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  • Level: GCSE
  • Subject: Maths
  • Word count: 10674

maths coursework-Height and Weight of Pupils and other Mayfield High School investigations

Extracts from this document...

Introduction

At a Mayfield High School

Introduction

This investigation is based upon the students of Mayfield High School, a fictitious school although the data presented is based on a real school. The total number of students in the school is 1183.

...read more.

Middle

43

1849

47

2209

39

1521

49

2401

59

3481

45

2025

43

1849

51

2601

68

4624

38

1444

32

1024

47

2209

60

3600

65

4225

35

1225

53

2809

44

1936

44

1936

75

5625

56

3136

75

5625

72

5184

54

2916

45

2025

1547

83663

µ = 51.56667

Standard deviation = √83663 – 51.56667²

                                        30

Standard deviation = 11.386 (3 d.p)

From this evidence I can see that the mean for boys’ weight is not a realistic way of interpreting the data and the mean is unreliable.

Year Group

Number of Boys

Number of Girls

Total

7

151

131

282

8

145

125

270

9

118

143

261

10

106

94

200

11

84

86

170

Boys Height

x

147

21609

164

26896

136

18496

171

29241

165

27225

151

22801

160

25600

162

26244

151

22801

170

28900

156

24336

152

23104

166

27556

165

27225

155

24025

160

25600

153

23409

170

28900

156

24336

169

28561

164

26896

156

24336

171

29241

163

26569

183

33489

174

30276

188

35344

179

32041

162

26244

192

36864

4911

808165

µ = 163.7

Standard deviation = √808165 – 163.7²

                                        30

Standard deviation = 11.88 (2 d.p)

I can see that my mean for boys’ height isn’t a good way to judge my data. It is unreliable as the standard deviation is quite high.

Girls Weight

x

47

2209

45

2025

53

2809

40

1600

47

2209

65

4225

38

1444

43

1849

50

2500

52

2704

51

2601

45

2025

40

1600

51

2601

72

5184

52

2704

51

2601

40

1600

40

1600

55

3025

48

2304

41

1681

52

2704

50

2500

52

2704

55

3025

42

1764

80

6400

64

4096

86

7396

1547

83689

µ = 51.56667

Standard deviation = √83689 – 51.56667²

                                        30

Standard deviation = 10.963 (3 d.p)

From the outcome of the standard deviation for girls’ weight, I can see that the mean for the girls’ weight isn’t a good way to interpret the data. The mean is unreliable.

Girls Height

x

161

25921

150

22500

172

29584

146

21316

148

21904

162

26244

143

20449

156

24336

160

25600

159

25281

162

26244

150

22500

143

20449

167

27889

165

27225

155

24025

145

21025

164

26896

153

23409

158

24964

170

28900

140

19600

152

23104

163

26569

178

31684

170

28900

173

29929

190

36100

189

35721

200

40000

4844

788268

µ=161.4667

Standard deviation = √788268 - 161.4667²

                                        30

Standard deviation = 14.287 (3 d.p)

The standard deviation for girls’ height is high and therefore I can not use the mean to judge my data. The mean is unreliable.

  1. From the results I have got for standard deviation I can see that the mean for girls and boy’s weights and heights isn’t a reliable way to interpret the data I have collected.

Product-moment correlation coefficient r (PMCC)

The product moment correlation coefficient is good for seeing how strong the correlations are on my scatter graphs. I can predict that the correlation for girls will be stronger than that for boys.

Formula: r =         Sxy    

√ (SxxSyy)

Sxy = ∑xy - ∑x∑y

                      n

Sxx = ∑x² - (∑x) ²

                      n

Syy = ∑y² - (∑y) ²

                      n

PMCC for boys

x

y

xy

1.47

41

2.1609

1681

60.27

1.64

50

2.6896

2500

82

1.36

45

1.8496

2025

61.2

1.71

49

2.9241

2401

83.79

1.65

64

2.7225

4096

105.6

1.51

59

2.2801

3481

89.09

1.60

43

2.56

1849

68.8

1.62

47

2.6244

2209

76.14

1.51

39

2.2801

1521

58.89

1.70

49

2.89

2401

83.3

1.56

59

2.4336

3481

92.04

1.52

45

2.3104

2025

68.4

1.66

43

2.7556

1849

71.38

1.65

51

2.7225

2601

84.15

1.55

68

2.4025

4624

105.4

1.60

38

2.56

1444

60.8

1.53

32

2.3409

1024

48.96

1.70

47

2.89

2209

79.9

1.56

60

2.4336

3600

93.6

1.69

65

2.8561

4225

109.85

1.64

35

2.6896

1225

57.4

1.56

53

2.4336

2809

82.68

1.71

44

2.9241

1936

75.24

1.63

44

2.6569

1936

71.72

1.83

75

3.3489

5625

137.25

1.74

56

3.0276

3136

97.44

1.88

75

3.5344

5625

141

1.79

72

3.2041

5184

128.88

1.62

54

2.6244

2916

87.48

1.92

45

3.6864

2025

86.4

49.11

1547

80.8165

83663

2549.05

r= (2549.05) - (49.11X1547)

                                 30                                .

