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### Relationship Between Weight and Height

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Relationship Between Weight and Height

My School is a secondary school and is of mixed genders from
ages 11 to 17 plus a 6th from which has students of 17-19years. The
school is based in the country side near to several small towns and
villages, most of the students around the area come to this school.

Preliminary Investigation

I have decided to investigate the following fields; Weight and Height.
I have decided upon these two values because I am curious if there is
any connection between them, as I know many people my age which are
around my weight but which are a lot shorter than me. The fields I
have chosen to investigate are both forms of continuous data, in
contrast to data that is known as discreet, for example, Gender or
Favourite sport. Discrete data usually occurs in a case where there
are only a certain number of values, or when we are counting something
(using whole numbers).

Continuous data makes up the rest of numerical data. This is a type of
data that is usually associated with some sort of physical
measurement, in this case, height and weight.

Data is discrete if there are only a infinite number of values
possible or if there is a space on the number line between each 2
possible values. Discrete data usually occurs in a case where there
are only a certain number of values, or when we are counting something
(using whole numbers), and so this form of data does not provide much
scope when concerned with comparability and analytical values.
Therefore I have chosen to investigate a type of data I will be able
to conclude un-limiting numerical data from as so I can make detailed
analogies and conclusions.

I predict that there will be a correlation between weight and height
as when you grow taller, your body mass increases as you have more
bone etc to make up your body. However it would be interesting to find
out is if you compare people of different ages weights when their
height is the same, and body maturity may make the person heavier due
to developed muscles etc. Also it may be intriguing to discover the
relationship between males and females of the same age.

Within my investigation I am concerned with gender difference as upon
examination of my data it is apparent that values for each sex are of
poles apart, in that values for males are of higher value and lower
for females, I believe this will become more apparent as my
investigate proceeds.

Within my preliminary investigation I will be analysing a sample of
random subjects, these subjects will be chosen methodically by
obtaining a range of thirty numbers from the range of each boys and
girls.

Year Group

Number of Boys

Boys to be surveyed

Number of Girls

Girls to be surveyed

Total

Total of Students surveyed

7

151

8

131

7

282

15

8

145

7

125

6

270

13

9

118

6

143

7

261

13

10

106

5

94

5

200

10

11

84

4

86

5

170

9

Total

604

30

579

30

1183

60

I have chosen to take a sample of 30 boys and girls as I have
calculated that 30 out of the total number of each gender produces a
result of about 5%, which I consider a good starting point for my
pre-test as it will produce quick, obtainable data and will give me a
good idea of how I can deal with the information as so I can improve
my testing further, for a more detailed analysis later on.

Percentage sample of boys

=30/604 x 100

=4.9669%

Percentage sample of girls

=30/579 x 100

=5.1813

I used Quota sampling to collect my data by simply choosing students
walking past me from the corresponding gender and year groups. This
may not be the fairest way of obtaining my data but I believe that is
was just as random as it would be if I had catalogued every single
student and picked them out at random.

I used this method to generate a sample of 30 boys and girls:

Girls

Sample Number

Year

Height (m)

Weight (Kg)

1

7

1.62

40

2

7

1.32

35

3

7

1.62

49

4

7

1.49

53

5

7

1.65

43

6

7

1.62

65

7

7

1.60

47

8

7

1.52

33

9

8

1.72

43

10

8

1.60

44

11

8

1.60

42

12

8

1.61

46

13

8

1.70

53

14

8

1.76

50

15

8

1.58

72

16

8

1.59

55

17

9

1.53

52

18

9

1.78

59

19

9

1.71

42

20

9

1.62

45

21

9

1.69

48

22

9

1.51

65

23

9

1.58

55

24

10

1.70

50

25

10

1.80

60

26

11

1.73

64

27

11

1.72

51

28

11

1.73

48

29

11

1.65

54

30

11

1.71

42

Boys

Sample Number

Year

Height (m)

Weight (Kg)

1

7

1.45

40

2

7

1.42

48

3

7

1.44

42

4

7

1.49

47

5

7

1.53

35

6

7

1.49

67

7

7

1.74

70

8

7

1.52

54

9

8

1.45

72

10

8

1.50

39

11

8

1.72

46

12

9

1.67

52

13

8

1.72

57

14

8

1.62

52

15

9

1.60

60

16

9

1.62

40

17

9

1.53

45

18

9

1.60

40

19

9

1.64

65

20

10

1.75

45

21

10

1.66

70

22

10

1.77

57

23

10

1.63

56

24

10

1.87

70

25

11

1.57

60

26

11

1.75

60

27

11

1.96

93

28

11

1.74

50

29

11

1.62

92

30

11

1.70

72

Handling the Data

I also examined accuracy of the data, as I can verify the truthfulness
of my data I will examine it on face value, I have ignored any data
that falls outside of 2 decimal places, weights that are not within
the parameters 30-100kg and heights outside 1-2 metres. I have decided
to eliminate these values as I consider them to be abnormal and
biased, I can conclude this by examining BMI, as I am aware that
any-one with a body mass Index lower than 18 or higher than 35 is not
typical. This applies for height as upon examining heights of my class
mates it is apparent that standard height is not blow 1m or above 2m.

