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• Level: GCSE
• Subject: Maths
• Word count: 5617

# In this coursework I want to find out what the average height, weight, and arm span of a Year 11 pupil is

Extracts from this document...

Introduction

Philip Davies 11B8th February 2005

The Average Year 11 Pupil

Hypothesis

In this coursework I want to find out what the average height, weight, and arm span of a Year 11 pupil is. I think that my height, and arm span will be 10% more than the average Year 11 pupil, and my weight will be 5% more than the average year 11 Pupil, I want to find out if this is correct. I will also compare the height, weight, and arm span of an average Year 11 male to an average Year 11 female, because I think that an average Year 11 male will be 10% taller, heavier, and have a 10% longer arm span than an average Year 11 female. Finally, I want to find out if an average Year 11 pupil is square, and if an average Year 11 male is squarer than an average Year 11 female. The calculations and graphs that I am going to make are;

 Histogram Interquartile Range Mean Box & Whisker Mode (Modal Group) Standard Deviation Cumulative Frequency Graph Scatter Diagram Median

Each pupil in Year 11 measured their own height, weight, and arm span, and all the data was collected and put in a table. So the data we are using is Primary Data because we have collected it. The sampling which I used to get my data was systematic sampling. To get better results I left out any sets of data with missing measurements. I chose every 4th set of data and recorded it in a table. I decided to choose 25 male, and 25 female sets of data, so I will have 50 sets of data. Here is the data that I have selected;

 Gender Height (cm) Arm Span (cm) Weight (kg) M 187 187 76 M 169 165 55 M 186.4 165 73 M 176 170 57 M 183.5 195 64 M 183 182 65 M 174.5 160 76 M 177.5 176 59 M 180 185 65 M 155 162 46 M 189 200 97 M 178 177 78 M 176.5 175 70 M 165 170 47 M 180 184 58 M 164 164 55 M 182.5 178 92 M 166 168 52 M 172 179 91 M 172.5 168 76 M 180 179 65 M 178 163 65 M 169 167 63.5 M 185 185 64 M 172 175 60 F 163 151 52 F 181 181 65 F 157 163 53 F 160 155 43 F 165 160 70 F 156.5 162 42 F 172 172 62.5 F 167 162 51 F 165 159 50 F 166.7 171 70 F 161 159 60 F 167 168 69.5 F 165.5 170 70 F 164 167 60 F 164 170 65 F 163 155 53 F 162.5 147 60 F 172 165 64.5 F 167 170 65 F 156 165 57 F 157 155 59 F 152 155 47 F 160 166 58.5 F 161 160 58 F 165 165 55

Here are my measurements;

 Gender Height (cm) Arm Span (cm) Weight (kg) M 182 185 68

There are no sets of data in the table which stand out, so I believe the data is all genuine. The data that I have collected is Continuous data so I will have to group my data for the 3 measurements.

1.Height (All data)

I will now do all my calculations and graphs for Height. I will firstly do the calculations and graphs for all of my data, and then I will split the data into male and female and complete the calculations and graphs for them. Here is my table of grouped data for height, including all my data;

 Height (cm) Frequency 150 ≤ h < 155 1 155 ≤ h < 160 5 160 ≤ h < 165 10 165 ≤ h < 170 12 170 ≤ h < 175 6 175 ≤ h < 180 5 180 ≤ h < 185 7 185 ≤ h < 190 4

1.Histogram

From the Histogram above I can see that there are more pupils with a smaller height than pupils with a larger height. I can also see that there is a slight negative skew in the Histogram which shows that there is a bigger distribution of smaller pupils in Year 11.

2.Mean

Mean =         Sum of Height  =        8491.6  =  169.832cm

No. of Pupils           50

Compared to my height of 182cm, the mean height for Year 11 pupils is 169.832cm, which would make me above average height by 12.1cm. As a percentage this would mean I am 6.6% taller than the mean height of a Year 11 pupil, so my hypothesis was almost correct.

