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Maths Year 10 Statistical coursework. Francis Newall
Introduction. I have been supplied with a database containing data about year 10 and 11 boys and girls in Mayfield School. I have to make up three Hypotheses, and prove them right or wrong throughout my project. I will use a large amount of various charts and graphs in order to do this. I was supplied with my data from Mayfield School, so the fact that I didn't collect it myself means that it is secondary data. There are many different methods of sampling, here are just a few
Random sampling - Randomly picking out samples.
Systematic sampling - This is where you have the whole population of the school and you pick out every 10th of 100th pupil.
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Stratified sampling This is where you divide the whole population of the school into groups of smaller sections and chose samples depending on the proportion of people in each group. You pick out these samples either randomly or systematically. I'm using Stratified Random Sampling, to make sure that the pupils have chosen are chosen totally by random. This is where you pick randomly from the groups you have made. The method of stratified random sampling is as follows
1. Firstly you need to work out how big your sample size is. For example, I have chosen to do a sample size of 41 this will mean that 41 of the pupils need to be stratified. Below is an example of one of the group's sample size.
Number of people in the group
_________________________ 100 x Eg.4
__ 100 = 5%
Total population of all the groups 70
Eg.100
__ = 4
5
Eg.71
__ = 17.75 = 18 pupils
4
. After you have worked out all of the group's sample sizes, you need to take one group at a time and pick pupils randomly. To dot his you either use the random button on a calculator or use excel.
. Once you have opened up an excel document, you need to click on a cell and type in, for example, '=4.' The number 4 can be changed depending on how many pupils are in the group.
4. You then need to click on 'Insert' and scroll down to 'function.'
5. Once you have entered 'Function', you need to locate the function name 'RAND', and double-click on it. A box will then appear from the top-left corner of the screen where you need to click on 'OK.'
6. A random number will now be in the cell that you first clicked on. If you click on that cell and then press 'ENTER', then another random number will appear. You need to note down however many numbers you need in your sample of one of the groups.
7. Now you need to open the group and find each random number along the left-hand side. You need to copy the 'Surname', 'Forename', 'Gender', 'Height' and 'Weight' for each of the randomly chosen students into a table. Finally you need to create a column for the BMI beside it. Into this table I have put all of my 41 samples.
Different types of data
I have been taught about two different types of data so far in my educational career, and they are Quantitative data and Qualitative data.
Quantitative data is data that is descriptive data, and Qualitative data, which can be either discrete or continuous. Discrete data for example, could be the number of children in a family, or shoe size (only certain values). Continuous data is data that can take any length, for example, hair length, weight or volume.
Hypothesis 1. - I predict that year 10 boys will have a larger body mass index than year 10 girls. Here are my sampled year 10 boys and girls.
Sampled Year 10 Boys
Student Number Surname Forename Gender Height(M) Weight(kg)
BMI
6 Smith Saf Male 1.8 64 17.1
5 Jones Nathan Male 1.54 76 .04
Dover David Male 1.1 6 16.
