To look at IQ and SATS results, I will construct a simple scatter graph to show instantly any correlation or link between the two. I will then construct a cumulative frequency graph for IQ and I will be able to see if IQ varies or the distribution in the sample is close to the mean. With these results, I will be able to infer a similar relationship between IQ and SATS results, depending on the correlation shown in the first scatter graph. The reason I do this, is because I cannot directly compare age and SATS results, as the data does not say when the SATS results were taken and I suspect all pupils took the exams at Year 6. This would mean that a direct comparison between age and results would be useless.
I will also compare box plots for IQ and SATS results to confirm or reject any relationship between them.
For the final hypotheses, I will use the quartiles established from the IQ cumulative frequency graph, which I will have already constructed. I will plot a histogram for the members of the population in the high quartile of IQ. I will put the average of their SATS results on the histogram and I will then do this for the members of the population with IQ within the IQR and for members with IQ in the low quartile. This should enable me to see how IQ affects individual results and if IQ can influence the career field that the particular member of the sample would excel in.
Collecting, Processing and Representing Data
I collected my sample as described above. It was a stratified sample. I used this method, as I believe that it makes my sample represent the whole population as accurately as possible. As a result of this, I hope, despite the sample’s small size, that I will be able to investigate my aims accurately.
My first hypothesis was to investigate the relationship between age and height. To do this, I decided to construct histograms for the heights of Year 7 pupils and the heights of Year 10 pupils. As the data is continuous, I decided to group the data and then draw the histogram. I also wanted to draw an additional frequency polygon on the histograms to make comparisons between the two relationships easier.
Using this technique, I will not only be able to see if age simply increases with height, but I will also be able to see the overall spread of the distribution. This will enable me to compare the heights of Year 7 and Year 10 pupils in more depth than an ordinary scatter diagram would.
The grouped frequency tables used to construct the histogram are shown below.
Grouped frequency table for the heights of Year 7 pupils
Grouped frequency table for the heights of Year 10 pupils
Once I had drawn the histograms, I was able to make comparisons between the heights of Year 7 and Year 10 pupils.
Firstly, the modal classes between the two distributions are different. For Year 7 it is1.45<x≤1.55, while for Year 10 it is 1.70<x≤1.80. When taking into account the fact that the two classes are separated by three years, I can say that as age increases so does height.
The spread of the two distributions is also significant. The Year 7 pupils’ height has a very slight spread, i.e. most of the values are close to the mean height. This means that there is not much variation in the heights of 12 year olds. However, in the Year 10 histogram and table, the spread is much larger, showing that there is much variation in the heights of 15 year olds. This is also significant, as I can also now say that as age increases, so does height and the variation of height between you and your peers.
Having discovered this, I decided to construct a cumulative frequency curve for the height of all pupils to see what the spread was like on the overall distribution of my sample. I could then also compare this with a cumulative frequency curve for weight, to see if the spread of values for height and weight is similar. To construct these two curves, it was necessary to find out the cumulative frequency for the heights and weights of the sample, and this was done by the use of tables.
Tables constructed to form a cumulative frequency curve for height
Tables constructed to form a cumulative frequency curve for weight
Once I had drawn the cumulative frequency curves, I worked out the median, Inter-quartile range (IQR), the modal class and the mean (from the grouped frequency table above) for both distributions (height and weight). All the working is shown above.
I performed all of these calculations so I could easily compare the two separate distributions of height and weight.
By looking at the shape of the curves, I can see instantly an important difference in the spread of the two distributions. The height graph has a much larger spread, indicating variation from the mean, whereas the weight graph has much less deviation from the mean. This means that the people in my sample are much closer related by weight than by height. This is indicated on the gradient of the lines. The weight curve has a much larger gradient throughout and is more ‘direct’ than the height graph. This suggests little standard deviation. However, the height graph has a much more wallowing gradient which indicates greater deviation from the mean and therefore a greater spread and range.
This relationship is also shown by the IQRs. The height graph has a large IQR and the weight graph a low IQR. To best illustrate this, I decided to plot a box and whisker diagram to show the differences in the IQR of both distributions.
