I am going to investigate the relationship between the IQ and the Key Stage 2 results within this school. This will help me make and prove different hypothesis using the data I have obtained.
8Mayfield High School
Maths Investigation
The following data was provided for Mayfield High school.
Year Group
Boys
Girls
Total
7
51
31
282
8
45
25
270
9
18
43
261
0
06
94
200
1
84
86
70
Total
604
579
183
I am going to investigate the relationship between the IQ and the Key Stage 2 results within this school. This will help me make and prove different hypothesis using the data I have obtained.
The data I have been given presents the Key Stage 2 results in three separate columns, maths, English and science, I will add the three results together to give me one number, this will make them easier to handle.
I will first take a random stratified sample of my data throughout the whole school of 100 boys and 100 girls; I will take a sample of 100 as this number allows me to make sufficient conclusions without there being too little to conclude anything or too high that it would take too long.
First I will investigate IQ and KS2 results throughout all the years of the school then I will investigate the relationships separately amongst males and females.
I will take a random stratified sample. My sample size will be 100. I have chosen to take a stratified sample as this method takes around about the correct proportion of students according to how many there are compared with the rest of the school so the investigation is fair.
The total number of students in the school is 1183.
In a stratified sample, values are sampled from a particular group in proportion to the group's size within the whole school.
Therefore, I will multiply 100 by the number o boys/girls in each
1183
Year group.
The table below shows my results.
I will now randomly select the correct number of boys/girls from each year group using the random number button on my calculator.
Year Group
Boys
Girls
Total
7
3
1
24
8
2
1
23
9
0
2
22
0
9
8
7
1
7
7
4
Total
51
49
00
I will now randomly select the correct number of boys/girls from each year group. I will use the random number button on my calculator.
Total of KS2 Results (Mixed)
Total of KS2 Results
Frequency
7
0
8
9
5
0
0
1
0
2
26
3
2
4
3
5
9
6
7
8
2
Total
00
Total of KS2 Results (Boys)
Total of KS2 Results
Frequency
7
0
8
9
2
0
4
1
7
2
2
3
6
4
6
5
0
6
7
8
Total
51
Total of KS2 Results (Girls)
Total of KS2 Results
Frequency
7
0
8
0
9
3
0
6
1
3
2
4
3
6
4
7
5
9
6
0
7
0
8
Total
49
The evidence from the tables of results, suggests that more boys have a total of KS2 results above 13, than girls do, however it is only by a small amount.
Total of KS2 Results
Mean
Mode
Median
Range
Boys
2.75
2
2
0
Girls
2.6
2
2
9
The mean for boys is very slightly higher than that of girls. The range for boys' results is also a bit bigger than that of girls.
This evidence tells me that boys' boys in this school did slightly better than the girls did in their KS2 tests, although this evidence is not enough to prove much as the gap between boy's and girls results is only small.
7 out of 51 boys or 14% have a total of KS2 results from 7 to 10. While, 9 out of 49 or 18 % of girls have a total of KS2 results from 7 - 10.
3 out of 51 boys or 25% have a total of KS2 results from 15 to 18.
0 out of 40 girls or 20% have a total of KS2 from 15 - 18.
This shows that more boys have higher results in their KS2 results compared to the girls in this school, who did worse.
IQ
Frequency (f)
Mid point (x)
fx
70-80
75
75
81-90
4
85.5
342
91-100
6
95.5
528
01-110
9
05.5
2004.5
11-120
0
15.5
155
21-130
0
25.5
0
31-140
35.5
35.5
51
5240
IQ
Frequency (f)
Mid point (x)
fx
70-80
75
75
81-90
3
85.5
256.5
91-100
9
95.5
814.5
01-110
6
...
This is a preview of the whole essay
Mid point (x)
fx
70-80
75
75
81-90
4
85.5
342
91-100
6
95.5
528
01-110
9
05.5
2004.5
11-120
0
15.5
155
21-130
0
25.5
0
31-140
35.5
35.5
51
5240
IQ
Frequency (f)
Mid point (x)
fx
70-80
75
75
81-90
3
85.5
256.5
91-100
9
95.5
814.5
01-110
6
05.5
688
11-120
9
15.5
039.5
21-130
25.5
25.5
31-140
0
35.5
0
?f = 49
?fx = 4999
Boys
x = ?fx = 5240 = 102.75
?f 51
Girls
x = ?fx = 4999 = 102.02
?f 49
IQ
Mean
Modal Class
Median
Range
Boys
02.75
01-110
01-110
53
Girls
02.02
91-100
01-110
45
By looking at these tables it is clear that the mean is slightly higher boys, however the medians are the same.