101 Stallin Joseph Male 1.84 6 18.1
65 Lister Kuta Male 1.6 7 7.4
15 Bonovan Liam Male 1.54 66 7.8
0 Singh Michael Male 1.68 64 .67
6 Leavy Thomas Male 1.7 71 .
Glintode Billy Male 1.55 50 0.81
1 George Wayne Male 1.75 68 .0
41 Hardy Jeff Male 1.7 75 .40
70 McManus Anthony Male 1.7 50 16.70
10 Taylor Lewis Male 1.0 80 .16
54 Kaura Kaz Male 1.66 6 .86
1 Slater Gordon Male 1.8 7 0.15
8 Smith Bill Male 1.55 7 46.45
51 Jones Paul Male 1.68 7 4.85
77 Pearce Robert Male 1.80 6 5.00
84 Robinson Mical Male 1.57 64 5.6
Carmichael Ryan Male 1.70 60 0.76
105 Urfon Homeed Male 1.6 57 1.71
4 Smith Arnold Male 1.54 54 .76
0 Browning Stephen Male 1.77 57 18.1
Aberdeen Richard Male 1.75 45 14.6
Edd Michael Male 1.68 5 5.11
57 Knott Mark Male 1.75 65 1.
6 Collins Alex Male 1.55 57 .7
8 Ray Adam Male 1.80 40 .
Smith Jason Male 1.68 4 14.88
41 Hardy Jeff Male 1.7 75 .40
104 Tumasi Rolo Male 1.50 65 8.88
75 Panjitsingh Mohammed Male 1.6 56 4.5
46 Honda Pablo Male 1.85 6 18.11
77 Pearce Robert Male 1.80 6 1.44
4 Hawk Matt Male 1.85 70 0.45
1 Black Joe Male 1.6 50 18.81
5 Fancropper James Male 1.81 56 17.0
78 Petit Neil Male 1.80 54 16.66
16 Brown Thomas Male 1.6 40 15.05
8 Gilroy Stuart Male 1.7 48 16.0
44 Hawks Tony Male 1.77 5 18.8
Sampled Year 10 Girls
Student Number Surname Forename Gender Height
(M) Weight(kg) BMI
14 Brown Emily Female 1.6 54 0.57
7 Sampras Paula Female 1.55 48 1.7
1 Chayse Erica Female 1.7 5 .05
50 Kelson Nina Female 1.80 60 18.51
85 Tahir Yasin Female 1.67 48 17.1
81 Slater Natalie Female 1.57 45 18.5
Connerly Jenny Female 1.70 48 16.60
78 Skully Josephine Female 1.60 66 5.78
7 Rogers Jade Female 1.65 5 1.67
Gorst Francesca Female 1.60 50 1.5
1 Brockley Maria Female 1.70 48 16.60
65 Owen Gemma Female 1.41 55 7.66
57 Martin Jane Female 1.67 48 17.1
7 Durst Freda Female 1.75 60 1.5
64 Morrison Nichole Female 1.6 48 18.06
5 Grimshaw Jane Female 1.6 7 7.0
4 Grimshaw Katie Female 1.7 50 16.0
87 Taylor-Wall Angela Female 1.70 55 1.0
80 Slater Sara Female 1.60 50 1.5
5 Lawson Karren Female 1.75 50 16.
Black Mia Female 1.75 57 18.61
5 Dean Samantha Female 1.70 50 17.0
18 Butt Sania Female 1.65 54 1.8
55 Long Anne Female 1.74 47 15.5
71 Roberts Sarah Female 1.54 45 18.7
1 Ali Aisha Female 1.5 45 1.47
17 Bullock Janice Female 1.7 51 17.
4 Ashiq Azra Female 1.60 56 1.87
0 Cell Jill Female 1.47 45 0.8
44 Hughes Donna Female 1.66 45 16.
4 Yo Rhonda Female 1.5 47 0.4
Fox Serena Female 1.0 40 11.08
46 Johnson Donna Female 1.68 50 17.71
0 Cell Jill Female 1.47 45 0.8
7 Bhatti Hannah Female 1.7 56 18.
65 Owen Gemma Female 1.41 55 5.45
81 Slater Natalie Female 1.57 45 18.5
6 Montogmerie Samantha Female 1.61 54 0.8
10 Blashaw Holly Female 1.7 51 17.04
Anderson Taz Female 1.80 60 18.51
1 Brandward Amy Female 1.65 5 1.46
As you can see, I have put a next to the last student on the list, Amy Brandward, as on the original data we were given, it said that she was 4.65 metres tall, which is very improbable! So I believe that this must have been a typing error, and I have changed it back to what it probably was originally, 1.65m.
As all of my hypotheses contain something about a BMI (Body Mass Index), I think I should explain how to work it out
Weight (kg)
BMI= __________
[Height (m)] ²
To try and prove my first hypothesis right I will show the Standard Deviation of the two sets of data, a stem-and-leaf diagram and a box and whisker diagram.
A Stem and leaf diagram to show year 10 boys' BMI against year 10 girls' BMI
Key for boys- 11 = 11.. 15 0 = 15.0. 14 6 8 = 14.6 and 14.8
Key for girls- 0 11 = 11.0. 5 15 = 15.5. 6 14 = 14.6 and 14.
