This table has the relevant and important pieces of data of 14 random Year 11 boys, which will be used in diagrams, charts, graphs and calculations to find the answers to my questions and to provide information for my original hypotheses.
Year 11 Girls
This table has the relevant and important pieces of data of 15 random Year 11 girls, which will be used in diagrams, charts, graphs and calculations to find the answers to my questions and to provide information for my original hypotheses.
E. Data Collection
I have collected all my data I planned to collect from Mayfield High School students. I didn’t experience any problems collecting my data as it was collected using a macro from a spreadsheet. I have included my data in a summary table as I have only included the relevant data for my hypotheses and questions that I will investigate.
My tables do have clear headings as they are typed out in bold. All my headings are relevant to what I need. I added another heading, ‘Average SATS Score’, as when a graph, chart or a diagram is shown, there are only two axes, not four. So I have to make an average of every three different SATS Scores for one student.
So in excel (a software package for a spreadsheet), I used a formula to find an average for three SATS Scores e.g. for the first student on the table for Year 7 Boys, I typed in:
=AVERAGE(I2:K2)
Here are my Scatter Graphs; Standard Deviations; Means; Spearman Rank Coefficient and Correlation Coefficient for my collected data.
KEY:
= Mean Point
= Best Fit
Number of points, n: 26 Centroid: (4.308, 101.4)
y-on-x Regression Line: y=9.44x+60.72 x-on-y Regression Line: x=0.07638y-3.436 Correlation Coeff, r: 0.8491 Spearman's Ranking Coeff: 0.9034
Mean, x: 4.308 Mean, y: 101.4
Standard Deviation, y: 7.406 Standard Deviation, x: 5.607
Straight Line: y= 9.44x+60.72
Summary of Scatter Graph for Year 7 Boys IQ vs. Average SATS Score
The Scatter Graph for Year 7 Boys of IQ vs. Average SATS Score shows that the higher the IQ of a Year 7 Boy, the higher the average SATS Score will be.
The Spearmans Rank Correlation Coefficient is 0.9034; and as 1 is the highest and -1 is the lowest; 0.9034 is very high as it is close to 1. This shows that the Correlation Coefficient is positive, as 0.9034 is high which means that the higher the IQ, the higher your Average SATS Score will be.
The mean of x (Average SATS Score) and the mean of y (IQ) tells us that; if a year 7 boy gets a level 4 as an average SATS Score, is IQ will be 101; or if his IQ is 101 he’ll get a level 4 as an average SATS Score.
The Line of Best Fit has been drawn accurately, as there is an equal amount of points on each side of the straight line, and the mean for y and x is on the line of best fit.
KEY:
= Mean Point
= Best Fit
Number of points, n: 22 Centroid: (4.308, 101.4)
Mean, x: 104 Mean, y: 4.091
Standard Deviation, x: 9.023 Standard Deviation, y: 0.668
Correlation Coeff, r: 0.6492 Spearman's Ranking Coeff: 0.712
y-on-x Regression Line: y=0.04807x-0.9062 x-on-y Regression Line: x=8.769y+68.08
Straight Line: y=9.44x+60.72
Summary of Scatter Graph for Year 7 Girls IQ vs. Average SATS Score
The Scatter Graph for Year 7 Girls of IQ vs. Average SATS Score shows that the higher the IQ of a Year 7 Girl, the higher the average SATS Score will be.
The Spearmans Rank Correlation Coefficient is 0.668; and as 1 is the highest and -1 is the lowest; 0.668 is high as it is close to 1. This shows that the Correlation Coefficient is positive, as 0.668 is high compared to the difference between -1 and 1 which means that the higher the IQ, the higher the Average SATS Score.
The mean of x (Average SATS Score) and the mean of y (IQ) tells us that; if a year 7 girl gets a level 4 as an average SATS Score, is IQ will be 104; or if his IQ is 104 he’ll get a level 4 as an average SATS Score.
The Line of Best Fit has been drawn accurately, as there are almost an equal amount of points on each side of the straight line, and the mean for y and x is on the line of best fit.