This suggests that the boys have a very slightly higher IQ than girls, but the results are similar to each other. The range suggests that boys' IQs are more spread out than the girls', the girls results are more closely packed than the boys who have results low and high.
57% of boys have an IQ between 101 and 110 and 39% of girls have an IQ between 91 and 100.
If you look at the frequency polygons you can see that the results are fairly similar.
Now, I will test a hypothesis.
In general, if a pupil has a high IQ they will have high KS2 results.
Looking at the scatter graph comparing IQ and KS2 results we can see that there is a moderate, good correlation between IQ and KS2 results. This means the hypothesis I predicted is correct and the higher the IQ of the person, the higher the KS2 results they obtain. However there were a few anomalous results, for example one pupil had an IQ of 108, and their total KS2 results were 18, this is one of the highest KS2 results but not the highest IQ. This may be due to the fact that this particular person didn't do as well as they could on the IQ test and so it does not properly reflect their capabilities.
Next I shall see how these results are affected by gender.
My next hypothesis is,
The correlation will be stronger when girls and boys are considered separately.
Looking at the scatter diagrams of IQ against Total of KS2 results, for boys and girls separately, it is clear that correlation is slightly stronger when boys and girls are considered separately, so my hypothesis is correct. The difference is much more clearly defined now.
Using this scatter diagram we can now predict the IQs or the totals of KS2 results for pupils, so if a pupil has an IQ of x we can predict what they may have got on their KS2 results.
The line of best fit predicts:
* A girl with an IQ of 100 would have a total of KS2 results equalling 11.02 (11 to 2s.f).
* A boy with an IQ 100 would have a total of KS2 results equalling 11.19 (11 to 2s.f.)
The results are very similar, but boys slightly better than girls even when they have the same IQ. This is also, why the boys' line of best fit is slightly steeper than that of the girls.
We can also use the equations of the lines of best fit to predict IQ or total of KS2 results.
Equations of the lines of best fit:
Mixed: y = 0.20x - 7.97
Boys: y = 0.21x - 7.03
Girls: y = 0.19x - 7.03
A boy with an IQ of 105:
y = 0.21x - 8.76
y = (0.21 x 105) - 8.76
= 13.29
Total of KS2 results = 13 (2s.f)
A girl with a total of KS2 results equalling 11:
y = 0.19x - 7.03
1 = 0.19x - 7.03
0.19x = 11+ 7.03
0.19x = 18.03
x = 94.89473684
IQ = 95 (2s.f)
The line of best fit is a very reliable estimation of relationship between two factors (IQ and total of KS2 results).
There are some exceptional values that fall outside the general trend, which affect the position of my line of best fit. In addition, rounding my results/equations of the line .etc. make my predictions less accurate.
I will draw a cumulative frequency graph to express IQ as it can be expressed as continuous data.
I will read off the median, upper quartile (UQ), lower quartile (LQ) and inter-quartile range (IQR).
IQ (Boys)
IQ
Frequency
IQ
Cumulative Frequency
70-80
70 up to but not including 81
81-90
4
70 up to but not including 91
5
91-100
6
70 up to but not including 101
21
01-110
9
70 up to but not including 111
40
11-120
0
70 up to but not including 121
50
21-130
0
70 up to but not including 131
50
31-140
70 up to but not including 141
51
IQ (Girls)
IQ
Frequency
IQ
Cumulative Frequency
70-80
70 up to but not including 81
81-90
3
70 up to but not including 91
4
91-100
9
70 up to but not including 101
23
01-110
6
70 up to but not including 111
39
11-120
9
70 up to but not including 121
48
21-130
70 up to but not including 131
49
31-140
0
70 up to but not including 141
49
IQ (Mixed)
IQ
Frequency
IQ
Cumulative Frequency
70-80
2
70 up to but not including 81
2
81-90
7
70 up to but not including 91
9
91-100
35
70 up to but not including 101
44
01-110
35
70 up to but not including 111
79
11-120
9
70 up to but not including 121
98
21-130
70 up to but not including 131
99
31-140
70 up to but not including 141
00
Fig. 8 is a cumulative frequency graph for IQ.