Obtaining evidence for analysis

My analysis will concern the differentiation of boys against girls, as
I am aware, from the age of 11 it is apparent both genders take on a
from of change and depending on their personal chemistry a change in
weight and height is one of the major factors within adolescence.

By obtaining data that I can tabulate I can carry out a series of
comparisons and conclude conclusions concerning my applied data.

There are a range of statistical calculations I can make use of.

I am aware I can obtain the following, for both genders and both
height and weight:

* Frequency distribution

* Mean;

* Mean deviation;

* Standard deviation;

* Median;

* Range;

* Inter-quartile range;

* Central tendency;

* Measure of dispersion or spread;

* Distribution.

Mode = Value that occurs most in a data set. Not a very useful measure
of central tendency.

Median = Middle value from a set of ranked observations. Useful for
highlighting the typical value of a data set.

Mean = Sum of a set of observations divided by the number of
observations in a data set, most widely used measure of central
tendency. Also can be calculated as a weighted mean for grouped data

Standard Deviation = Measures or depicts the amount of spread or
variability in a data set; how typical of a whole distribution the
mean actually is. It is apparent, the larger the Standard Deviation,
the greater the spread of observations and the less typical the mean.

Standard Deviation or Variance to compare locations or regions is an
absolute measure.

Mean = A measure of central tendency calculated by dividing the sum of
the scores in a distribution by the number of scores in the
distribution. This value best reflects the typical score of a data set
when there are few outliers and/or the dataset is generally
symmetrical.

Box plot = Summary plot based on the median, quartiles, and extreme
values. The box represents the inter-quartile range which contains the
50% of values. The whiskers represent the range; they extend from the
box to the highest and lowest values, excluding outliers. A line
across the box indicates the median.

Skewness = Measures the degree to which data values are evenly or
unevenly distributed on either side of the mean. If a majority of the
values in a data set fall below the mean, then data are positively
skewed with the tail of the histogram falling to the right. If a
majority of the values fall above the mean, then data are negatively
skewed and the tail of the histogram will fall to the left.

As I have limited amount of time given for this investigation I will
consider the importance of the above actions before I carry any of
them out.

Histogram to compare the Heights for boys and girls

Histogram = Graphic representation of grouped data along two axes

I have chosen to draw a histogram of the heights of the boys and girls
within my sample. This will require the grouping of both boys and
girls as so I can accumulate frequency and frequency density. Class
widths need to be appropriate as so their individual frequency are of
tabulating range, as so I can conclude a clear distribution.

I have created a table below which displays the frequency distribution
and frequency density for my samples.

Frequency = frequency density x class width interval

Boy's heights (m)

Class width

1.30+

1.50+

1.55+

1.60+

1.65+

1.70+

1.75+

Frequency

6

5

0

6

3

4

6

Frequency density

0.3

1

0

1.2

0.6

0.8

2.4

Girl's heights (m)

Class width

1.30+

1.50+

1.55+

1.60+

1.65+

1.70+

1.75+

Frequency

2

3

3

8

3

8

3

Frequency density

0.1

0.6

0.6

1.6

0.6

1.6

1.2

Boy's weights (kg)

Class width

30+

40+

50+

60+

65+

70+

75+

Frequency

2

9

7

3

2

5

2

Cumulative frequency

2

11

18

21

23

28

30

Girl's weights (kg)

Class width

30+

40+

50+

60+

65+

70+

75+

Frequency

2

13

10

2

2

1

0

Cumulative frequency

2

15

25

27

29

30

30

The above sets of data were utilised within figure 1.1, which concerns
each genders heights. I have plotted my Histograms as so boys and
girls exist within the same graph as so I can make direct comparisons.
A histogram allows me to assimilate distribution of the concerned
information more quickly than if I were to simply examine the above
tables, a graph demonstrates the information clearly and conclusions
such as, for example a lack in symmetry, or skew can be concluded.

What is the mean and deviation?

I wish to calculate the means and standard deviations using raw data
as so I can obtain additional statistics for my comparison.

I used my calculator to obtain the mean and standard deviation for
both genders, using my data for heights and then weights.