KEY:
= Mean Point
= Best Fit
Number of points, n: 14 Centroid: (4.308, 101.4)
Mean, x: 100.4 Mean, y: 4
Standard Deviation, x: 12.07 Standard Deviation, y: 0.9258
Correlation Coeff, r: 0.9141 Spearman's Ranking Coeff: 0.922
y-on-x Regression Line: y=0.07012x-3.042 x-on-y Regression Line: x=11.92y+52.76
Straight Line: y=9.44x+60.72
Summary of Scatter Graph for Year 11 Boys IQ vs. Average SATS Score
The Scatter Graph for Year 11 Boys of IQ vs. Average SATS Score shows that the higher the IQ of a Year 11 Boy, the higher the average SATS Score will be.
The Spearmans Rank Correlation Coefficient is 0.922; and as 1 is the highest and -1 is the lowest; 0.922 is high as it is close to 1. This shows that the Correlation Coefficient is positive, as 0.922 is extremely high which means that the higher the IQ, the higher the Average SATS Score.
The mean of x (Average SATS Score) and the mean of y (IQ) tells us that; if a year 11 boy gets a level 4 as an average SATS Score, is IQ will be 100; or if his IQ is 100 he’ll get a level 4 as an average SATS Score.
The Line of Best Fit has been drawn accurately, as there are almost an equal amount of points on each side of the straight line, and the mean for y and x is on the line of best fit.
KEY:
= Mean Point
= Best Fit
Number of points, n: 15 Centroid: (4.308, 101.4)
Mean, x: 100.3 Mean, y: 4.067
Standard Deviation, x: 9.456 Standard Deviation, y: 0.7717
Correlation Coeff, r: 0.7735 Spearman's Ranking Coeff: 0.7384
y-on-x Regression Line: y=0.06312x-2.267 x-on-y Regression Line: x=9.478y+61.79
Straight Line: y=9.44x+60.72
Summary of Scatter Graph for Year 11 Girls IQ vs. Average SATS Score
The Scatter Graph for Year 11 Girls of IQ vs. Average SATS Score shows that the higher the IQ of a Year 11 Girl, the higher the average SATS Score will be.
The Spearmans Rank Correlation Coefficient is 0.7384; and as 1 is the highest and -1 is the lowest; 0.7384 is high as it is close to 1. This shows that the Correlation Coefficient is positive, as 0.7384 is high which means that the higher the IQ, the higher the Average SATS Score.
The mean of x (Average SATS Score) and the mean of y (IQ) tells us that; if a year 11 girl gets a level 4 as an average SATS Score, is IQ will be 100; or if his IQ is 100 he’ll get a level 4 as an average SATS Score.
The Line of Best Fit has been drawn accurately, as there is an equal amount of points on each side of the straight line, and the mean for y and x is on the line of best fit.
Here are my Histograms for my collected data:
Grouped Data Statistics:
Total Frequency, n: 26
Mean, x: 104.615
Standard Deviation, x: 10.4627
Modal Class: 100-
Lower Quartile: 100.5
Median: 107
Upper Quartile: 113.5
Semi I.Q. Range: 6.5
Grouped Data Statistics:
Total Frequency, n: 22
Mean, x: 109.091
Standard Deviation, x: 9.49119
Modal Class: 100-
Lower Quartile: 102.941
Median: 109.412
Upper Quartile: 115.882
Semi I.Q. Range: 6.47059
Summary of Year 7 Boys and Year 7 Girls Histogram for Average SATS Score
Both the histograms have a distribution with no axis of symmetry. They show a lean towards the right – hand side, which means that their distribution has a negative skew.
The Histogram of Year 7 Girls’ Average SATS Score tells me that Year 7 Girls do better in their SATS than Year 7 Boys; as the Girls have a higher mean.
The Year 7 Girls have a smaller standard deviation which means the data isn’t spread out as much as the Year 7 Boys, which means the Girls, are more consistent.
The girls have a higher upper quartile which means that the Year 7 Girls have done better than Year 7 Boys in the SATS, as the statistics have proved it.
The girls had a higher mean which means there average was better so they did better than the Year 7 Boys.
However as there are more boys than girls in the sample taken, because the sample is proportioned to the total population; so is there were more girls than boys in the population and the stratifies sample would have been different, the Year 7 boys could have done better than the Year 7 Girls in the SATS; but all we know that, looking at the statistics, the Year 7 Girls have done better.