3.Mode (Modal Group)

Mode (Modal Group) = 165 ≤ h < 170 cm

The modal group height for a Year 11 pupil is 165 ≤ h < 170 which is

17cm < h ≤ 12cm smaller than my height which means I am 9.3% ≤ h < 6.6% taller than the average Year 11 pupil.

4.Cumulative Frequency Graph

Here is the table and graph for Cumulative Frequency;

 Height (cm) Frequency Cumulative Frequency 150 ≤ h < 155 1 1 155 ≤ h < 160 5 6 160 ≤ h < 165 10 16 165 ≤ h < 170 12 28 170 ≤ h < 175 6 34 175 ≤ h < 180 5 39 180 ≤ h < 185 7 46 185 ≤ h < 190 4 50

From the Cumulative Frequency graph above I can see that the gradient of the curve is steeper below the median height, and that the gradient is not as steep for the taller people above the median. This shows that the heights of the smaller pupils seem to be closer together, and the heights of the taller people seem to be more spread out.

5.Median

Median = 169cm

The median height of a Year 11 pupil is 169cm, this is 13cm (7.1%) smaller than my height. This shows my prediction of 10% is almost correct.

6.Interquartile Range

Interquartile Range   =   Q3 – Q1   =   178cm – 163cm   =   15cm

From this diagram and the Interquartile Range I can see that the middle 50% of the data covers almost exactly half of the total data range, so it has got a normal spread. I can also see that the lower 50% of the data is closer together, and the higher 50% of the data is more spread out.

7.Standard Deviation

Standard Deviation = 9.52

The Standard Deviation measures how the data are dispersed from the mean. So from this figure I can see that the data values are spread widely from the mean because the standard deviation is high, this means that there is a large dispersion.

From the calculations I have made for Height I have found out that I am between 6.6% and 9.3% taller than the average Year 11 Pupil. In my hypothesis I predicted I was 10% taller, so I was not far off.

1.Height (Male & Female)

I will now split the data into male and female because I want to find out if an average Year 11 Male has a 10% bigger height than an average Year 11 Female. I will repeat the calculations and graphs, and compare the two sets of answers I get as I go along. Here are the two tables of grouped data for Male and Female Heights;

 Male Data Female Data Height (cm) Frequency Height (cm) Frequency 150 ≤ h < 155 0 150 ≤ h < 155 1 155 ≤ h < 160 1 155 ≤ h < 160 4 160 ≤ h < 165 1 160 ≤ h < 165 9 165 ≤ h < 170 4 165 ≤ h < 170 8 170 ≤ h < 175 4 170 ≤ h < 175 2 175 ≤ h < 180 5 175 ≤ h < 180 0 180 ≤ h < 185 6 180 ≤ h < 185 1 185 ≤ h < 190 4 185 ≤ h < 190 0

Middle

Height (cm)

Frequency

Cumulative Frequency

150 ≤ h < 155

0

0

155 ≤ h < 160

1

1

160 ≤ h < 165

1

2

165 ≤ h < 170

4

6

170 ≤ h < 175

4

10

175 ≤ h < 180

5

15

180 ≤ h < 185

6

21

185 ≤ h < 190

4

25

Here is my table and graph for Female Cumulative Frequency for Height;

 Female Data Height (cm) Frequency Cumulative Frequency 150 ≤ h < 155 1 1 155 ≤ h < 160 4 5 160 ≤ h < 165 9 14 165 ≤ h < 170 8 22 170 ≤ h < 175 2 24 175 ≤ h < 180 0 24 180 ≤ h < 185 1 25 185 ≤ h < 190 0 25

From these graphs for Cumulative frequency I can see that the gradient of the curve for Males is steeper towards the end half of the curve, and the gradient of the curve for Females is steeper at the beginning and then the gradient is very low at the end. This shows that the concentration of smaller pupils is greater for Females, and the concentration for taller pupils is much greater for Male Year 11 pupils. I can also see from the Female Cumulative Frequency graph that there are very few tall female Year 11 pupils, and from the Male Cumulative frequency graph I can see that there are very few small male Year 11 pupils.

