Investigating the Relationship Between Height and Weight of Students


Collecting the data

Mayfield High School is a school for 11-16 year olds. The line of
inquiry, which I will use to investigate the school, is the relation
between height and weight.

I will take a random sample of the school by giving each student a
number and creating a random number between 0 and 1 using the random
number button on my calculator (RAN#). Then I will multiply this
number by 1183 to create a number between 1 and 1183.

I will do this until I get a sample of 30 boys and 30 girls.

The Data

· Number-0.89 x 1183 = 1052.87 = 1053 (f)

· Number-0.476 x 1183 = 563.108 = 563 (f)

· Number - 0.95 x 1183 = 1123.85 = 1124 (m)

· Number - 0.855 x 1183 = 859.74525 =860 (m)

· Number - 0.678 x 1183 = 802.074 =802 (m)

· Number - 0.611 x 1183 = 722.813 =722 (m)

· Number- 0.071 x 1183 = 83.993 = 84 (f)

· Number - 0.651 x 1183 = 770.113 = 770 (m)

· Number - 0.993 x 1183 = 1174.719 = 1175 (m)

· Number - 0.856 x 1183 = 1012.648 = 1013 (f)

· Number - 0.692 x 1183 = 818.636 = 819 (f)

· Number - 0.935 x 1183 = 1106.105 = 1106 (f)

· Number - 0.52 x 1183 = 615.16 = 615 (m)

· Number - 0.981 x 1183 = 1160.523 = 1161 (m)

· Number - 0.58 x 1183 = 686.14 = 686 (m)

· Number - 0.348 x 1183 = 411.684 = 412 (m)

· Number - 0.24 x 1183 = 283.92 = 284 (m)

· Number - 0.564 x 1183 = 667.212 = 667 (m)

· Number - 0.975 x 1183 = 1153.425 = 1153 (m)

· Number - 0.426 x 1183 = 503.958 = 504 (m)

· Number - 0.803 x 1183 = 949.949 = 950 (m)

· Number - 0.47 x 1183 = 556.01 = 556 (m)

· Number - 0.815 x 1183 = 964.145 = 964 (f)

· Number - 0.747 x 1183 = 883.701 = 884 (f)

· Number - 0.185 x 1183 = 218.855 = 219 (m)

· Number - 0.266 x 1183 = 314.678 = 315 (f)

· Number - 0.12 x 1183 = 141.96 = 142 (m)

· Number - 0.768 x 1183 = 908.544 = 909 (m)

· Number - 0.667 x 1183 = 789.061 = 789 (m)

· Number - 0.345 x 1183 = 408.135 = 408 (f)

· Number - 0.936 x 1183 = 1107.288 = 1107 (f)

· Number - 0.534 x 1183 = 631.722 = 632 (f)

· Number - 0.285 x 1183 = 337.155 = 337 (f)

· Number - 0.575 x 1183 = 680.225 = 680 (m)

· Number - 0.237 x 1183 = 280.371 = 280 (m)

· Number - 0.471 x 1183 = 557.193 = 557 (m)

· Number - 0.837 x 1183 = 990.171 = 990 (f)

· Number - 0.269 x 1183 = 318.227 = 318 (f)

· Number - 0.193 x 1183 = 228.319 = 228 (f)

· Number - 0.457 x 1183 = 540.631 = 541 (m)

· Number - 0.533 x 1183 = 630.539 = 631 (m)

· Number - 0.27 x 1183 = 319.41 = 319 (m)

· Number - 0.19 x 1183 = 224.77 = 225 (m)

· Number - 0.377 x 1183 = 445.991 = 446 (f)

· Number - 0.639 x 1183 = 755.937 = 756 (m)

· Number- 0.676 x 1183 = 799.708 = 800 (f)

· Number - 0.57 x 1183 = 674.31 = 674 (f)

· Number - 0.127 x 1183 = 150.241 = 150 (m)

· Number - 0.927 x 1183 = 1096.641 = 1097 (f)

· Number - 0.329 x 1183 = 389.207 = 389 (m)

· Number - 0.572 x 1183 = 676.676 = 677 (m)

· Number - 0.172 x 1183 = 202.293 = 202 (f)

· Number- 0.955 x 1183 = 1129.765 = 1130 (f)