The box plots are there so that I can easier compare the IQRs of the two distributions. Consistent with the cumulative frequency curves, it is evident that the IQR for the height of the pupils is larger and so the spread is greater. The opposite is true for weight, where pupils tend to have similar weights. This would mean that there is a slight relationship between height and weight. The reason I suggest this is because as people get taller, they are likely to get heavier. However, on the other hand, they may get taller and skinnier and their weight may not change. I was more interested in this relationship than that of simply age and weight, so I decided to investigate it instead.
In order to get a good idea of the relationship between height, weight and age I would have to combine two of the variables. I decided to combine height and weight, as they seem to fit together best. When combined they form a new unit of ‘density’ which is cm/Kg. I wrote a formula to this ‘height to weight ratio’ and it is shown below.
Having found the cm/Kg value for all members of the sample, I had to find a way to compare this new unit to age. I did not want to do a scatter graph against age as I had done previous scatter graphs for age (years and months) with all 60 pupils, and due to the large number of points on the graph correlations became difficult to see and the graph just appeared cluttered.
Therefore, I decided to try and plot a line graph with a smooth line to illustrate any relationship (not correlation) between age and cm/Kg. I also decided to use school year group as a measure of age rather than years and months. I felt this would simplify the graph and make it easier to see a relationship between age, weight and height. To do this, I found the mean value for the cm/Kg of each year group, as shown in this table. This would give me an idea of how cm/Kg changed as people went through their secondary school years.
From this table, I can see that generally, people get denser as they get older. I can tell this because they have less kilos of bodyweight per centimetre they are tall. A decreasing number of cm per kilo, means that people get heavier. This is shown by the table, as in Year 7, the average pupil has 3.35 cm/Kg, but in Year 11 they only have 2.9cm/Kg. When looking at figures like this in a table it is more difficult than necessary to see overall trends, so I decided to use ICT to put the above table into a graph.
From the graph, it is easier seen that cm/Kg decreases as age increases. This means that as the average pupil gets older he/she has less cm/Kg. This would fit with pupils growing up as they tend to fill out ads they get older, but not necessarily get any taller. This would account for the shape of the graph.
The rise in the cm/Kg value for the Year 8 children is particularly interesting. This would indicate that the average pupil has a rise in cm/Kg. This would very likely mean that they get taller but not necessarily heavier. This is probably the average child having a growth spurt and it is not until year 9 that their weight can catch up with their height. This time span would obviously vary from child to child with the suddenness and severity of the growth spurt.
Therefore, so far, I have established that physical factors seem to increase with age, but what about mental capacity. Does this also increase through the secondary school years?
The next hypothesis to investigate was if age increases with IQ. I was particularly interested in this as I was wondering how exams could increase in difficulty, learning potential and ability to understand more complex material without an increase in IQ. However, I must remember what IQ is. It is not how clever somebody is, but a measure of intelligence and there is a subtle difference between the two.
To see if there was a link between age and IQ, I used ICT to plot a scatter graph and I observed of any correlation was present. To plot age, I used a simple formula to convert all the members of the populations ages into months. Where y is the number of years old they are and x is how many months into that year they are, to calculate their age in months I used the formula: Age (months) = 12y + x. I then plotted this against IQ.
I used a scatter graph as it shows correlation instantly when a line of best fit is drawn. However, I cannot draw a line of best fit on this particular scatter graph, as there is no correlation
Having drawn this graph, I can tell that there is no correlation between IQ. I would just like to confirm this by using numerical averages in a simple table.
This table confirms to me that there is no correlation between age and IQ. The averages are all over the place. This table shows that particular year groups are more intelligent than others. It shows that Year 8 is a particularly intelligent year, whereas Year 11 is not.
As a result of these calculations, I can say that IQ is not affected by age. But what is IQ in the real world? Who cares about a number that is on a piece of paper? What are the benefits of a high IQ? Does a low IQ hinder you at all in the real world? One way to investigate the importance of IQ is to see if IQ affects exam results. I would like to see if actually affects your SATS results. Does intelligence actually have an effect on performance?