5.Median

Male Median Height = 177.5 cm

Female Median Height = 164 cm

From these figures I can see that the Male median Height for a Year 11 pupil is 13.5 cm or 7.6% taller that the median Height for a Year 11 Female, which is not far off my prediction of 10%.

6.Interquartile Range

Male Interquartile Range

= Q3 – Q1 = 183 cm – 170 cm = 13 cm

Female Interquartile Range

= Q3 – Q1 = 167.5 cm – 160.5 cm = 7 cm

Here are my Box and Whisker Diagrams;

From the Interquartile Range and the Box and Whisker diagrams I can see that the middle 50% of the Female height data has a much tighter distribution than the middle 50% of the male height data. From the diagrams I can see that the middle 50% of the male data covers the taller heights, and the middle 50% of the female data covers mainly smaller heights.

7.Standard Deviation

Male Standard Deviation = 8.255

Female Standard Deviation = 6.018

From these calculations I can see that the spread of the data from the mean for Females is much tighter than the spread of the data from the mean for Males.

From these calculations I have found out that an average Year 11 Male is between 7.1% and 11.1% taller than an average Year 11 Female. I will now move on to Weight.

2.Weight (All data)

I will now do all my calculations and graphs for Weight. I will firstly do them for all the data, I will then split it into male and female and repeat the calculations and graphs. Here is my table of grouped data for the weight of Year 11 pupils, including all my data;

 Weight (Kg) Frequency 40 ≤ w < 45 2 45 ≤ w < 50 3 50 ≤ w < 55 6 55 ≤ w < 60 10 60 ≤ w < 65 9 65 ≤ w < 70 8 70 ≤ w < 75 5 75 ≤ w < 80 4 80 ≤ w < 85 0 85 ≤ w < 90 0 90 ≤ w < 95 2 95 ≤ w < 100 1

1.Histogram

From this Histogram I can see that it has a negative skew, there are three outliers that can be seen on the histogram between 90 and 100. Also, the weights of Year 11 pupils have a tight distribution without the outliers.

2.Mean

Mean = Sum of Weight =        3129.5 = 62.59 Kg

No. of Pupils           50

Compared to my weight of 68 Kg, the mean weight of a Year 11 pupil is 5.41cm smaller than my weight. As a percentage this is 8.0%, which makes my prediction of 10% almost correct.

3.Mode (Modal Group)

Mode (Modal Group) = 55Kg ≤ w < 60Kg

The modal weight for an average Year 11 pupil is 55Kg ≤ w < 60Kg which is 13Kg ≤ w < 8Kg smaller than my weight of 68%. This means my weight is 19.1% ≤ w < 11.8% above average.

4.Cumulative Frequency Graph

Here is my table and graph for the Cumulative frequency for weight;

 Weight (Kg) Frequency Cumulative Frequency 40 ≤ w < 45 2 2 45 ≤ w < 50 3 5 50 ≤ w < 55 6 11 55 ≤ w < 60 10 21 60 ≤ w < 65 9 30 65 ≤ w < 70 8 38 70 ≤ w < 75 5 43 75 ≤ w < 80 4 47 80 ≤ w < 85 0 47 85 ≤ w < 90 0 47 90 ≤ w < 95 2 49 95 ≤ w < 100 1 50

From this graph I can see that the gradient of the curve is steep up to 80Kg, and then it hardly goes up at all. The flat line in the curve shows that no pupils from my data have a height of between 80Kg and 90Kg.

5.Median

Median = 62 Kg

The median weight of a Year 11 pupil is 6 Kg or 8.8% smaller than my weight of 68 Kg.

6.Interquartile Range

Interquartile Range   =   Q3 – Q1   =   69.5Kg – 56Kg   =   13.5Kg

From the Box and Whisker diagram and the Interquartile range I can see that the middle 50% of the data covers 25% of the total data, which shows that the data in the middle 50% is tightly distributed.