Median
LQ
UQ
IQR
Mixed
01.5
95
08
3
Boys
02
95.5
08
2.25
Girls
01
95
08.5
3.5
The box - and - whisker diagrams (fig. 9) provide a very clear comparison of the different measures of spread.
The box - and - whisker diagrams show that the boys' IQR is 2 less than the girls' IQR, suggesting that the boys have many results clustered about the median. This proves that the range is unreliable; as if the range was considered alone; we would conclude that the boys have many results spread out around the median. The IQR tells us that this s not the case.
28 girls have an IQ less than the boys' median (102). Therefore, 22 girls or 45% have an IQ above the boys' median. More than half of the girls' IQs are below the boys' median IQ. (Fig. 10)
In general, boys have greater IQs (and therefore, greater totals of KS2 results) than girls. However, quite a large proportion of 44% have an IQ greater than the boys' median, 102.
There is a positive correlation between IQ and the total of KS2 results. The higher the IQ a pupil has, the higher the total of their KS2 results.
The points on the scatter diagram for girls (fig. 7) are less dispersed about the line of best fit than for boys (fig. 6). This means that the correlation is better for girls and that boys' results vary more. The points on the scatter diagram for boys and girls (fig. 5) are more dispersed about the line of best fit, than for fig. 6 or fig 7. This means that the correlation is stronger when boys and girls are considered separately - gender affects the relationship between IQ and KS2 results.
The points on the scatter diagram for boy and girls (fig. 5) can be approximated by a curve (in red) instead of a line (fig 11). However, the curve is very flat and is not that much different to the line. I think that the relationship is linear. The scatter graphs can be used to give reasonable predictions. Either by reading off the graph or using the equations of the lines of best fit.
The cumulative frequency curves (fig. 8) prove that boys' and girl's IQs are very similar but boys are slightly higher, as the median IQ for boys is slightly higher than that of girls.
The box - and - whisker diagrams show that range is not a reliable measure of spread. The boys' range is bigger than the girls' are, but the boys' IQR is smaller.
A larger sample would have been more representative of the school, but would have been more difficult to work with.
My conclusions are based on general trends. However, there were outstanding values that do not follow my trend and would have affected some of my outcomes. I chose not too ignore any exceptional values because none of them were extremely odd.
I will now look at one year group, I will take a 10% sample of year 7.
I will need:
0% x 151 = 15.1
5 Year 7 Boys
0% x 131 = 13.1
3 Year 7 Girls
Total of KS2 Results (Boys and Girls)
Total of KS2 Results
Boys
Girls
7
0
0
8
0
0
9
0
0
1
3
2
2
3
4
3
2
0
4
5
3
5
2
6
0
0
7
0
0
8
0
0
Total
5
3
By looking at the tables above and fig. 12
Total of KS2 Results
Mean
Mode
Median
Range
Boys
2.9
5
3
6
Girls
3
5
4
6
Two out of three measures of average are greater for girls than for boys. The mode result is the same for both and so is the range.
The evidence suggests that in general, year 7 girls have higher totals of KS2 results than boys do. If the same trend occurs here as in the rest f the school, year 7 girls will also have greater IQs.
IQ (Boys)
IQ
Frequency (f)
Midpoint(x)
fx
70-80
0
75
0
81-90
85.5
85.5
91-100
3
95.5
286.5
01-110
1
05.5
160.5
11-120
0
15.5
0
21-130
0
25.5
0
31-140
0
35.5
0
?f = 15
?fx = 1532.5
IQ (Girls)
IQ
Frequency (f)
Midpoint (x)
fx
70-80
0
75.5
0
81-90
85.5
85.5
91-100
6
95.5
573
01-110
4
05.5
422
11-120
15.5
15.5
21-130
25.5
25.5
31-140
0
35.5
0
Total
?f = 13
?fx = 1321.5
Boys
x = ?fx = 1532.5 = 102.2
?f 15
Girls
x = ?fx = 1321.5 = 101.7
?f 13
By looking at the above and fig. 13
IQ
Mean
Mode
Median
Range
Boys
02.2
01-110
01-110
8
Girls
01.7
91-100
91-110
39
The three measures of average are greater for boys than for girls. Suggesting, like in the rest of the school, boys do slightly better than girls do. The range is extremely high for girls, suggesting that in my sample of year 7 girls, there are a few pupils with extreme results. However, it is not consistent because the boys' median is still higher than the girls' are.
