Maths Coursework - Mayfield School

George Smart (Candidate: 1626) Abbs Cross School (12815) Page 1 of 19 Mathematics Coursework (Mayfield School)

Introduction For this investigation, I will be using some fictitious data from Mayfield High School’s student database. Although the school is fictitious, the data is based on real pupils. This database contains information on all the students in the school and was provided by the school, so it is classed as secondary data – this is because the data was given to me, and I did not collect it myself. I will stratospherically sample 120 pupils from the original 1183 entries in the database – the data must be stratospherically sampled because there are 5 separate year groups (years 7, 8, 9, 10 and 11), as well as the year groups being spilt into male and female. Through the use of tables, charts and graphs, I will look for correlations to help validate my hypotheses.

Hypothesis Throughout this coursework investigation, I aim to find out whether my hypotheses are correct or incorrect. I hypothesise that:

• The taller the pupil, the heavier. • Older pupils are taller and heavier than younger pupils. • Males, on average, weigh more than females.

This is true in real life as when you see males and females together, usually the male is taller. If there is relationship between height and weight, males should prove to be heavier than females.

Aim Although my hypotheses are true in real life, I hope to prove them true or false using mathematics. I will be using the methods below to prove my hypotheses.

• Averages • Scatter Diagrams • Cumulative Frequency Diagrams • Box and Whisker Diagrams • Histograms • Standard Deviation

Calculations To keep the results a true representation of what the school’s data is like, I need to randomly and stratospherically sample the database. In order to achieve this, I will use Stratified Random Sampling. This


means that I need to work out how many pupils of each gender I need to take from each year group, in order for it to be a true representation of the original data, and to keep all the proportions correct. This means that the data I have sampled MUST have the same proportions of each year group and gender as the original database had. If it does not, it will seriously affect the results.

Calculating the Sample First of all, I needed to know the total number of pupils in the database. This was done by using the =count function in Microsoft Excel – this counts all the entries in the spreadsheet. This was done to check that there were the correct number of entries in the database, i.e. that the number of pupils in the spreadsheet matched that of the amount the school had told us there were. The total number of pupils in the spreadsheet was 1183, which matched exactly with the information supplied by the school. I then took the number of girls in year 7 (131) and divided it by the total number of pupils in the school (1183). The answer calculated to be 0.11073541842772612003381234150465 (or 0.111 when rounded to 3 decimal places). This answer was then multiplied by the total sample (120) to give us how many year 7 girls needed to be sampled. This calculated to be 13.288250211327134404057480980558 (or 13.288 when rounded to 3 decimal places). As we cannot have .288 of a pupil, I rounded up the number again to the closest integer (whole number). This was 13, and so tells us that we need to randomly select 13 year 7 girls from the Mayfield School Database. The following formula can be used to calculate the remaining answers. Sample = (No of Pupils from a year group and gender ÷ 1183) × 120 This method was then repeated for all genders and year groups, until I had calculated how many pupils to sample from each gender for year groups 7, 8, 9, 10 and 11. The results are shown in Table 1 (below).

Table 1 Boys Boys Girls Girls Total Year

Group Boys

Sample Rounded Girls

Sample Rounded Rounded

7 151 15.3169907 15 131 13.2882502 13 28

8 145 14.7083686 15 125 12.6796281 13 28

9 118 11.9695689 12 143 14.5054945 15 27

10 106 10.7523246 11 94 9.5350803 10 21

11 84 8.5207101 9 86 8.7235841 9 18

Totals 62 60 122


Table 1 shows how many boys and girls need to be sampled from each year group. As you can see from the table, the total of the sample data totals to 122 pupils – this is a result of rounding. Since the results have been rounded up, I have gained another 2 people. I have chosen to ignore this rather than remove the two people, as it makes the sampled data more representative of original data. The final results are shown in table 2 (below).

Table 2 Boys Girls Total Year

Group Rounded Rounded Rounded

7 15 13 28

8 15 13 28

9 12 15 27

10 11 10 21

11 9 9 18

Totals 62 60 122

Now that I have calculated the sampling data, I can then go onto randomly sampling the original database. This is described in the Data Sampling section below.

Data Sampling – Getting the Data Firstly, I created a new sheet in Microsoft Excel. To do this, I needed to go to the bottom left of Excel, and where the list of worksheets are located, right click on one of the tabs, and click Insert… This is shown in figure 1 below.

Figure 1

Then Click Here

Right-Click here

When the Insert box comes up (figure 2 - below) press OK as this will default to inserting a Worksheet. I then got a new worksheet.