I predict that I will have a strong positive correlation as I think that a person will have a high IQ to be able to understand, interpret and deploy the material necessary to achieve a good exam result. Due to exam technique, I feel that there will be people with a high IQ, but poor exam technique so their marks will be lower, and at the same time, people with lower IQ could have better marks due to excellent exam technique.
I used combined SATS results, as I would get a true picture of the results and I would avoid influence of extreme or freak data.
As the last scatter graph was a success, i.e. it gave a definitive judgement on the correlation between two variables, I have decided to use another. I use the scatter graph to give me a graphical assessment of the, if any, correlation between IQ and SATS results.
N.B. For this graph, I have drawn the best-fit line through the co-ordinates determined by the mean of the x-axis and the y-axis. This will be the same for all scatter graphs that
I draw.
Having drawn this scatter graph, it is clear that there is a very strong positive correlation. This means that as IQ increases, so does your combined grade. This clearly shows the value of IQ. If you have a high IQ, you are very likely to do better in exams. I think it would also be interesting to determine how well you do if you have any IQ value.
To do this, I would have to find the equation of my best fit line. I cannot do this on the graph I showed the initial correlation on because the axis do not start at zero and the y-intercept will not be on the scale. Therefore, I made a graph with more appropriate axis using ICT.
I will find the equation of the best fit line. This equation should be the same as the equation linking IQ and SATS results. To find the equation of the line I will use the equation y = mx + c, where m is the gradient and c is the y-intercept.
Using the graph with appropriate scales, I can withdraw the following information.
y-intercept ≈ -15
Gradient ≈ 0.27
Using the above information, I am able to find the equation for the line of best fit and so the relationship between IQ and SATS results.
y = mx + c
y = 0.27x + (-15)
The equation in bold is the equation of the line. It can be tested with two known values. I will use the averages of IQ and SATS results as my testing co-ordinates (100,12). I will use the IQ of 100 to try and estimate the combined SATS level using my formula.
y = mx + c
y = (100 x 0.27) + (-15)
y = 27 + (-15)
y = 12
This estimation is correct as it has produced a y value which when plotted against the x value on my graph, lines up on the line of best fit exactly. This means I have found the correct formula for predicting combined SATS grades, based on IQ. It is shown below, along with the re-arranged formula to find an estimation of IQ based on combined SATS results.
(Where x = IQ and y = Combined SATS results)
y = 0.27x + (-15)
-and/or-
x =_______
Therefore, I have discovered that IQ and combined SATS results are closely related, and can be found by the formula, Combined SATS results = 0.27IQ + (-15).
I also wanted to look at a cumulative frequency curve for IQ. This would show me the range, IQR and spread of the distribution and give me the idea of the deviation. I had to construct the following table to make the cumulative frequency curve.
I could then construct this cumulative frequency curve.
The graph shows a moderate distribution, meaning that there is a significant range of readings. This is reflected in the IQR, which is quite large. Due to the very close correlation between IQ and SATS results, I am able to compare the two distributions on this one graph. As the correlation is strong on the scatter graph, I know that the cumulative frequency curves would be very similar. This means that the shape of the two distributions will be very similar. Therefore, the spread of the distributions of IQ and SATS results are both normal, with moderate IQRs and modal classes with frequencies not too much larger than most other groups.
From the graph, I have also now found the quartiles of IQ throughout my sample and this will be very useful when I progress to test my last hypothesis.
While looking at IQ, I thought it might be interesting to look at which of the sexes was more intelligent, and, based on the findings when investigating the last hypothesis, which sex would do better in exams.
To look at this, I decided to use a stem-and-leaf diagram. This is good as it will show the overall shape of the distribution and, like a bar chart, will let me see roughly how many of each sex have a certain IQ. However, the reason I use this diagram and not a bar chart is because I want to directly compare two distributions, and so I chose a stem-and-leaf diagram.
The stem-and-leaf diagram shows that the boys and girls IQs that I tested were very similar distributions. However, I think that the boys had marginally better IQs, as they had nothing in the 70s, and four in the 110s. Elsewhere, the boys and girls were closely matched. It is for this reason that I think the boys have marginally better IQs, but it is so close that in another sample of 60, the results might be very different. In the particular diagram the boys seemed to have a higher IQ, but there were more of them and it might happen to be a sample in which the boys have high IQs.
