I am going to calculate the standard deviation of each stratums height. The formula for calculating the standard deviation is:
Standard deviation = √ {(∑fx2/∑f) – Mean2}
First, I will calculate the boys’ height deviations.
Mean= ∑fx/∑f = 23.625÷15 = 1.575
Variance = (∑fx2/∑f) – Mean2 = 37.299÷15 – 2.481=0.006
Standard deviation = √ Variance = 0.080
Mean= ∑fx/∑f = 23.375÷15 =1.558
Variance = (∑fx2/∑f) – Mean2 = 36.639÷15 - 2.428 = 0.0146
Standard deviation = √ Variance = 0.121
Mean= ∑fx/∑f = 19.8 ÷ 12 = 1.65
Variance = (∑fx2/∑f) – Mean2 = 32.748 ÷ 12 - 2.723 = 0.006
Standard deviation = √ Variance = 0.080
Mean= ∑fx/∑f = 19.325 ÷ 11 = 1.757
Variance = (∑fx2/∑f) – Mean2 = 34.002 ÷ 11 - 3.086 = 0.005
Standard deviation = √ Variance = 0.070
Mean= ∑fx/∑f = 13.4 ÷ 8 = 1.675
Variance = (∑fx2/∑f) – Mean2 = 22.51 ÷ 8 – 2.806 = 0.008
Standard deviation = √ Variance = 0.089
I will now calculate the standard deviation of the girls’ heights.
Mean= ∑fx/∑f = 20.175 ÷ 13 = 1.552
Variance = (∑fx2/∑f) – Mean2 =31.383 ÷ 13 - 2.408 = 0.006
Standard deviation = √ Variance = 0.077
Mean= ∑fx/∑f = 20.725 ÷ 13 = 1.594
Variance = (∑fx2/∑f) – Mean2 = 33.168 ÷ 13 - 2.542 = 0.009
Standard deviation = √ Variance = 0.095
Mean= ∑fx/∑f = 22 ÷ 14 = 1.572
Variance = (∑fx2/∑f) – Mean2 = 34.649 ÷ 14 - 2.469 = 0.006
Standard deviation = √ Variance = 0.077
Mean= ∑fx/∑f = 15.95 ÷ 10 = 1.595
Variance = (∑fx2/∑f) – Mean2 = 25.496 ÷ 10 - 2.544 = 0.006
Standard deviation = √ Variance = 0.077
Mean= ∑fx/∑f = 13.3 ÷ 8 =1.662
Variance = (∑fx2/∑f) – Mean2 = 22.18 ÷ 8 - 2.764 =0.009
Standard deviation = √ Variance = 0.095
This table shows the standard deviations of all the stratums.
All the stratum show their data within two standard deviations of their mean apart from Yr8 Boys and Girls which have their data within three standard deviations of their mean. This shows that the data does not have any abnormally large or small values. This suggests the data is accurate.
I have plotted a bar chart of the standard deviations for ease of interpretation.
Analysis of aim 1 results:
My aim 1 was:
- To investigate the height change between year groups
To do this I have grouped the data, tallied the frequency and plotted cumulative frequency graphs with box and whisker plots. The box and whisker plots where then draw on to two sheets of paper one for boys and one for girls. I have done this as boys grow at different times to girls. I also took the standard deviation of the values and plotted each stratums deviation on to a bar chart.
Girls:
I can see clearly from my girls’ box and whisker plots that they have grown steadily from Year 7 to Year 11. All though the ranges do not necessarily reflect this observation, I can see that the inter-quartile ranges are slowly creeping up, with the age. There is one exception in between Year 8 girls and Year 9 girls however; I think this can be accounted for as they are close, Year 8 shows a large range of heights, (three standard deviations from the mean) and this may have caused the discrepancy in the data. (Outliers may have affect the median point, upper and lower quartile points on which the box and whisker is based.) Or it could simply be that Year 8 is exceptionally tall whilst Year 9 small for their age. I think the former explanation is the more plausible of the two having studied the charts and the standard deviations.
In the Year 7 girls, the middle 50% - inter-quartile range, is far larger than that of the Year 10 or Year 11 girls. Year 7 has some largely differing heights within the middle 50% where as in Year 10 and 11 the heights of the middle 50% of the girls has come a lot closer. I think this is probably because the girls in Year 7 are split generally into those who have come into puberty and those who have not. Some of them will have had growth spurts where as others will have that to come. This means that the inter-quartile range will be further spread. However, when analysing the Year 10 and 11 charts the inter-quartile range is far smaller. I believe the reason for this is that the Year 10 and 11 girls have more or less, all been through the puberty growth spurts. There is still some differing individual difference nevertheless; they have largely achieved a roughly similar height.