Figure 2

Now Click Here

Finally, I renamed the new worksheet “KS3+KS4” from its default name (sheetx). To do this, I right clicked on the new tab, and select rename from the list. This is shown in figure 3 below.

Figure 3

Click Here to Rename Rename to “KS3+KS4” Right-Click

here

The next step involves moving data around in Microsoft Excel. I first opened the Microsoft Excel (xls) file containing the data from Mayfield School. I then selected the KS3 tab from the bottom left of the screen. I needed to then copy all of the data from that table, including the column headings (i.e. Excel Lines Line 1 to Line 815). I then clicked Edit > Copy in Excel. I then switched to the worksheet I just made, and pasted the copied information into the new sheet (note: use copy, not cut – using cut will remove the data from the KS3 worksheet). Then I switched to the KS4 worksheet, and copied all the data from the KS4 worksheet, but this time, do not copy the column headings.


Next I sorted the data on the KS3+KS4 worksheet. I sorted the data by year group, gender, and surname. I did this by switching again into Microsoft Excel, and in the very top left of the worksheet (where the letters row meets the numbers column) there is a blank square (shown in figure 4 – below). Click this, and it will highlight all the data.

Figure 4

Click here once

Clicking on the square as shown in figure 4 will highlight all the data on the current worksheet. Then I clicked on Data > Sort, and a box like the one shown in figure 5 came up. Firstly I selected to sort by Year Group, then second sort was Gender, and the last sort was Surname. This is also shown in figure 5.

Figure 5

Pressing OK to this dialog box will sort all of the data into year groups, genders, and surname. I then renumbered column A, so that all of the pupils from each year group and gender, for example Year7 Girls should be numbered from 1 to 131. Year7 Boys should be numbered from 1 to 151. They need to be consecutive, for example … 130, 131, 1, 2 … so there are no gaps, and no spare lines. Next I put my scientific calculator (Casio (Model: fx-85wa) S-V.P.A.M) into Fix mode. This tells the calculator to display only the digits before the decimal place. I did this by Pressing Mode 3 times, until the screen displayed “Fix(1) Sci(2) Norm(3)”. I then pressed the 1 key, to select Fix


mode. The screen then displayed “Fix: 0~9?”. I pressed the 0 key to tell the calculator to display no decimal places, and the calculator goes back to ready. I typed in the last number for a year7 girls, which is 131. This was followed by a Rand# command which generates a random number between 0 and the value you type in (in this case 131). Then with the KS3+KS4 Excel Worksheet on my computer screen, I pressed the equals button on my calculator. The screen displayed 46. I then scrolled through the worksheet, until I found the 46th year7 girl. Using the Fill Tool ( ), I filled the row yellow. I pressed the equals button once again on my calculator, and the next random number was 102. I then scrolled to the 102nd year7 girl, and highlighted that yellow with the fill tool. I repeated this until I had highlighted yellow 13 year7 girls, which was the number required by the sample. This was then repeated for year7 boys, and again for year 8, 9, 10, and 11, boys and girls. Once I had highlighted all the boys and girls from each year group, I deleted all the un-highlighted entries. I was left with 122 entries (as noted earlier).

Preliminary Testing I decided to conduct a preliminary test to establish if there were any trends or relationships between weight and height, and to help determine the values of data. To obtain the data for the preliminary test, I used the first and last record for each year group and gender, so that I had 2 year7 girls, 2 year7 boys, and so on. Overall I ended up with 20 entities. With these records, I used Microsoft Excel’s chart wizard to draw a scatter graph. Graph 1 shows the correlation between the weight and height of pupils, from years 7 to 11. Female data is in Pink, Male data is in Blue.

Graph 1

Preliminary Testing - Weight to Height Correlations

1.301.401.501.601.701.801.90

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Weight of Pupil (kg)

Hei

ght o

f Pup

il (m

)

Famale Male Female Weight Male Weight

George Smart (Page 7 of 19

Male Gradient: y = 0.0053x + 1.3626 Female Gradient: y = 0.0128x + 0.9706

Candidate: 1626) Abbs Cross School (12815)

Mathematics Coursework (Mayfield School)

As you can see in Graph 1, there is a positive correlation between pupil weight and height - the more the pupil weighs, the taller they are. This is true for both males and females. I also carried out another test, to establish whether older pupils weighted more and were taller than younger pupils. Graph 2 shows the relationship between Age (year group) and Height (meters).

Graph 2

Relationship between Age and Height

1.301.401.501.601.701.801.90

6 7 8 9 10 11

Yeargroup

Hei

ght (

met

ers)

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Male Height Female Height Male Height Female Height


As you can see from Graph 2, the older the pupil (or the higher the year group) the taller the pupil. This is proven by the trend line gradients. The gradients are both positive (Male = +0.0225x, Female = +0.0405x).