I have established that IQ is closely related to SATS results. This is quite a basic comparison. However, based on this, I would like to expand the project in more detail and see if I can investigate something that has some relation to the real world. I would like to investigate, for my final hypothesis to see if a high IQ or low IQ can give strengths in certain subjects and ultimately guide you towards a certain field of employment. If someone was good at maths, they would be likely to be able to pursue a mathematics involved career such as an accountant or a statistician.
On thought on how to investigate this, I decided to use the quartiles of IQ established when drawing the cumulative frequency curve which divide up the population into three groups based on their IQ. These groups are <Q1, IQR and >Q3. I will plot a bar chart for each of the members of each quartile, of IQ against average SATS results for each individual subject. I will do three bar charts to compare the average individual results of each quartile for each subject against IQ. This will enable me to see and compare what subjects people with differing IQs are best at.
I will also use frequency polygons on each bar chart to compare each distribution. I am using bar charts rather than the preferred histograms, as the data I am dealing with is not frequency orientated so is not suitable for a histogram. The polygons will also enable me to compare the distributions easier.
The following table was used to help me plot the histogram.
The histogram shows that the better your IQ, the better you do in all subjects with nobody from a lower quartile having a better performance in their best subject than that of the worst subject of the person in the quartile above them
Also clear is that as your IQ increases, you tend to become better at maths and science, rather than English. People with a low IQ are best at science, probably because science is a factual subject and mostly requires learning rather than intelligence. It is clear, that as you get a higher IQ, your best subject becomes maths, then science and then English. Therefore, people with a higher or medium IQ are best suited to technical subjects like maths, the more technical the better. People with an IQ in the lower quartile are not nearly as good at the subject as people with an IQ higher than theirs. Although they do not have stronger or weaker subjects trend would suggest than in more complex exams, like GCSEs rather than SATS they would do best in English.
Interpreting and discussing findings
Conclusions and how diagrams support conclusions
From the histograms and tables I constructed, I was able to say that as height increases with age. I also concluded that the spread or distribution of Year 7 pupils was much greater than that of Year 11 pupils. I can therefore also say that as you get older, you get taller and the variation in height between you and your peers also increases.
When I constructed my cumulative frequency curves for height and weight I was able to make valid comparisons. The distribution for height has a much larger spread and range than weight indicated by the loose shape of the cumulative frequency curve. The box plots also confirm this, illustrating a larger IQR for height than weight.
From my table and graph concerned with cm/Kg, I was able to state that cm/Kg decreases with age, i.e. as you get older, you get denser. This is almost certainly due to the fact that you fill out and become generally physically denser as you progress through Year 7 to 11. At Year 8, there is a rise in the cm/Kg value shown by the rise in the graph. This means that you have a growth spurt at Year 8 but do not necessarily get heavier. This would explain the rise in the graph.
The scatter graph of age and IQ shows me there is no correlation between these two variables.
My calculations and graphs relating to SATS results and IQ have helped me to draw firm conclusions relating these two variables. I can say that there is a strong positive correlation between IQ and SATS results showing that as IQ increases, so does your combined exam result. I am also able to say that the two variables are related in the equation y = 0.27x + (-15)
My cumulative frequency curve for IQ helped me to state that the distribution of IQ is a normal distribution. Due to the close relationship between IQ and SATS results, I can also say that the distribution of SATS results, which would be difficult to draw a graph for, is very similar to that of IQ i.e. both distributions for IQ and combined SATS results are normal distributions.
My ordered stem-and-leaf diagram helped me to state that the IQ’s of girls and boys are very similar and with the sample I have I cannot say which sex is more intelligent as the results are too close and would probably vary with another sample.
The final bar chart helped me to state that the better your IQ, the better you do in individual SATS subjects and you are very likely to better any score that the person with an IQ in a lower quartile than you achieved.
Also, as your IQ increases, you become better at more technical subjects – maths and science. You also get better at English, but your best subjects become maths and science. Therefore, I can say that people with a high IQ tend to have their strongest subjects as maths and people with a lower IQ tend not to have best subjects as very even scores are achieved in all subjects.