Grouped Data Statistics:
Total Frequency, n: 14
Mean, x: 97.1429
Standard Deviation, x: 14.357
Modal Class: 80-
Lower Quartile: 86.25
Median: 95
Upper Quartile: 107.5
Semi I.Q. Range: 10.625
Grouped Data Statistics:
Total Frequency, n: 15
Mean, x: 104.667
Standard Deviation, x: 11.4698
Modal Class: 100-
Lower Quartile: 95
Median: 105.556
Upper Quartile: 113.889
Semi I.Q. Range: 9.44444
Summary of Year 11 Boys and Year 11 Girls Histogram for Average SATS Score
The Year 11 Boys histogram has a distribution with no axis of symmetry. It shows a lean towards the right – hand side which means the distribution has a negative skew.
The Year 11 Girls histogram has a distribution with no axis of symmetry. It shows a lean towards the left – hand side which means the distribution has a positive skew.
The Histogram of Year 11 Girls’ Average SATS Score tells me that Year 11 Girls do better in their SATS than Year 11 Boys; as the Girls have a higher mean.
The Year 11 Girls have a smaller standard deviation which means the data isn’t spread out as much as the Year 11 Boys, which means the Girls, are more consistent.
The girls have a higher upper quartile which means that the Year 11 Girls have done better than Year 11 Boys in the SATS, as the statistics have proved it.
The girls had a higher mean which means there average was better so they did better than the Year 11 Boys.
However as there are more boys than girls in the sample taken, because the sample is proportioned to the total population; so is there were more girls than boys in the population and the stratifies sample would have been different, the Year 11 boys could have done better than the Year 11 Girls in the SATS; but all we know that, looking at the statistics, the Year 11 Girls have done better.
Here are the Cumulative Frequencies for IQ’s and SATS.
∑f = 26
∑fx = 125
∑fx² = 612.5
Mean = 4.808
Standard Deviation = 0.6662
Variance = 0.4438
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 20 10 20 0 0
20 ≤ x < 40 30 20 0 0
40 ≤ x < 60 50 20 0 0
60 ≤ x < 80 70 20 1 1
80 ≤ x < 100 90 20 5 6
100 ≤ x < 120 110 20 20 26
120 ≤ x < 140 130 20 0 26
140 ≤ x < 160 150 20 0 26
160 ≤ x < 180 170 20 0 26
180 ≤ x < 200 190 20 0 26
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 1 0.5 1 0 0
1 ≤ x < 2 1.5 1 0 0
2 ≤ x < 3 2.5 1 0 0
3 ≤ x < 4 3.5 1 3 3
4 ≤ x < 5 4.5 1 12 15
5 ≤ x < 6 5.5 1 11 26
∑f = 22
∑fx = 101
∑fx² = 473.5
Mean = 4.591
Standard Deviation = 0.668
Variance = 0.4463
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 20 10 20 0 0
20 ≤ x < 40 30 20 0 0
40 ≤ x < 60 50 20 0 0
60 ≤ x < 80 70 20 0 0
80 ≤ x < 100 90 20 3 3
100 ≤ x < 120 110 20 17 20
120 ≤ x < 140 130 20 2 22
140 ≤ x < 160 150 20 0 22
160 ≤ x < 180 170 20 0 22
180 ≤ x < 200 190 20 0 22
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 1 0.5 1 0 0
1 ≤ x < 2 1.5 1 0 0
2 ≤ x < 3 2.5 1 0 0
3 ≤ x < 4 3.5 1 4 4
4 ≤ x < 5 4.5 1 12 16
5 ≤ x < 6 5.5 1 6 22
∑f = 14
∑fx = 1360
∑fx² = 1.35E+005
Mean = 97.14
Standard Deviation = 14.36
Variance = 206.1
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 20 10 20 0 0
20 ≤ x < 40 30 20 0 0
40 ≤ x < 60 50 20 0 0
60 ≤ x < 80 70 20 1 1
80 ≤ x < 100 90 20 8 9
100 ≤ x < 120 110 20 4 13
120 ≤ x < 140 130 20 1 14
140 ≤ x < 160 150 20 0 14
160 ≤ x < 180 170 20 0 14
180 ≤ x < 200 190 20 0 14
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 20 10 20 0 0
20 ≤ x < 40 30 20 0 0
40 ≤ x < 60 50 20 0 0
60 ≤ x < 80 70 20 1 1
80 ≤ x < 100 90 20 8 9
100 ≤ x < 120 110 20 4 13
120 ≤ x < 140 130 20 1 14
140 ≤ x < 160 150 20 0 14
160 ≤ x < 180 170 20 0 14
180 ≤ x < 200 190 20 0 14
∑f = 15
∑fx = 1570
∑fx² = 1.663E+005
Mean = 104.7
Standard Deviation = 11.47
Variance = 131.6
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 20 10 20 0 0
20 ≤ x < 40 30 20 0 0
40 ≤ x < 60 50 20 0 0
60 ≤ x < 80 70 20 0 0
80 ≤ x < 100 90 20 5 5
100 ≤ x < 120 110 20 9 14
120 ≤ x < 140 130 20 1 15
140 ≤ x < 160 150 20 0 15
160 ≤ x < 180 170 20 0 15
180 ≤ x < 200 190 20 0 15
Table of Values of Raw Data:
Class Int. Mid. Int. (x) Class Width Freq. Cum. Freq.