· Number - 0.963 x 1183 = 1139.229 = 1139 (f)

· Number - 0.941 x 1183 = 1113.203 = 1113 (f)

· Number-0.868 x 1183 = 1026.844 = 1027 (f)

· Number - 0.153 x 1183 = 180.999 = 181 (f)

· Number - 0.214 x 1183 = 253.162 = 253 (f)

· Number - 0.203 x 1183 = 240.149 = 240 (f)

· Number - 0.307 x 1183 = 363.181 = 363 (f)

· Number - 0.397 x 1183 = 469.651 = 470 (f)

· Number - 0.491 x 1183 = 580.853 = 581 (f)

· Number - 0.559 x 1183 = 661.297 = 661 (f)

Tables of Data

Height (m)

Weight (kg)

1.63

38

1.58

36

1.63

52

1.60

45

1.67

52

1.65

59

1.65

54

1.80

50

1.54

50

1.62

46

1.58

72

1.30

36

1.68

50

1.54

54

1.64

40

1.52

50

1.67

48

1.56

63

1.57

52

1.60

54

1.62

35

1.65

59

1.75

50

1.71

54

1.49

53

1.32

48

1.60

46

1.57

52

1.47

38

1.69

42

Boys Girls

Height (m)

Weight (kg)

1.65

50

1.75

60

1.53

32

1.70

110

1.73

47

1.62

57

1.52

47

1.90

80

1.64

59

1.50

49

1.52

45

1.54

40

1.87

70

1.66

46

1.55

50

1.34

42

1.61

45

1.60

50

1.78

67

1.62

42

1.52

45

1.50

50

1.60

40

1.65

69

1.55

47

1.60

41

1.65

51

1.55

38

1.76

55

1.72

46

Height, h (cm)

Frequency

1.20-1.30

1

1.31-1.40

1

1.41-1.50

2

1.51-1.60

11

1.61-1.70

12

1.71-1.80

3

1.81-1.90

0

Boys Girls

Height, h (cm)

Frequency

1.20-1.30

0

1.31-1.40

1

1.41-1.50

2

1.51-1.60

11

1.61-1.70

9

1.71-1.80

5

1.81-1.90

2

Weight, w, kg

Frequency

30-40

6

41-50

11

51-60

11

61-70

1

71-80

1

Weight, w (kg)

Frequency

30-40

4

41-50

16

51-60

6

61-70

3

71-80

0

81-90

0

91-100

0

101-110

1

Heights

Graphs

[IMAGE]

[IMAGE]

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The mode height in my sample was higher for girls than boys.

The evidence from the sample suggests the range of boys' heights will
be larger than the range for girls' heights.

Mean height for girls and boys

Mean height for boys=all boys' heights/30= 48.73m/30= 1.624m

Mean height for girls=all girls' height/30=47.9m/30= 1.596m

Modal class interval for girls and boys

Modal class interval for boys = 1.51-1.60m

Modal class interval for girls = 1.61-1.70m

Median height for girls and boys

Median height for boys= 1.615m

Median height for girls = 1.61m

Range of heights for boys and girls

Range of heights for boys= 1.90-1.34m=0.56m

Range of heights for girls= 1.80-1.30m=0.5m

Height (m)

Mean

Mode

Median

Range

Boys

1.624m

1.51-1.60m

1.615m

0.56m

Girls

1.596m

1.61-1.7m

1.61m

0.5m

All 3 measurements of average (mean, mode and median) are greater for
boys apart from the modal class value. Also the range of heights is
slightly larger for boys. In general I believe boys are taller
generally as the mean which is often the most accurate average is
substantially higher for boys than for girls. Evidence from the data
shows that 27/30 boys (90%) are taller than 1.51m while only 25/30
(83%) of girls are taller than 1.51m.