Summarised Findings
To conclude, I have been able to state the following:
- Height increases with age
- There is a greater variation in heights of Year 10 pupils than Year 7 pupils
- Pupils are often closer related by weight rather than height as the spread of distribution in terms of height tends to be large
- Year 11 pupils have a smaller cm/Kg value that Year 7 pupils showing that pupils get denser and fill out after Year 8.
- Year 8 seems to be the year in which a sudden growth spurt occurs.
- There is no correlation between age and IQ
- IQ and combined SATS results are closely related within the equation y = 0.27x + (-15) and the distributions between SATS results and IQ are similar
- Intelligence does not seem to be affected too much by sex
- The higher your IQ the better you are all subjects especially technical subjects like maths
- The lower your IQ the worse you are at all subjects and you are much less likely to have a significantly stronger or weaker subject
Comment on conclusion relating to hypothesis
Of the hypothesis I started with, all of them agree with their representative findings, except that relating age and IQ. The hypothesis was that IQ was affected by age but this was proved to be incorrect, as IQ was not affected by age.
Limitations in findings
There are not too many limitations in the findings, apart from the intelligence between the sexes investigation where I am unsure on my findings due to my limited sample size and lack of any obvious conclusion. It is due to the subtle differences in IQ between the sexes and the size of the sample, which make me feel that there is a limitation to this conclusion/finding.
One other small point is that of the use of ICT to draw scatter graphs. Although the point plotting when using a computer is very accurate, when there are two identical points to be plotted, the computer ignores the repeated one. This accounts for there not being 60 points on one of my graphs. However, it is not too big a problem, as the shape of the distribution remains unchanged.
Limitations and reliability of conclusions
There were some limitations to my sample and findings. My sample, I think, was too small. The best evidence for this is my uncertainly about my IQ findings with regard to gender. I am extremely unsure over whether the results are due to my sample being too small or if the results are actually correct. To improve this, I would like to take a much larger sample size. Ideally I would like to have a sample of a couple of hundred pupils to analyse. Also the sample was possibly too stratified which meant inconsistencies with the addition within the sample and multiplication to find the same value. An example was the addition of Year 7 pupils and the working out of all Year 7 pupils, which differed by one.
The data itself was very unreliable in places. There was quite a lot of missing and freak data, which I replaced with the mean value. However, I am unsure of the accuracy of the data. For instance, I do not know my exact IQ or my exact height or weight. If the data was collected by a questionnaire I think members of the population may have just guessed their IQ or exact height, which may affect my conclusions. However, I think, due to the accuracy in the readings (most have been rounded to 2.d.p. which is quite detailed) that these problems were accounted for and that the data that is given is quite accurate.
Despite these potential problems I think that my conclusions are reliable because they make sense if you think about them. It is logical that height increases with age and that you get better exam results if you have a higher IQ. I also think that I have performed the statistical techniques to a degree of accuracy necessary to form valid and reliable conclusions.
I think that some of the conclusions that I have discovered are significant to the real world. For example the findings about height variation increasing with age would be very significant to a school shop ordering trousers or jackets. Based on my fin dings they would need to order a greater range of sizes for Year 11 than they would for Year 7.
To extend this project, I would like to focus on my last conclusion. I would like more detailed data than just SATS levels. I would like to have data on employment figures for schoolchildren for deeper analysis into this last idea. E.g. individual percentage marks for GCSE results, ideal career choice, eventual career, OASIS test results etc.
I think that the project went well, but next time I would reduce the amount of work I had to do as I spent too much time working on the project. To do this, I would focus on just one or two hypothesis in the next project. Despite this, I think that the project went smoothly and I was able to make inferences linking back to every hypothesis, which is pleasing. I also think that the conclusions that I made are accurate and reliable and are significant to the real world. I think that the project was a success, but next time I would have fewer hypotheses. This would enable me to go into further detail to form more detailed conclusions. I would also take a larger sample to make the conclusions more reliable. Overall the investigation was a success because I was able to make valid and reliable conclusions and come to a judgement on every one of my hypotheses.