Girls Boys
0 11
1
1
14 6 8
5 15 0
6 6 16 0 6 7
0 0 7 17 0
6 5 5 5 0 18 1 1 8 8
8 5 5 5 4 4 4 1 4
8 8 8 5 0 1 4 7 8
8 6 1 7
1 6 7 8
4 4 7
4
7 5
6
6 0 7 4 8
8 8
0
0
1
0
4
5 0 1
6
7
8
40
41
4 8
4
44
45
46 4
Box and whisker diagrams to show year 10 boys' BMI and year 10 girls' BMI
Boys
Min = 14.6
Max = 46.45
Median Q = 1.45
Q1 = Lower Quartile = 18.1
Q = Upper Quartile = .7
Girls
Min = 11.0
Max = .0
Median Q = 18.75
Q1 = Lower Quartile = 17.
Q = Upper Quartile = 1.
Standard Deviation, workings on year 10 boys and year 10 girls data.
Year 10 Boys data
The formula most commonly used to find the standard deviation is
Where N is the number of data points in my data set, and xj is the jth data point. The x with the bar over it is the average value of the data. This is also the formula used here.
If your data is known to follow the bell shaped curve (or is normally distributed or Gaussian distributed data), then 68% of your data points should fall within ± 1 standard deviation of your datas average.
In this set of year 10 boys data, I have 41 total data points. If you examine it carefully, you will see that that 7 data points are between 15.87 and 0.8.
The variance is the standard deviation squared. So for this set of data, the variance is 5.605
Year 10 Girls data
In this set of year 10 girls data, I have 41 total data points. If you examine it carefully, then you will see that 7 data points are between 16.05 and .5.
The variance is the standard deviation squared. So for this data, the variance is 11.5
Hypothesis . I predict that boys in year 11 will be taller than boys in year 10. To be able to do this, I need to have some year 11 boys samples. I shall use the same year 10 boys from hypothesis 1 to compare to the year 11 boys.
Here is my sample of year 11 boys.
Student number Surname Forename Gender Height (m) Weight (kg)
6 Berk Stephan Male 1.77 57
66 Paul Niel Male 1.7 64
47 Khan Assad Male 1.68 6
7 Horney Anthony Male 1.81 54
11 Brown Kevin Male 1.85 7
16 Cripp Justin Male 1.67 50
1 Cunning Kenneth Male 1.51 40
7 Singh Norman Male 1.51 8
5 Major William Male 1.8 68
8 Hossany Selim Male 86
18 Cullen Sam Male 1.55 54
1 Cunning Kenneth Male 1.51 40
77 Solomons Ian Male 1.7 7
81 Warne Michael Male 1.84 76
45 Kent Philip Male 1.86 80
7 Simmons Russell Male 1.65 50
Dixon Graham Male 1.6 5
80 Vincent Nigel Male 1.8 6
4 Fairfax Jacob Male 1.6 51
14 Chidgley Steven Male 1.6 5
4 Johnes Jimmy Male 1.4 80
1 Dixon Rico Male 1.6 8
50 Little James Male 1.65 47
5 Major William Male 1.8 68
74 Slim Andre Male 1.7 50
58 Mole Adam Male 1.64 60
70 Rottecth Amine Male 1.61 4
Ballson James Male 1.5 60
64 Oliver Marcus Male 1.57 4
75 Smith Michael Male 1.5 45
47 Krane Assad Male 1.68 6
Hughes Mark Male 1.65 58
1 Armstrong Simon Male 1.67 66
6 Fasworth John Male 1.7 7
10 Boggart john Male 1.75 60
48 Lee Brett Male 1.8 75
5 Bentley James Male 1.1 8
4 Lewis James Male 1.68 56
15 Chinny Anthony Male 1.6 56
51 Madalin Joseph Male 1.6
56 McDonald James Male 1.6 50
6 Olderson Stuart Male 1.6 48
To prove this hypothesis correct or incorrect, I shall do a scatter diagram and a bar chart.
