 3 and grades U and G are close at 12% and 11% respectively. These levels and grades that I have mentioned correspond in terms of success or failure in either exam.
To carry on this idea I will find the average values of the data. I will use three methods:
i. Mean
ii. Mode
iii. Median
There are many reasons for using this method to see if there is any correlation between results - Mean - it is easy to calculate, uses all the data and is useful for further calculation, the only drawback of this is that it is affected by extreme data.
The mode on the other hand is not affected by extreme data, it's easily understood, but it may not be unique. Again, the median is not affected by extreme data, but it is harder to work out as arranging all the data in order can be tedious. Since the GCSE grades are not numeric, we cannot find the mean. From the frequency table (table 2, page 3) I will be able to find the mode and median.
KEY STAGE 3: mode = 6 median = 6
GCSE: mode = D median = D
From these answers it is possible to predict that a KS3 level 6 correlates to a grade D at GCSE.
We can use scattergraphs to examine this correlation further. A correlation is an association or link between two variables (in this case the KS3 level and GCSE grade). A scattergraph can tell us:
> If any association exists,
> The strength of the association,
> The direction (+ve or -ve).
If there is a correlation then I will be fit to draw a 'line of best fit', this is drawn so that the points are evenly distributed on either side of the line, using this it is possible to make predictions or complete any missing data. Again I have used Microsoft Excel to produce a scattergraph (chart 5, page 9). Seeing I had to deal with non-numeric data (GCSE grades) I have coded it accordingly:
U - 1 D - 5
G - 2 C - 6
F - 3 B - 7
E - 4 A - 8
The scattergraph when a line of best fitted shows that there is a positive correlation, which is very weak. Using the line of best fit it is again possible to prove that a KS3 level 6 corresponds to a GCSE grade D.
CONCLUSION
I am now able to accept hypothesis 1 and can say that there is a positive, but quite weak correlation between the results of KS3 and GCSE.
Hypothesis 2: Girls are achieving higher grades at both KS3 and GCSE.
To either prove or disprove this hypothesis I will have to represent the data for boys and girls separately, but we have to remember that there is a higher number of girls in the school which will to a certain extent effect the results.
Again, using Excel I will generate dual bar charts for KS3 levels (chart 6, page 12), and for GCSE (chart 7, page 13). The dual bar charts are the best way of comparing the data. Also, I will use pie charts showing the boys and girls results at KS3 (chart 8, page 14) and GCSE (chart 9, page 15).
From the charts I can see:
CHART 6 - Dual Bar Graph for KS3
* Girls achieving higher at 3 of the 5 levels
* There are equal numbers of girls and boys at level 7
* Equal amounts of boys and girls are achieving the two lowest grades
* The highest number of boys and girls are getting grades 5 and 6
CHART 7 - Dual Bar Graph for GCSE
* Girls are achieving higher at 4 of the 8 grades
* There are equal numbers of girls and boys at grade G and grade A
* 19 girls achieved a pass (grade A-C) compared to 4 boys
* 17 girls failed (grade D-U) compared to 20 boys
CHART 8 - Pie Charts for KS3
* The percentage of boys and girls achieving level 7 is equal at 8%
*this is due to the larger number of girls
* A larger percentage of girls achieved a level 6
* The percentage of girls and boys achieving a level 5 is close at 28% and 29% respectively
* Just over a 1/4 (26%) of the boys achieved the lowest two levels, compared to 17% of the girls
CHART 9 - Pie Charts for GCSE
* Exactly 1/4 of the boys got the lowest three grades (F-U), compared to only 12% of the girls
* Over half (52%) of the girls passed (grade A-C), compared to only 16% of the boys
Now I will use cumulative frequency to further discover if this hypothesis is
true or false. Firstly I will construct a frequency table.
FREQUENCY TABLE (table 3)
KS3
Level 3
Level 4
Level 5
Level 6
Level 7
Boys
3
6
3
22
24
Girls
4
6
6
34
36
TOTAL
7
2
29
56
60
GCSE
U
G
F
E
D
C
B
A
Boys
4
5
6
0
20
20
22
24
Girls
2
4
1
7
29
34
36
TOTAL
5
7
0
21
37
49
56
60
From this table I will construct cumulative frequency curves for the KS3 (chart 10, page 17) and GCSE (chart 11, page 18) results.
From the KS3 curve it is clearly seen that apart from level 4 results the girls achieving higher grades than boys. From the GCSE curve it is not as clearly cut, because there are quite a few places where boys outperform the girls i.e. grade U, G, F and D. But at these points the two groups are quite close, when the girls do outperform the boys they do by a large margin. So, indeed there is evidence that boys are under-achieving compared to girls.
Now, I will rearrange table 1 into the girls and boys separately, so I will be able to generate scattergraphs for the girl's (chart 12, page 20) and boy's (chart 13, page 21) results separately.
RAW DATA (table 4 as table 1, but arranged in terms of gender)
GENDER
KS3 LEVEL
GCSE GRADE
GENDER
KS3 LEVEL
GCSE GRADE
BOY
7
A
GIRL
6
C
BOY
7
A
GIRL
6
C
BOY
6
D
GIRL
6
C
BOY
6
D
GIRL
6
C
BOY
6
D
GIRL
6
C
BOY
6
D
GIRL
6
C
BOY
6
D
GIRL
6
C
BOY
5
E
GIRL
6
C
BOY
3
G
GIRL
6
C
BOY
3
U
GIRL
3
D
BOY
3
U
GIRL
5
D
BOY
4
U
GIRL
5
D
BOY
6
B
GIRL
6
D
BOY
6
B
GIRL
6
D
BOY
5
D
GIRL
6
D
BOY
5
D
GIRL
3
E
BOY
5
D
GIRL
5
E
BOY
5
D
GIRL
5
E
BOY
6
D
GIRL
5
E
BOY
5
E
GIRL
5
E
BOY
5
E
GIRL
5
E
BOY
6
E
GIRL
6
E
BOY
4
F
GIRL
3
F
BOY
4
U
GIRL
4
F
GIRL
6
A
GIRL
3
G
GIRL
7
A
GIRL
4
U
GIRL
6
B
GIRL
6
B
GIRL
6
B
GIRL
6
B
GIRL
7
B
GIRL
5
C
GIRL
5
C
GIRL
5
C