      √ (80.8165) – (49.11)² X (83663) – (1547) ²

  1. 30

r = 16.611

  40.58170188

r = 0.409332

  1. I can see from calculating the PMCC, that my strength for the correlation between the two variables, height and weight, for boys is weak.

PMCC for girls

x

y

xy

1.61

47

2.5921

2209

75.67

1.50

45

2.25

2025

67.5

1.72

53

2.9584

2809

91.16

1.46

40

2.1316

1600

58.4

1.48

47

2.1904

2209

69.56

1.62

65

2.6244

4225

105.3

1.43

38

2.0449

1444

54.34

1.56

43

2.4336

1849

67.08

1.60

50

2.56

2500

80

1.59

52

2.5281

2704

82.68

1.62

51

2.6244

2601

82.62

1.50

45

2.25

2025

67.5

1.43

40

2.0449

1600

57.2

1.67

51

2.7889

2601

85.17

1.65

72

2.7225

5184

118.8

1.55

52

2.4025

2704

80.6

1.45

51

2.1025

2601

73.95

1.64

40

2.6896

1600

65.6

1.53

40

2.3409

1600

61.2

1.58

55

2.4964

3025

86.9

1.7

48

2.89

2304

81.6

1.4

41

1.96

1681

57.4

1.52

52

2.3104

2704

79.04

1.63

50

2.6569

2500

81.5

1.78

52

3.1684

2704

92.56

1.70

55

2.89

3025

93.5

1.73

42

2.9929

1764

72.66

1.90

80

3.61

6400

152

1.89

64

3.5721

4096

120.96

2.00

86

4

7396

172

48.44

1547

78.8268

83689

2534.45

r= (2534.45) - (48.44X1547)

                                 30                                .

      √ (78.8268) – (48.44)² X (83689) – (1547) ²

 30                              30

r = 36.56066667

     48.96490299

r = 0.74667

  1. I can see from the answer that my prediction was right. The correlation for girls’ height and weight is definitely stronger than that for boys. This tells me that there is a better relationship between height and weight for girls more than boys.

Conclusion from random sampling

  1. There is a positive correlation between height and weight. In general tall people will weigh more than smaller people.
  1. The points on the scatter diagram for the girls are less dispersed about the line of best fit than those for boys. This suggests that the correlation is better for girls than for boys.
  1. The points on the scatter diagrams for boys and girls are less dispersed than the points on the scatter diagram for mixed sample of boys and girls. This suggests that the correlation between height and weight is better when girls and boys are considered separately.
  1. I can use the scatter diagrams to give reasonable estimates of height and weight. This can be done either by reading from the graph or using the equations for the line of best fit.
  1. The cumulative frequency curves confirm that boys and girls have quite a close height and weight, with girls being slightly higher in weight and boys slightly higher in height.
  1. The median for boys is higher in height and the median for girls is higher in weight.
  1. From the box and whisker diagrams I can conclude that, in general boys are taller than girls, but not exclusively so. The cumulative frequency curves can be used to estimate that 23% of girls have a higher height than 172 cm, the upper quartile height of boys.
  1. Also from the box and whisker diagrams I can conclude that in general girls weigh more than boys but not exclusively so. The cumulative frequency curves can be used to estimate that 23% of boys have a higher weight than girls above 60 kg. This could also be a result of my sampling which has more students from year 7 and 8 then 9, 10 or 11. This could mean more lighter people than heavier people
  1. I could have had a greater confidence in these results if we had taken larger samples. Also, my predictions are based on general trends observed in the data. In both samples there were exceptional individuals whose results fell outside the general trend.
  1. When age is taken to consideration, the correlation between height and weight will be better than when age is not considered.

This was based upon 60 students sampled at random. To ensure that the students from different age groups are represented equally I will now take a stratified sample.

Stratified Sample

Year Group

Number of Boys

Number of Girls

...read more.

Conclusion

  1. The standard deviation showed me that the mean isn’t a reliable way of interpreting my data.
  1. The product-moment correlation coefficient shows that the correlation between height and weight is stronger for girls than for boys.

Final conclusion

  1. In general the taller a person is, the more they will weigh.
  1. There is a positive correlation between height and weight. In general tall people will weigh more than smaller people.
  1. The points on the scatter diagram for the girls are less dispersed about the line of best fit than those for boys. This suggests that the correlation is better for girls than for boys. Also, the points on the scatter diagrams for boys and girls are less dispersed than the points on the scatter diagram for mixed sample of boys and girls. This suggests that the correlation between height and weight is better when girls and boys are considered separately.
  1. There therefore is a positive correlation between height and weight across the school as a whole. This correlation seems to be stronger when separate genders are considered
  1. I can use the scatter diagrams to give reasonable estimates of height and weight. This can be done either by reading from the graph or using the equations for the line of best fit.
  1. There is a better relationship between height and weight when people in the school are taken into proportion in each year.
  1. I could have had a greater confidence in these results if we had taken larger samples. Also, my predictions are based on general trends observed in the data. In both samples there were exceptional individuals whose results fell outside the general trend.

This coursework was both interesting and enjoyable to do although it was hard work. I have learnt a few things from this coursework such as standard deviation and product-moment correlation coefficient, both of which I had previously not known about.

...read more.

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