I will now test a hypothesis, to see if the same trend occurs in year 7 as in the rest in the school.
"In general, the higher the IQ a pupil has, the higher the total of their KS2 results."
Fig.14 is a scatter diagram of the IQs and totals of KS2 results of the year 7 pupils. There is a positive correlation, proving my hypothesis correct.
I predict, "The correlation will be stronger when boys and girls are considered separately (like in the rest of the school)."
Fig. 15 and 16 are scatter diagrams of the IQs and totals of KS2 results of year 7 pupils, considering boys and girls separately. Therefore, again, my hypothesis is correct and year 7 follows the same trend as the rest of the school.
The gradient for boys was greater than the gradient for girls, when taking the whole school into account (fig. 6 and 7). When the year 7 pupils are considered alone, the boys' gradient is again greater than that of the girls. However, it considerably greater, this may suggest that year 7 boys do a lot better than year 7 girls, compared to boys and girls generally in the school.
There is one extreme value (IQ- 128, Total of KS2 results - 15), if ignore it, the position of the line of best fit is affected (fig. 16i). The extreme result made the gradient of my line of best fit considerably smaller. I will choose to ignore this point, as I think following results will be more realistic and give better observations of the year group.
The new results, when the extreme has been ignored.
IQ
Mean
Mode
Median
Range
Boys
02.2
01-110
01-110
8
Girls
99.7
91-100
91-110
23
I will now draw a cumulative frequency graph of the IQs for Year 7 pupils.
Boys
IQ
Frequency (f)
Cumulative Frequency
70-80
0
70 up to and including 81
0
81-90
70 up to and including 91
91-100
3
70 up to and including 101
4
01-110
1
70 up to and including 111
5
11-120
0
70 up to and including 121
5
21-130
0
71 up to and including 131
5
31-140
0
72 up to and including 141
5
Girls
IQ
Frequency (f)
Cumulative Frequency
70-80
0
70 up to and including 81
0
81-90
70 up to and including 91
91-100
6
70 up to and including 101
7
01-110
4
70 up to and including 111
1
11-120
70 up to and including 121
2
21-130
0
71 up to and including 131
2
31-140
0
72 up to and including 141
2
Mixed
IQ
Frequency (f)
Cumulative Frequency
70-80
0
70 up to and including 81
0
81-90
2
70 up to and including 91
2
91-100
9
70 up to and including 101
1
01-110
5
70 up to and including 111
26
11-120
70 up to and including 121
27
21-130
0
71 up to and including 131
27
31-140
0
72 up to and including 141
27
The cumulative frequency graph shows:
Median
LQ
UQ
IQR
Mixed
01.5
96
05
9
Boys
02.25
98.5
05
6.5
Girls
97
94
04.5
0.5
The IQR for the year 7 girls is quite a lot larger than that of the boys. This suggests that in year 7 girls' IQs are spread out across the median, while boys' IQs are quite clustered around it. This is quite similar to the results for the whole school. Although, the girls has a smaller range in that case, whereas here the IQR and range are larger than the boys' are. We can easily compare the measures of spread by looking at the box-and-whisker diagrams, fig. 18.
There is a positive correlation between IQ and the total of KS2 results, like in the whole school. The higher the IQ a pupil has, the higher the total of their KS2 results.
The points on the scatter diagram for girls (fig. 16i) are very slightly less dispersed about the line of best fit than for boys (fig. 6). This means that the correlation is better for girls and that boys' results vary more. This is the same as in the whole school. Again, gender affects the relationship between IQ and total of KS2 results, as the correlation is stronger when boys and girls are considered separately.
I think that the relationship, within year 7 is linear. Fig. 19 shows in red how the relationship could be expressed as a curve, but it is extremely similar to the line of best fit.
The cumulative frequency curves (fig. 17) show that there more girls with lower IQs than there are boys, even though my sample includes more boys than girls.
Again, a larger sample would have been more representative of the school.
I chose to ignore an anomalous value because it affected my line of best fit considerably.