0 ≤ x < 20 10 20 0 0
20 ≤ x < 40 30 20 0 0
40 ≤ x < 60 50 20 0 0
60 ≤ x < 80 70 20 0 0
80 ≤ x < 100 90 20 5 5
100 ≤ x < 120 110 20 9 14
120 ≤ x < 140 130 20 1 15
140 ≤ x < 160 150 20 0 15
160 ≤ x < 180 170 20 0 15
180 ≤ x < 200 190 20 0 15
Summary of Cumulative Frequency Diagram for Year 7 Boys IQ and Average SATS Score
For the IQ Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows and a straight line is produced.
For the Average SATS Score Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows, but in this diagram there isn’t a straight line being produced as the maximum value is 6 and the cumulative frequency curve stops at 6.
Summary of Cumulative Frequency Diagram for Year 7 Girls IQ and Average SATS Score
For the IQ Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows and a straight line is produced. The line is very steep.
For the Average SATS Score Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows, but in this diagram there isn’t a straight line being produced as the maximum value is 6 and the cumulative frequency curve stops at 6.
Summary of Cumulative Frequency Diagram for Year 11 Boys IQ and Average SATS Score
For the IQ Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows and a straight line is produced.
For the Average SATS Score Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows, but in this diagram there isn’t a straight line being produced as the maximum value is 6 and the cumulative frequency curve stops at 6.
Summary of Cumulative Frequency Diagram for Year 11 Girls IQ and Average SATS Score
For the IQ Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows and a straight line is produced. The line is very steep.
For the Average SATS Score Cumulative Frequency Diagram, the IQs start to increase and so does the Frequencies. However, at the very end, the frequencies stop, as the graph shows, but in this diagram there isn’t a straight line being produced as the maximum value is 6 and the cumulative frequency curve stops at 6.
F. Data Processing
I have carried out all my calculations to plan as I have used Autograph 3.2 (a software package) to complete my calculations, and as this software package is computerized, so all the calculations will be accurate; except that I have hand drawn one scatter graph, cumulative frequency diagram, box plot and histogram, so that the reader understands that I know the calculations. I have also done one standard deviation and Spearmans rank correlation coefficient calculation, for the same particular reason. The rest of the diagrams and drawings have been completed neatly and accurately by the computer, so there isn’t a huge mess on the drawings. I have double checked that the results of my calculations to make sure that they are accurate and that the results are reasonable in the context of my investigation. I have rounded the sampling sensibly, e.g.
151 / 1183 x 200 = 25.56 25.56 is rounded up to 26 as the first decimal place is 5 or over.
I have also rounded some calculations in the Spearmans rank correlation coefficient, for finding out the square of the differences to one decimal place e.g.
1.4 squared = 1.96 1.96 is rounded up to 2 as the first decimal place is 5 or over.
1.6 squared = 2.56 2.56 is rounded up to 2.6 as the first decimal place is 5 or over.
I have presented a summary for all my results. I have also described what the results show, and they provide more support for my original hypotheses.
G. Data Representation
I have included all diagrams, charts and graphs as accordingly to my plan; such as histograms to find the area of each piece of data; scatter graphs to find the correlation and relationship of two pieces of data.
All the hand drawn diagrams, charts and graphs have been completed neatly and accurately, and the rest were made on computer using a software package called Autograph 3.2.
They all include a clear, bold title on top of the drawings, and all the axes and bars have been labeled. I have provided a key for my scatter graphs, as they include different colours, so I have shown a key for them to make the reader understands the graph.
I didn’t include any extra diagrams, charts or graphs, as I didn’t need to add them to find the answers to my questions and to provide extra information for my original hypotheses.