[IMAGE]


Stem leaf diagram for boys

Stem

Leaf

Frequency

120

0

130

4

1

140

0

150

0, 0, 3, 2, 2, 4, 5, 2, 5, 5

10

160

5, 2, 4, 6, 1, 0, 2, 0, 5, 0, 5

11

170

5, 0, 3, 8, 6, 2

6

180

7

1

190

0

1

Stem leaf diagram for girls

Stem

Leaf

Frequency

120

0

130

0, 2

2

140

7, 9

2

150

8, 4, 8, 4, 2, 6, 7, 7

8

160

3, 3, 0, 7, 5, 5, 2, 8, 4, 7, 0, 2, 5, 0, 9

15

170

5, 1

2

180

0

1

190

0

Weights

Graphs

[IMAGE]

[IMAGE]


Mean weight

Mean boy's weight= 1570/30=52.3kg

Mean girl's weight= 1478kg/30= 42.3kg

Modal class interval

Modal boy's weight=51-60kg

Modal girl's weight=41-50 and 51-60kg

Median weight

Median boys weight=49.5kg

Median girls height=50kg

Range of boys and girls weights

Range of weights for boys= 110-32 kg = 78kg

Range of weights for girls= 72-36kg=36kg

Weight (kg)

Mean

Modal class interval

Median

Range

Boys

52.3

51-60

49.5

78

Girls

42.3

41-50 and 51-60

50

36

All three measurements for boys and girls were either higher for boys
or equal though the range for the boys was double the range of the
girls with a range of 78kg to 36k. The evidence suggests that there
are many heavy boys as the mean average is quite high and range are
high which means that the small number of heavy boys are pushing up
the average.

[IMAGE]


Extension

I think that the taller a person, the heavier he or she will weigh. I
will test this hypothesis by taking a new sample of 30 students by
using the random number function again.

1.82

64

1.51

48

1.68

53

1.50

52

1.55

57

1.50

57

1.53

44

1.56

60

1.54

54

1.84

76

1.62

72

1.59

47

1.41

40

1.68

59

1.55

50

1.48

39

1.62

49

1.6

42

1.32

38

1.67

51

1.54

57

1.42

29

1.72

51

1.71

60

1.68

50

1.60

49

1.62

45

1.55

67

1.55

36

1.63

50

[IMAGE]


[IMAGE]

A Person

[IMAGE]

There is a positive correlation between height and weight. This
suggests the taller someone is, he more they will weigh. My line of
best fit suggests that someone who is 1.6m tall will weigh 52kg. There
will be a better correlation it I consider boys and girls separately.

Text Box: A Person

[IMAGE]

[IMAGE]


[IMAGE]

There is a stronger correlation if boys and girls are considered
separately. The line of best fit suggests that a girl of 1.6m would
weigh 51kg when a boy of 1.6m would weigh 56kg.

Cumulative Frequency Graphs

Height, h (cm)

Boys

Height, h (cm)

Girls

Height, h (cm)

Mixed

1.20-1.30

0

1.20-1.30

1

1.20-1.30

1

1.31-1.40

1

1.31-1.40

2

1.31-1.40

3

1.41-1.50

3

1.41-1.50

4

1.41-1.50

7

1.51-1.60

14

1.51-1.60

15

1.51-1.60

29

1.61-1.70

23

1.61-1.70

27

1.61-1.70

50

1.71-1.80

28

1.71-1.80

30

1.71-1.80

58

1.81-1.90

30

1.81-1.90

30

1.81-1.90

60

[IMAGE]


Text Box:

[IMAGE]


[IMAGE]

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Summary of results

According to my results, there is a positive correlation between
height and weight so therefore tall people will weigh more than short
people. Since the points on the boys scatter gram are more dispersed
about the line of best fit than for girls. This suggests that the
correlation is better for girls and that girls' heights are more
predictable than boys. Also the relationships between height and
weight do not seem to be linear so a curved line of best fit would be
preferred. The cumulative frequency graphs prove that girls are
heavier than boys. However I think these correlations will be clearer
if I take a stratified sample and analyse the heights by gender and
year. Also if I analyse age and create 10% samples for every year
group I will get a better set of results.


New Hypothesis
--------------

I think that that girls will weigh less than boys for the same heights
but that

girls heights will be more predictable i.e. on the scatter gram the
results will be more bunched together and for boys the results will be
more spread out e.g. there will be boys at either extremes (some tall,
some short). I also think that older people will weigh more than
younger people. I will try and prove my hypothesises by taking a
stratified sample of the school so that age and gender bias are
eliminated and by taking a 10% sample of every year and analysing the
data.


Stratified samples
------------------

A stratified sample means that all the results are in ratio to each
other and reflect the spread of people in the school. To do this
stratified sample, I will find out what fraction of the population is
in each year and base my results on that. By taking a stratified
sample you can eliminate bias from gender and age.