Two scatter diagrams, one showing year 10 boys and the other showing year 11 boys, both according to height
Average height of year 10 boys = 1.80m
Average height of year 11 boys = 1.7m
Hypothesis . I predict that the taller you are, the higher your BMI will be. Here are my randomly chosen samples that I will need to do this hypothesis. I have not included any additional information except for the information that I will need to know
Year 10 boys
Weight (kg) Height (m) BMI
57 1.8 17.
57 1.6 1.7
58 1.7 1.6
40 1.6 15.1
60 1.80 18.5
56 1.61 1.6
56 1.60 1.8
64 1.55 6.6
75 1.7 .4
5 1.77 14.
68 1.80 1.0
57 1.77 18.
Year 10 girls
Weight (kg) Height (m) BMI
55 1.70 1.0
66 1.67 .7
57 1.67 0.4
54 1.6 0.
5 1.65 1.5
45 1.7 15.
56 1.56 .0
54 1.71 18.5
54 1.65 1.8
48 1.67 17.
Year 11 boys
Weight (kg) Height (m) BMI
6 1.68 .
54 1.55 .5
5 1.50 15.6
57 1.77 18.
54 1.58 1.6
7 1.78 16.7
6 1.80 1.1
51 1.6 1.4
54 1.70 18.7
Year 11 girls
Weight (kg) Height (m) BMI
54 1.65 0.0
44 1.6 16.6
45 1.60 17.6
48 1.58 1.
44 1.6 16.6
51 1.7 17.
8 1.6 14.
60 1.75 1.6
51 1.6 1.7
4 1.5 16.6
54 1.65 1.8
44 1.6 16.6
I shall do four separate scatter diagrams, one for each table, and then I shall see of there is any correlation between height and BMI.
Here are the four scatter diagrams.
Conclusions of all three hypotheses.
Hypothesis 1- I predicted that year 10 boys would have a larger body mass index than year 10 girls.
This prediction was indeed, correct, as we can see from the scatter diagrams, the stem and leaf diagram, and especially the box and whisker diagram. One of the key points to note, is that the boys' minimum BMI was 14.6, whereas the girls' minimum BMI was 11.0, a difference of .6, and the largest clue to the end result of boys having a larger BMI than girls is the fact that the boys' maximum BMI was 46.45, compared to the girls' maximum BMI of .0, a huge difference of 17.45!
One of the reasons that boys have a bigger BMI than girls may be the fact that boys tend to be stronger than girls, which means that they have more muscle. Muscle weighs more than fat, which may be why a lot of the boys had quite high BMI's, as the range in which a healthy BMI lies is 18-5.
Hypothesis - I predicted that boys in year 11 would be taller than boys in year 10.
This prediction was incorrect, as the average height of the year 10 boys was 1.80m, and the average height of year 11 boys was 1.7m.
This was a very unexpected result, as usually older people tend to be slightly heavier than people 1 or years younger than them.
I don't think this was a case of randomly picking small year 11's and tall year 10's, as one of my friends who predicted that year 11 girls would be taller than year 10 girls found out that the year 10 girls were taller than the year 11's, just like me, although I have not studied or looked at her graphs so I do not know how different they were.
Hypothesis - I predicted that the taller you are, the higher your BMI will be.
As you can see from my scatter graphs, there was no correlation between height and BMI, so therefore if someone is tall doesn't necessarily mean that they have a high BMI, as we need to take into account that weight is also included in the BMI, and we need to realise that someone who is tall probably has the same body structure as someone who is an average height. This means that even though they are taller, they are heavier because of this. This is why people who are taller do not necessarily have a higher BMI than an average person.
Possible Bias
Not all the pupil's measurements would have been measured at the same time and so they may have varied. If someone's weight was measured just after they had eaten then it would obviously be higher.
Final note
I feel I need to point out, that unless we take a survey including everyone in the whole world, the results I have come up with only suggest that, for example, boys have a bigger hand-span than girls. Also, I may have just randomly picked pupils that would tend to swing the results one way or the other, for example, for my second hypothesis, where I predicted that year 11 boys would be taller than year 10 boys, I may have just randomly picked big year 10 boys, and small year 11 boys, as I have already pointed out in my conclusion of the nd hypothesis.
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