7.Standard Deviation

Standard Deviation = 11.671

From this figure I can see that the data has a high dispersion, and is spread quite far from the mean.

From all the calculations and graphs I have made for weight I have found out that I am between 8.0% and 19.1% heavier than the average Year 11 Pupil. In my hypothesis I predicted I would be 10% heavier, so my prediction was correct.

2.Weight (Male & Female)

I will now split the data into male and female because I want to find out if an average Year 11 Male is 10% heavier than an average Year 11 Female, I will do my calculations and graphs for both and compare the figures and results I get as I go along. Here are the two tables of grouped data for Male and Female Weights;

 Male Data Female Data Weight (Kg) Frequency Weight (Kg) Frequency 40 ≤ w < 45 0 40 ≤ w < 45 2 45 ≤ w < 50 2 45 ≤ w < 50 1 50 ≤ w < 55 1 50 ≤ w < 55 5 55 ≤ w < 60 5 55 ≤ w < 60 5 60 ≤ w < 65 4 60 ≤ w < 65 5 65 ≤ w < 70 4 65 ≤ w < 70 4 70 ≤ w < 75 2 70 ≤ w < 75 3 75 ≤ w < 80 4 75 ≤ w < 80 0 80 ≤ w < 85 0 80 ≤ w < 85 0 85 ≤ w < 90 0 85 ≤ w < 90 0 90 ≤ w < 95 2 90 ≤ w < 95 0 95 ≤ w < 100 1 95 ≤ w < 100 0

1.Histogram

From these two Histograms I can see that the data for females seems to show a tighter distribution than the data for males. I can also see from these graphs that the outliers with a high weight all seem to be males. Overall, the gender with mainly small weights seems to be females, and the Year 11 males seem to have heavier weights.

2.Mean

Mean Male Weight = Sum of Weight = 1669.5 = 66.76 Kg

No. of Pupils        25

Mean Female Weight = Sum of Weight = 1460 = 58.4 Kg

No. of Pupils        25

From these results I can see that the mean weight for a Year 11 male is 8.36 Kg more than the mean weight for a Year 11 Female, which is 12.52% heavier on average.

3.Mode (Modal Group)

Male Modal Weight = 55 ≤ w < 60 Kg

Female Modal Weight = 50 ≤ w < 65 Kg

From these figures I can see that the mode for females covers a larger range of weight than the mode weight for males, this is because the frequency for females is the same for three columns in my Histogram.

4.Cumulative Frequency Graph

Here is the table and graph for Male Cumulative Frequency for Weight;

 Male Data Weight (Kg) Frequency Cumulative Frequency 40 ≤ w < 45 0 0 45 ≤ w < 50 2 2 50 ≤ w < 55 1 3 55 ≤ w < 60 5 8 60 ≤ w < 65 4 12 65 ≤ w < 70 4 16 70 ≤ w < 75 2 18 75 ≤ w < 80 4 22 80 ≤ w < 85 0 22 85 ≤ w < 90 0 22 90 ≤ w < 95 2 24 95 ≤ w < 100 1 25

Here is the table and graph for Female Cumulative Frequency for Weight;

 Female Data Weight (Kg) Frequency Cumulative Frequency 40 ≤ w < 45 2 2 45 ≤ w < 50 1 3 50 ≤ w < 55 5 8 55 ≤ w < 60 5 13 60 ≤ w < 65 5 18 65 ≤ w < 70 4 22 70 ≤ w < 75 3 25 75 ≤ w < 80 0 25 80 ≤ w < 85 0 25 85 ≤ w < 90 0 25 90 ≤ w < 95 0 25 95 ≤ w < 100 0 25

Conclusion

If I was to do this coursework again I would take all of the measurements myself and measure them in the same way to make it fair. I would also take a larger set of data which would hopefully give more accurate results.

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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