Height

Gender

Standard deviation

mean

Boys

0.13

1.63

Girls

0.098

1.63

Weight

Gender

Standard deviation

mean

Boys

14.45

56.53

Girls

8.94

50.17

Cumulative curves of weights of both genders

Within the above tables concerning frequency and frequency density for
both boys and girls weights I have also calculated the cumulative
frequency as so I could create a histogram of these results (figure
1.2)

I can use my graph to analyse the relationship between the data for
boys and girls. These two cumulative corves will be plotted within the
same graph as so I can carry out a better analysis

Median weight (kg):

Boys = 55

Girls = 48.5

I was able to obtain the median values for each sex easily as my
results from my sample were put into a table within a spreadsheet and
so I was able to arrange the data in ascending weight order and
pinpoint the middle value.

Once the data is arranged in order of ascending height, I can
conclude:

Weight range (kg):

Boys = 35-93

Girls = 33-72

Attaining Lower, Upper and inter-quartile range

I was able to obtain these results by pinpointing the intervals 25
percentile, 50 percentile and 75 percentile on my graph and reading
off the corresponding data, this was simplified by the fact that
weight is a continuous variable, it is a continuous approximation of
the distribution of values.

Scatter diagrams of weight against height for boys and girls

This graph enables me to look at any possible correlation between the
two variables, height and weight. I can deuce the coefficient,
depending on the degree of correlation with line of best fit and
plotted points. (Figure 1.3)

Summary of findings from preliminary investigation

My results lead me to believe that in general terms that the central
tendency for boys heights is within the range 1.60-1.65 metres and
girls, both ranges 1.60-1.65 and 1.70-1.75 metres. I can conclude from
the standard deviation for boys concerning both weight and height,
suggests that the boy's values vary more than the girls thus meaning
their mean is less typical. It appears that in fact the girls vary
less concerning height. Interestingly the boys and girls mean height
is the same and their mean weights do not vary too greatly.

These values indicate that a typical weight for boys within Lytchett
Minster School, aged from 11-16 is greater than for girls within the
same parameters.

As observed by the product of standard deviation, boys are more spread
out and have a wider range compared to girls.

These results indicate, in general that both sexes follow a trend
concerning weight until their weights until they reach 50 kg. Evidence
suggests that 15% of boys are above 68kg, and 15% girls are above 57kg
a difference of 11kg. I can also conclude that 50% of boys are above
the median weight of 55kg and 73% of girls are above the median weight
of 48.5kg. The plotted points on my scatter graphs for both sexes
demonstrating weight dependant on height. Both sexes demonstrated a
lack of correlation and the deviation from the line of best fit
illustrates a wide spread, especially concerning boys.

Analysis

Using my provisional conclusions, I have collected some issues I feel
need further investigation. My Histogram indicated that both sexes
follow a trend in increasing height until their weight exceed around
50kg, therefore there could be a point where boys weights and heights
exceed girls, or that girls start to grow in weight at a more
proportional weight to each other and reach a steady weight before
boys.

My original approach to sampling of pupils involved a Quota sampling
of 5% from each sex and 10% of the whole school.

I would like to take a more detailed look at the school, this time
exceeding the percentage by a further 10% so an over all school sample
of 20%.

Considering my proposed hypothesis I am going to conduct a survey
across year groups rather than the school as a whole, as I do not
believe a further look at the generalised patterns of height and
weight for boys and girls will help my theory any further than before.

I am going to examine year groups 7 and 11 as so I can pinpoint
whether in fact:

* Girls appear to mature at a steadier rate than boys after a
certain point, and finish their growth spurt before boys reach
their full potential.

I have chosen these two year groups as a comparison as they are at
each end of my available age range and I believe will produce the most
promising results.

Depending at the outcome of this search I will consider whether it is
necessary for me to take a further look at year groups in more detail.

To ensure my testing is fare and candidates have an equal chance of
being picked I have decided to use stratified random sampling dividing
up the school into years and genders. This is basically finding the
ratio of the total number of values you want from each group.

A Sample of 50% sample of each boys and girls from the combined year
groups 7 and 11 which have 282 and 170 pupils

Total number of pupils = 170+282= 452

Within this I wish to take a sample of 10% sample in total.

Therefore the following calculations correspond to my sample:

Number of boys from year 7 = 151/452 x 120=40

Number of girls from year 7 = 131/452 x 120=35

Number of boys from year 11= 84/452 x 120=22

Number of girls from year 11 = 86/452 x 120= 23

This then is used to find 40 random pupils from year 7 boys, 35 random
pupils from year7 girls and so on.