Year group

Number of boys

Number of girls

Total

7

4

3

7

8

4

3

7

9

3

4

7

10

3

2

5

11

2

2

4

Total

16

14

30

Gender

Year group

Height(cm)

Weight(kg)

Male

7

1.54

38

Male

7

1.51

45

Male

7

1.40

40

Male

7

1.49

47

Male

8

1.57

51

Male

8

1.48

40

Male

8

1.72

53

Male

8

1.63

55

Male

9

1.67

60

Male

9

1.70

65

Male

9

1.72

90

Male

10

1.77

80

Male

10

1.57

64

Male

10

1.50

65

Male

11

1.52

45

Male

11

1.51

38

Female

7

1.61

45

Female

7

1.65

45

Female

7

1.42

34

Female

8

1.61

60

Female

8

1.57

45

Female

8

1.58

58

Female

9

1.8

57

Female

9

1.68

57

Female

9

1.51

65

Female

9

1.57

62

Female

10

1.73

42

Female

10

1.80

74

Female

11

1.59

45

Female

11

1.60

55

Mean boy's height = 1.58m


Mean boy's weight = 51kg
------------------------

Modal class interval of height=1.51-1.6m

Modal class interval of weight=31-40kg and 51-60kg

Mean girl's height=1.62m

Mean girl's weight=53kg

Modal class interval of height= 1.51-1.6m

Modal class interval of weight=41-50kg

Range of boy's heights= 0.37m

Range of boy's weight's= 52kg

Range of girl's heights= 0.38m

Range of girls weights= 32kg

Even now we do not have enough information to make valid statements
about each year group. I will rectify this problem by making a 10%
sample of each year group. I will start this by taking 10% of the Year
7 pupils.

[IMAGE]

[IMAGE]


Height(m)



Mean height: 1.6m
=================

Mean weight: 44.7kg

Median height: 1.52m

Median weight: 43.5kg

Modal height: 1.52m

Range of heights: 0.29m

Range of weights: 44kg

Weight(kg)

1.45

43

1.48

44

1.55

53

1.62

48

1.60

40

1.42

26

1.59

45

1.56

53

1.45

40

1.52

40

1.42



Deviation
=========

Another way to find the measure of spread is to measure the deviation
of a result from the mean. I will do this by subtracting the mean from
the height of the person.

41

1.50

41

1.41

45

1.36

38

1.52

37

1.54

43

1.52

25

1.64

50

1.61

45

1.62

40

1.61

47

1.65

45

1.63

45

1.58

50

1.56

35

1.48

42

1.49

38

1.65

69

Year 7

Height (m)

The mean deviation for the year 7 pupils is - 1.71 which shows many
people are below the average height of 1.6m

Deviation (m)

1.36

-0.24

1.36

-0.23

1.41

-0.18

1.41

-0.18

1.42

-0.17

1.42

-0.17

1.45

-0.14

1.45

-0.14

1.48

-0.11

1.48

-0.11

1.49

-0.1

1.5

-0.09

1.52

-0.07

1.52

-0.07

1.52

-0.07

1.54

-0.05

1.55

-0.04

1.56

-0.03

1.56

-0.03

1.58

-0.01

1.59

0

1.6

0.01

1.61

0.02

1.61

0.02

1.62

0.03

1.62

0.03

1.63

0.04

1.64

0.05

1.65

0.06

1.65

0.06

[IMAGE]

[IMAGE]


Final Summary

The stratified sample of 30 students shows there was a mean height for
boys of 1.58m and 1.62 for girls. However the range of heights for the
boys was much greater than that from the girls. This suggests there
will be many boys who are taller than the girl's mean of 1.62 but
there will be even more boys who are shorter. The correlation between
height and weight is not very clear in year 7 but the correlation is
still positive but only slightly positive. To find out a better and
more accurate result a sample from every year should be taken.

There is also a positive correlation between height and weight across
the school but it is strongest when genders and year groups are
considered separately. However it seems that the relationship is not
linear so a curved line of best fit should be more appropriate but
more analysis and sampling is needed for that as the number of people
sampled is not very high so a few freak results could upset the whole
collection of results e.g. in my main sample there was a girl who was
1.7m tall and 110 kg which changed the scatter gram's line of best fit
considerably.