The largest growth spurt on the charts seems to be between Year 7 and Year 8 for the girls. However, I bring this result into question as the Year 8 girls have such a large range of heights compared to the other year groups, and this may have some affect on the inter-quartile range.
Boys:
The boys’ box and whisker plot jump a lot more than the girls however, this is to be expected as boys are generally taller than girls by the end of puberty.
Boys do show a general increase in height from Year 7 to 11 however, the Year 11’s do appear to be shorter than the Year 10’s. I think that the data shows no reason for this and it may just be that the Year 11’s are small for their age and the Year 10’s tall.
The range of the Year 8 boys is large compared to the other stratum (data in 3 standard deviations instead of 2). They also have a wide spread inter-quartile range (middle 50%). This suggests to me that like the girls in Year 7, the boys in Year 8 are split generally into those who have come into puberty and those who have not. Some of them will have had growth spurts where as others will have that to come. This explains why the inter-quartile range is further spread. This wide spread has occurred later in the boys (Year 8) than the girls (Year 7) This follows most scientific theories that state boys puberty developments happen at an older age to that of girls.
The boys’ biggest growth spur seems to be between Year 9 and Year 10. The spur is so great that the two inter-quartile ranges do not over lap at all. But, I think this may have something to do with the Year 10’s exceptionally tall factor as the Year 11’s appear to be shorter than them.
Conclusion:
In answer to my aim, I think that I have sufficiently proved that there is indeed a relationship between age and height at this age.
Overall, the girls’ biggest growth spur seems to be between Year 7 and Year 8 where as the boys’ biggest growth spurt seems to be between Year 9 and Year 10. These results also seem to correlate with the scientific theory of later male development however to make a solid conclusion of the issue I would need stronger results.
To achieve this I would use a larger sample of data to gather evidence that is more conclusive. It might also have been more beneficial to have used a different sampling technique, as this technique gave me different numbers of students from each stratum. This meant I only had 8 Year 11 girls whilst far more Year 8 girls, which may have been the cause of the increased range of their data. A systematic sample might have been better. (E.g. Every third person.)
Results for Aim 2:
From these tables I have plotted this graph in Microsoft Excel Application.
The graph clearly shows there is no correlation between the two variables. It seems there is one outlier in the Yr7G stratum. I have not plotted a line of best fit, as there is no correlation.
It is possible that this outcome is due to a fault in the random sample; it maybe that I have selected high ability Yr7’s and low ability Yr11’s. I think this is unlike however, to test it I will compare the Yr11 KS2 results with that of the Yr7’s, as these test where sat at the same age. If the ability of the Yr7’s is similar to that of the Yr11’s the results should be in roughly the same place. I have added each child’s KS2 results together to get one value.
From this graph I can see that the students of a similar IQ where close to the same KS2 results. There is a definite positive correlation. As these tests, where taken at the same age it is fair to say that the random sample has represented each Year fairly.
I am going to put a line of best fit on the charts.
The mean co-ordinates are (1dp):
Yr7B: (100.8, 12.4)
Yr7G: (98.1, 12.7)
Yr11B: (105.6, 12.9)
Yr11G: (100.8, 11.9)
Analysis of aim 2 results:
My aim 2 was:
- To observe correlation between IQ and age
To do this I have calculated each student’s age as a decimal created from the months and years of their age. I have placed these ages in a table with the students IQ. From these tables I have plotted a scatter diagram in Microsoft Excel Application using the entire sample population. The graph shows IQ against age. As this graph has no correlation at all, I have used the Year 7 and Year 11 KS2 Results to assess the intelligence of the strata. I created a single value for the KS2 Result by adding the 3 numbers together. I then plotted this summed value against the IQ of the students. One this graph I plotted lines of best fit from a mean point of each stratum. (So one for Year 7 girls, one for Year 7 boys, one for Year 11 girls and one for Year 11 boys.)
As I previously stated it is possible that this outcome is due to a fault in the random sample; it maybe that I have selected high ability Year 7’s and low ability Year 11’s. So to eliminate this possibility I plotted the students KS2 Results (because they where sat at same age.) against their IQ to see if they appear in the same general area. It turns out the variables correlated and where in the same general area, abolishing the idea that the Year 7’s where of a higher ability than the Year 11’s.