Year 7 Boys Heights and Weights:

Sample number

Height (m)

Weight (kg)

1

1.35

39

2

1.36

38

3

1.36

45

4

1.41

45

5

1.42

36

6

1.45

40

7

1.46

53

8

1.47

41

9

1.47

50

10

1.48

35

11

1.49

38

12

1.5

41

13

1.5

43

14

1.52

36

15

1.52

42

16

1.52

37

17

1.53

40

18

1.53

44

19

1.54

43

20

1.54

43

21

1.55

38

22

1.55

50

23

1.55

53

24

1.56

35

25

1.58

40

26

1.59

45

27

1.59

52

28

1.6

38

29

1.6

40

30

1.6

44

31

1.6

50

32

1.62

48

33

1.62

48

34

1.62

49

35

1.63

59

36

1.64

50

37

1.65

64

38

1.65

69

39

1.68

45

40

1.71

49

Year 7 girls Weights and Heights:

Sample number

Height (m)

Weight (kg)

1

1.32

35

2

1.42

41

3

1.42

34

4

1.45

52

5

1.46

45

6

1.48

42

7

1.48

39

8

1.48

46

9

1.5

44

10

1.5

40

11

1.51

50

12

1.51

50

13

1.52

40

14

1.54

40

15

1.55

50

16

1.56

53

17

1.57

45

18

1.58

50

19

1.58

48

20

1.58

43

21

1.59

54

22

1.59

41

23

1.59

38

24

1.61

45

25

1.61

47

26

1.61

48

27

1.62

40

28

1.62

42

29

1.62

49

30

1.62

54

31

1.63

45

32

1.63

50

33

1.65

45

34

1.65

40

35

1.73

62

Year 11 Boys Weights and Heights:

Sample number

Height (m)

Weight (kg)

1

1.32

45

2

1.51

40

3

1.52

60

4

1.55

54

5

1.6

38

6

1.61

47

7

1.62

52

8

1.62

56

9

1.67

66

10

1.67

50

11

1.68

50

12

1.69

55

13

1.71

57

14

1.72

63

15

1.72

58

16

1.75

60

17

1.77

57

18

1.78

67

19

1.8

60

20

1.81

72

21

1.85

73

22

1.86

56

23

1.88

75

24

1.91

82

Year 11 Girls Weights and Heights:

Sample number

Height (m)

Weight (kg)

1

1.55

36

2

1.56

50

3

1.6

54

4

1.61

54

5

1.61

54

6

1.62

56

7

1.62

54

8

1.63

44

9

1.63

44

10

1.63

47

11

1.63

48

12

1.65

66

13

1.65

54

14

1.65

42

15

1.67

52

16

1.68

54

17

1.68

48

18

1.69

51

19

1.72

51

20

1.72

51

21

1.73

64

Median weight (kg):

Year 7 Boys = 46

Year 7 Girls = 38

Year 11 Boys = 56

Year 11 Girls =48

Year 7 Height

Gender

Standard deviation

mean

Boys

0.13

1.54

Girls

0.098

1.55

Year 7 Weight

Gender

Standard deviation

mean

Boys

14.45

44.88

Girls

8.94

45.34

Year 11 Height

Gender

Standard deviation

mean

Boys

0.13

1.69

Girls

0.098

1.64

Year 11 Weight

Gender

Standard deviation

mean

Boys

14.45

58.04

Girls

8.94

51.14

The below Graphs are not to the same height scale, this was because
the results were either to close together and hard to see the separate
results, or spread out over a range of height.

The above scatter diagrams Indicate to me what to expect when I
compile my scatter diagrams in more detail.

Equation for line of best fit

y = mx + c (m = gradient and c = y intercept).

[IMAGE][IMAGE][IMAGE]To find the equation for the line of best fit on
any graph you need to find the gradient of the line and the y
intercept. So you can substitute them into the equation for every line
which is y = mx + c. So therefore I need to find them for my graphs.

For example if the gradient on my graph was 40, I could substitute
this into my equation: y = 40x + c

[IMAGE]Now because my line doesn't go through the y-axis we have to
work out where about it would normally go so I need to substitute in 2
values off my graph. I have chosen them as:

x = 1.4

y = 42

So I need to replace x and y to find c:

42 = 40 x 1.4 + c

42 = 56 + c ...