I have explained what each table, diagram, chart and graph shows, so that it is clear to the reader, and for each explanation it provides more information for the original hypotheses.
H. Overall Summary and Interpretation
My first original hypothesis was that:
-Boys do better than Girls in SATS
The calculations to prove this hypothesis were to find the mean of Year 7 Boys and Girls; and the mean of Year 11 Boys and Girls. To find this I needed to draw histograms and use statistics to explain the histograms.
The Year 7 girls had a higher mean which means there average was better so they did better in their SATS than the Year 7 Boys.
The Year 11 girls had a higher mean which means there average was better so they did better in their SATS than the Year 11 Boys.
This means that the Girls, who are the youngest in the school, and the eldest, did better than the boys, who are the youngest and eldest in the school, in their SATS.
My first hypothesis was not correct, as the mean and histograms provided information to explain the reason for it to be incorrect.
My second original hypothesis was that:
-The higher the IQ, the higher the average SATS results.
The calculations to prove the hypothesis were to find the correlation and relationships between the IQs of Year 7 Boys and Girls, Year 11 Boys and Girls; and also the SATS scores of Year 7 and 11 Boys and Girls. Also I needed to see what happens to the frequencies as the IQ and SATS scores goes higher?
To find the correlation and relationship between Year 7 Boys’ and Year 7 Girls’ IQs and Average SATS Scores, I needed to draw a scatter graph.
The Scatter Graph for Year 7 Boys of IQ vs. Average SATS Score shows that the higher the IQ of a Year 7 Boy, the higher the average SATS Score will be., the statistic that supports this is the Spearmans Rank Correlation Coefficient as it is 0.9034; and as 1 is the highest and -1 is the lowest; 0.9034 is very high as it is close to 1.
The Scatter Graph for Year 7 Girls of IQ vs. Average SATS Score shows that the higher the IQ of a Year 7 Girl, the higher the average SATS Score will be, and the statistic that supports this is the Spearmans Rank Correlation Coefficient as it is 0.668; and as 1 is the highest and -1 is the lowest; 0.668 is high as it is close to 1 compared to -1.
The Scatter Graph for Year 11 Boys of IQ vs. Average SATS Score shows that the higher the IQ of a Year 11 Boy, the higher the average SATS Score will be, and the evidence that supports this is the Spearmans Rank Correlation Coefficient, as it is 0.922; and as 1 is the highest and -1 is the lowest; 0.922 is high as it is close to 1.
The Scatter Graph for Year 11 Girls of IQ vs. Average SATS Score shows that the higher the IQ of a Year 11 Girl, the higher the average SATS Score will be, and the statistic that supports this is the Spearmans Rank Correlation Coefficient as it is 0.7384; and as 1 is the highest and -1 is the lowest; 0.7384 is high as it is close to 1.
Finally to see what happens to the frequencies as the IQs and Average SATS Scores go up, I needed to draw a cumulative frequency diagram.
The Cumulative Frequency Diagrams showed us that as the IQs and Average SATS Scores go up, the Frequencies go up, but stop at the end. This was the same in both IQ Cumulative Frequency Diagram and Average SATS Score Cumulative Frequency Diagram. So as the IQ increases, it showed us that the higher the IQ, the better you’ll do in your SATS; or the higher your Average SATS Score, the better your IQ will be.
This means that my second hypothesis was correct, as the scatter graphs and cumulative frequency diagrams provided information to explain the reason for it to be correct.
Evaluation
My strategy was affective as I planned what I was going to do, and I explained how I was going to do that. I got a sample from a population by using stratified sampling and put the samples in different tables. I have hand drawn one scatter graph, cumulative frequency diagram, box plot, and histogram, so that the reader understands that I know the method for my calculations. I have also done one standard deviation and Spearmans rank correlation coefficient calculation, for the same particular reason. Then I did the rest of my calculations and diagrams on Autograph, and I made a summary of each result.
Finally that had got my answers, and I could see whether my original hypotheses were right. Unfortunately one of them was wrong.
The thing I could have changed is to get both of my hypotheses right and to get my data from a real high school instead of a fictitious, however, at least the pieces data collected, were from real people which was important thing.
The things that I am proud of are that I used the correct sampling method, so that I wasn’t biased, and the sample sizes were not too small, as they were to proportion of the population.