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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.

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 x² 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 x² 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 x² y² 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 x² y² 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

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.

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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