Calculating correlation coefficients

To make accurate comparisons of the two sets of data for each sex and
age, I will use spearmans rank. Spearmans rank will show me how
closely related height is to weight. Because I have to do the ranking
four times, I have decided to only use a sub-sample of ten random
people from years 7 and 11 for boys and girls. I will choose the
people for my sample via a random number generator on my calculator
however, my calculator goes into the second decimal place so I will
round up to the nearest whole number. After I have worked out the
difference between the ranks squared (dÂ²) I will then use the
following equation to calculate the correlation coefficient:

1-6Î£dÂ² / n(nÂ²-1)

Year 7 boys

1

2

3

4

5

6

7

8

9

10

Weight (w)

45

38

36

44

52

36

64

53

50

43

Height (h)

141

136

142

160

159

152

165

146

164

154

Weight rank

5

8

9.5

6

2

9.5

1

3

4

7

Height rank

9

10

8

3

4

6

1

7

2

5

Difference (d)

4

2

-1.5

-3

2

-3.5

0

4

-2

-2

dÂ²

16

4

2.25

9

4

12.25

0

16

4

4

The sum of dÂ² is 71.5

1 - 6 x 71.5/10(10Â²-1) = 0.56

Year 7 Girls

n

1

2

3

4

5

6

7

8

9

10

Weight (w)

42

42

45

45

41

40

35

44

39

52

Height (h)

148

162

157

163

142

150

132

150

148

145

Weight rank

5.5

5.5

2.5

2.5

7

8

10

4

9

1

Height rank

6.5

2

3

1

9

4.5

10

4.5

6.5

8

Difference (d)

1

-3.5

0.5

-1.5

2

-3.5

0

0.5

-3.5

7

dÂ²

1

12.25

0.25

2.25

4

12.25

0

0.25

12.25

49

The sum of dÂ² is 93.50

1 - 6 x 93.50/10(10Â²-1) = 0.43

Year 11 Boys

1

2

3

4

5

6

7

8

9

10

Weight (w)

60

57

47

72

57

55

40

58

54

82

Height (h)

180

171

161

181

177

169

151

172

155

191

Weight rank

3

5.5

9

2

5.5

7

10

4

8

1

Height rank

3

6

8

2

4

7

10

5

9

1

Difference (d)

0

0.5

-1

0

-1.5

0

0

1

1

0

dÂ²

0

0.25

1

0

2.25

0

0

1

1

0

The sum of dÂ² is 5.50

1 - 6 x 5.50/10(10Â²-1) = 0.96

1

2

3

4

5

6

7

8

9

10

Weight (w)

54

42

47

50

54

66

52

56

36

51

Height (h)

161

165

163

156

168

165

167

162

155

172

Weight rank

3.5

9

8

7

3.5

1

5

2

10

6

Height rank

8

4.5

6

9

2

4.5

3

7

10

1

Difference (d)

4.5

-4.5

-2

2

-1.5

3.5

-2

5

0

-5

dÂ²

20.25

20.25

4

4

2.25

12.25

4

25

0

25

Year 11 Girls

The sum of dÂ² is 117

1 - 6 x 117/10(10Â²-1) = 0.29

Conclusions

I can tell from these scatter graphs first of all that the girls have
a poorer correlation than the boys, this is also proved by my
spearmans ranking where I discovered that the correlation coefficient
is much greater for boys than for girls. Although the boys have better
correlation, the girls have a closer height bracket whereas the boys
have a bigger height bracket and a better correlation. In both of the
graphs the majority of people are in the 1.6m to 1.8m range. The boys
height however continues after 1.8m unlike the girls.

Very like in year 7 the girls weight is not spread out and the
majority are still compacted into the 40kg to 60 kg weight range,
although it is now more spread out because before it was all nearly on
the 40kg line whereas now it is more spread out but still very
compact. The boy's weight is spread out mainly in the 40kg to 80kg
range. This shows us that while the girls are fairly uniform in weight
the boys are a lot more varied. Like my first prediction the boys are
heavier then the girls by year 11.

I can tell from the year 11 cumulative frequency graph for weight that
the boys are heavier because the graph ends later for boys then the
girls. Also the boys have a greater inter-quartile range. I can tell
this because of the shape of the curves. The girls have a tight
distribution. I can also tell from the year 11 cumulative frequency
graph for height that boys are taller since because many more boys are
still on the graph even after the tallest girl has been counted for.

Evaluation

I think that my investigation has been a complete success in proving
my original hypothesis, however I do think that I should have made my
original samples a little larger, also I think that my sub-sample for
spearmans rank was too small as the correlation would have greatly
varied if one result was different as there was such a small number of
samples. Apart from that I think I have proved that females tend to be
of the same weight, but with varied heights, making them have poor
correlation, and men tend to be spread over a wide range of heights
and weights, but have both strongly related.

MLA Citation:
"Relationship Between Weight and Height." 123HelpMe.com. 20 Apr 2014
<http://www.123HelpMe.com/view.asp?id=121192>.