This leads me to believe either that there is not a relationship between IQ and age, although another reason for the lack of correlation between IQ and age maybe that the IQ tests where taken at the same age. If this was the case, then obviously the chart would show no correlation.
Conclusion:
I answer to my aim I have observed the lack of correlation between these two variables however, the evidence would need more support to draw a firm conclusion.
If there is no relationship between IQ and age at this point that maybe because the children have already developed the IQ section of the brain before this age so they will not change after that point.
However to draw I firm conclusion on this I would have to repeat the study checking first, that the IQ tests are current. To further, the investigation I would compare the results of younger children to see if that had an effect on the correlation.
Results for aim 3:
My spearman’s rank correlation is 0.05, a positive number however very close to 0 showing that the correlation between the two variables is weak to none. I think this maybe because my sample is not large enough. So I will take a new sample using every third person within the total population.
This correlation is very positive correlation showing the children’s choices where much the same.
This correlation is very positive correlation showing the children’s choices where much the same.
This correlation is again positive correlation showing the children’s choices where much the same.
I have drawn some pie charts to further illustrate the changes in the children’s preferences.
Analysis of aim 3 results:
My aim 3 was:
- To investigate changes in favourite TV genre choice
To do this I placed the student’s favourite TV show choices into different genres. From this, I found the spearman’s rank correlation between the Year 7 girls and the Year 11 girls and the Year 7 boys and the Year 11 boys. I also plotted some pie charts using Microsoft Excel Application to show the percentage of student’s choices. I did this for every stratum group.
The first spearman’s rank I calculated was between the boys. This showed my hat there was little to no correlation. From this, I decided to re sample the Year 7 boys’ and the Year 11 boys’ data as the other sample was simply too small to show anything of fairness, I used systematic sample-taking every third student. After this, the larger sample showed a strong positive correlation. I repeated this process for the girls although, I did find that the smaller sample showed an equally strong positive correlation I took the re-sample just to be sure. I think that the girls may have shown more of a correlation – even in the small sample because girls tend to develop before boys and they may have developed these TV preferences before Year 7 age group where as the boys are later with the changes which show up in my sample age.
This is re-iterated by the fact that the Year 11 boys data it is far more spread than the Year 7 boys data. By looking at the frequency charts you can see that the Year 11 boy’s have branched out their choices, they have less people watching cartoons and soaps and more people watching educational programs and sports. This shows that the Year 11 boy’s have a wider range of interests and are more open to new things, this makes them more mature than they where in Year 7. The pie charts show this is true of both sexes. Although they chose the same top programmes for each age group, they have branched out their choices.
Conclusion:
In answer to my aim, I think the student’s favourite choices have not differed much nonetheless; the students have become more diverse suggesting a more mature outlook.
To further my investigation I would like to test the TV preferences of younger children, in particular girls to see if there is much change in favourite programme. I think this would be useful because the girls may have made preference changes at a younger age.
Overall conclusion:
My main hypothesis was:
Children between Year 7-11 change physically and mentally as they mature.
I think I have proved this statement correct. The children clearly change heights (physical) as they grow older. They don’t appear to change their favourite TV programme choice however; they do appear to diversify slightly with maturity. The data shows no relationship between IQ and age, suggesting that the children do not get brighter with age, but this maybe questionable as the data does not say when the IQ tests are sat.
Evaluation:
I think my investigation has been successful. I have answered my hypothesis in fair depth and sighted future tests that could be of interest to this topic.
In my first aim, it may have been better to use a larger sample of data to gather evidence, as the sample was very small in places. It might also have been more beneficial to have used a different sampling technique, as this technique gave me different numbers of students from each stratum. This meant I only had 8 Year 11 girls whilst far more Year 8 girls, which may have been the cause of the increased range of their data. A systematic sample might have been better. (E.g. every third person.)
In my second aim, I considered using a weighted mean to calculate the KS2 overall number however decided against it as it would be unfair to place more importance to different subjects.
In my third aim, I had to resort to re-sampling the data totally, as the results showed nothing due to lack of information.
If I was to further my investigation I would probably widen my age range as this would be beneficial to all areas of my study.
- Heights (show spurts particularly girls)
- IQ (see if this part of brain develops earlier)
- TV programme preference (see if the change occurs at an earlier date.)