Graph 3

Relationship between Age and Weight

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Yeargroup

Wei

ght (

kilo

gram

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Male Weights Female Weights Male Weights Female Weights

Male Gradient: y = 7.95x - 21.05 Female Gradient: y = 2.35x + 29.05


Graph 3 then shows that the older the pupil the more they weigh (on average). This is not a true representation, as the results have not been averaged. In the real test, I will use averages to make the results more representative of the whole year group instead of just various single pupils. However, these tests have given me some idea of the sorts of data that will be involved, and how to prove my hypothesis true or false. Table 3 below shows the mean heights and weights (from the preliminary data) for each year group. I am only interested in the average heights, but the average weights are included also. From the results you can see that they appear to follow a weak correlation, much weaker than I had thought. This could be due to how I sampled the data, and that there may not be enough people from each year group (2 pupils) to make the averages true averages. I will continue to try and prove my hypothesis true or false in the main testing section…

Table 3 Year 7 Year 8 Year 9 Year 10 Year 11 Averages Boys Girls Boys Girls Boys Girls Boys Girls Boys Girls

Height (m) 1.631 1.575 1.660 1.515 1.690 1.590 1.685 1.700 1.710 1.685Taller Males Males Males Females Males

Weight (kg) 32.00 42.50 44.50 50.50 54.00 51.00 56.00 55.00 66.00 52.00Heavier Females Females Males Males Males

When trying to find this out on the real sample, it will be more difficult, as there may not be an equal number of males to females. Therefore I may be averaging young males against older females, and the females will be heavier, or visa-versa.

Graph 4

Comparison in Height between Male and Female pupils

1.4001.4501.5001.5501.6001.6501.7001.750

7 8 9 10 11

Yeargroup

Ave

rage

Hei

ght (

met

ers)

Male

Female



Graph 4 shows the comparison between the average height of males and females. This is shown as the male trend line is above the female trend line, and also, on the formula, the point of intersection with the Y axis is higher (male = 1.6203, female = 1.4915).

Main Testing For the main testing, I decided to take the preliminary test work and expand on it. I used the whole sample of 122 pupils to give me averages where possible. This is because the averages from the 2 pupils (as used in preliminary work) were not a true representation of the year groups and genders.

• Do Taller Pupils Weight More? Chart 5 shows that there is a strong correlation between male height and weight for the samples in Mayfield School. This is shown by the trend line gradients. The male trend line is much stepper, and therefore the male trend line has a large positive number (y = 43.158x). The female trend line on the other hand also shows a positive correlation, but nowhere near as strong as the males. This is due to the results having a weaker correlation (much more spread out – especially towards the lower height end). The gradient for the female trend line is still clearly positive (y = 3.0679x), and therefore also shows that the taller the pupil, the more they weigh.

Graph 5

Scatter Graph showing relationship between Pupil Heights and Weights

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100120

1.00 1.50 2.00Pupil Height (meters)

Pupi

l Wei

ghts

(k

ilogr

ams)

Male Sample Female Sample


Male Gradient: y = 43.158x - 17.439 Female Gradient: y = 3.0679x + 43.961

Therefore this proves my first hypothesis true. The taller the pupil, the more they weigh.

• Are Older Pupils Heavier and Taller than Younger Pupils? To find whether older pupils are taller than younger pupils, I found the mean height for the males and females for each year group. I then plotted these on a Graph 6, shown below. It shows the mean height for males and females from each year group.

Graph 6

Chart Showing Relationship between Mean Height and Age (year group)

1.450

1.500

1.550

1.600

1.650

1.700

1.750

6 7 8 9 10 11

School Year group

Ave

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Male Height Female Height Mean Male Height Mean Female Height

Male Gradient: y = 0.0304x + 1.3707

Female Gradient: y = 0.0113x + 1.4696 From Graph 6, you can clearly see that there is a strong relationship between the age and the height of the pupils at Mayfield School. Both the males and females grow taller as they age. By looking at the trend lines and the trend line gradients, the males appear increase in height quicker (y = 0.0304x) than the girls (y = 0.0113x), but never the less, they both show that older pupils are taller, as both the gradients are a positive number.

[This space is left intentionally blank]


Graph 7 backs up what is shown on Graph 6. It shows the relationship between pupils’ age and weight. As is evident from the graph, the mean male weight increases steadily (y = 2.9379x) with age. However, the mean female weight varies only slightly (y = 0.0769x) over the 5 years. One possible cause of this is the female hormones. Whilst their bodies are changing, their weight increases, and once their bodies have changed and their hormones settle down their bodies become shapelier and their weight stabilises. One other possible cause is that around the age of 13-14 (year9) the females start to take more interest in their personal appearance, and are more weight conscious than males of that age.

Graph 7

Chart Showing Relationship between Mean Weight and Age (year group)

44.0046.0048.0050.0052.0054.0056.0058.0060.00

6 7 8 9 10 11 12

School Year group

Ave

rage

Wei

ght

(kilo

gram

s)

Male Weight Female Weight Male Weight Female Weight


Using Graphs 6 and 7, I have drawn the conclusion that older pupils are, on average, taller and heavier than younger pupils.

Therefore this proves my second hypothesis true. Older pupils are heavier and taller than younger pupils.

[This space is left intentionally blank]


• Do Males, on average, Weigh more than Females? To prove or disprove my hypothesis, I decided to use a Cumulative Frequency Graph. Before I could draw the graph, I needed to tally how many pupils had the same weight, and then to group them. Before I could do this however, I needed to know the minimum and maximum weight values. These are shown in Table 3.

Table 3 Weight (kilograms)

Male Female Both M & F Min Max Min Max Min Max Range 26 110 35 74 26 110 84

Table 3 shows the minimum and maximum weights for males and females, as well as the overall minimum and maximum weights. I then decided on what sized groups I would need to use. Below, in Tables 4 and 5 are the tallies for male and female weights.

Table 4

Male Weight Tally Table To From Tally Total 25 29 130 34 035 39 540 44 1245 49 1050 54 655 59 1060 64 965 69 170 74 775 79 080 84 085 89 090 94 095 99 0100 104 0105 109 0110 114 1

Total: 62

Table 5

Female Weight Tally Table To From Tally Total 25 29 030 34 035 39 340 44 745 49 2150 54 1955 59 660 64 165 69 070 74 175 79 080 84 085 89 090 94 095 99 0100 104 0105 109 0110 114 0

Total: 58

After checking that both the tally tables (tables 4 and 5) were correct, i.e. all the pupils in the sample were included in the tally table, and that none were left out or had been added in, I drew a histogram showing the popularity of each group. This is shown on Graph 8 on the following page.


Graph 8

Chart Showing the Popularity of each Weight Group

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25-29

30-34

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Weight Group (kg)

Pupi

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Male Weight Female Weight

From Graph 8, you can see that the male weights are much more spread out over the groups, as opposed to the female weights which are much more concentrated from 35kg to 64. I will look into the spread of the weights in the standard deviation section. I then went on to making a cumulative frequency table, for my cumulative frequency graph from the data in tables 4 and 5. Tables 6 and 7 contain the data to be plotted onto the Cumulative Frequency Graph.

[tables 6 and 7 on the next page]


Table 6 Male Cumulative Weight Group CF (kg) +

29 1 1 34 1 0 39 6 5 44 18 12 49 28 10 54 34 6 59 44 10 64 53 9 69 54 1 74 61 7 79 61 0 84 61 0 89 61 0 94 61 0 99 61 0 104 61 0 109 61 0 114 62 1

Table 7 Female Cumulative Weight Group CF (kg) +

29 0 034 0 039 3 344 10 749 31 2154 50 1959 56 664 57 169 57 074 58 179 58 084 58 089 58 094 58 099 58 0104 58 0109 58 0114 58 0

Graph 9

Cumulative Frequency Graph showing a Comparison between Male and Female Weights for the Pupils in Mayfield School

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29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114

Weight Groups (kg)

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Male Weight Female Weight

Graph 8 (shown above) is the cumulative frequency graph showing a comparison between male and female weights. This graph has only been drawn to check that my hand-drawn graph was correct. My hand-drawn graph can be found on a sheet of squared graph paper later in


this coursework. However, I included the computer rendered graph (Graph 9) here, as it makes it easier to explain. Next I found the averages of all the weight data – shown in Table 8. These will be needed when drawing the cumulative frequency graph, and also the box and whiskers diagram.

Table 8 Weight

(KG) Mean Median Mode

Male 53.15 51 40 Female 48.78 49 45

From the averages table I was able to find the upper quartile and lower quartile, and from there, the inter-quartile range. Cumulative Frequency Graph From the cumulative frequency graph I got a graphical representation of the data from tables 6 and 7. This enabled me to compare the male weight data against the female weight data. The Cumulative Frequency graph shows mixed results. At the start of the male series they had more people in the lower groups when compared with the female series, until the weight group ≤43, where the two series cross. Through the main part of the graph, the females have more in the groups when compared to the male series. Once again, at the higher end of the, around ≤73, the male and female series cross over once again. This shows us that the spread of female weights is less than the spread of male weights. I decided to draw a box and whiskers diagram to show this data in a different way. Box and Whiskers Diagram The box and whiskers diagram also shows the data in a graphical format; however, from the box and whiskers diagram I cannot see the data, only the spread and ranges of the data. From the box and whiskers diagram I can see that the females have a much smaller spread, and that their weights are much more concentrated into a smaller area (also shown on Graph 8). Standard Deviation – The Spread of the Data Standard Deviation (or ∂) is a statistic that tells me how tightly various examples are clustered around the mean in a set of data. I am going to use Standard Deviation to help me find out how tightly the male and female pupils weights are arranged around the mean. This will then tell me how much of a spread there is. Looking at the standard deviation can help explain why the data is the way it is. To calculate Standard Deviation, I have used Formula 1 – this is explained on the next page.


Formula 1

Formula 1 is shown above in its simplest form. Although it looks complex, it is actually relatively simple. In formula 1 (above), the “∂” is the symbol for standard deviation. The “X” is the weight of the pupil in Kilograms, and the “X” is the mean for the data set. “n” is the number of examples, in our case, pupils.

To calculate the formula, I needed to firstly, for each value of “X”, subtract the mean (or “X”) of the dataset from it and then square (multiply by itself) the result of the previous subtraction. Then I needed to sum up all of the squared values. Then I divided all the summed up squared values by “n” (number of pupils). This left me with a positive number – this is the standard deviation for the set of data (“X”). The lower the Standard Deviation value, the less the data deviates from the mean, or in other words, the less spread there is. My Standard Deviation calculations are below… Male:

Firstly I subtracted the mean from all the weights and then squared the number. I then added all the squared numbers together, and the result is shown below.

∑(X-X)2 = 10785.69 I divided 10785.69 by 62 (total of males in sample).

10785.69 / 62 = 173.9628

I finally found the Square Root of 173.9628 and this left me with the Standard Deviation for the Male Sample.

√173.9628 = 13.1895

Rounded to 2dp = 13.20

Female:

Firstly I subtracted the mean from all the weights and then squared the number. I then added all the squared numbers together, and the result is shown below.

∑(X-X)2 = 2290.086 I divided 2290.086 by 58 (total of females in sample).

2290.086 / 58 = 39.48424

I finally found the Square Root of 39.48424 and this left me with the Standard Deviation for the Female Sample.

√39.48424 = 6.283649

Rounded to 2dp = 6.28


As is evident from the calculations on the previous page, the female weight sample deviates less from the mean average, than the males. As mentioned above, the lower the standard deviation value, the less deviation from the mean is present. In practice, I assumed that data is approximately normally distributed. If that assumption can be justified, then 68% of the values are at most 1 standard deviation away from the mean, 95% of the values are at most two standard deviations away from the mean, and 99.7% of the values lie within 3 standard deviations of the mean. This is known as the "68-95-99.7 rule". Graph 10 is the Graph for Standard Deviation. The graph shows that the female series comes to a much sharper point, and is much narrower than the male series, which is broader and flatter. This shows that the female series (being narrower) has a lower value of standard deviation, a lower range, and therefore are closer to the mean than the male series.

Graph 10

Standard Deviation Comparison between Male and Female Weights

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Weight (kg)

Stan

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Male Std.DevFemale Std.Dev

From this standard deviation graph, I can see that there are very few pupils which are to either extreme. This means that I can trust the averages to prove whether the males are heavier than others. The averages are shown in Table 9 on the following page.


Table 9 (copy of Table 8)

Weight (KG) Mean Median Mode

Male 53.15 51 40 Female 48.78 49 45 Heavier Males Males Females

From Table 9 I can see that both the Mean Average and the Median Average show that the Males are heavier than Females. However the Mode Average shows that the Females are heavier than the Males. I have chosen to say that Males are, on average heavier than Females, based on the results of the averages table (Table 9), combined with the results from the standard deviation graph (Graph 10). Although the Mode Average showed that the females were heavier than males, the Mean and Median averages are more significant for this investigation.

Therefore this proves my third hypothesis true. Males do, on average, weigh more than females.

Conclusion In conclusion to the statistical work I carried out on Mayfield High School, I can prove, using the above texts and graphs that my 3 hypotheses are correct. I enjoyed the statistical challenges which were brought up by the Mayfield School coursework, and managed to mathematically prove my hypotheses true.


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Maths Coursework - Mayfield School