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• Level: GCSE
• Subject: Maths
• Word count: 5507

Investigating the relationship between height and weight in a sample of 60 pupils.

Extracts from this document...

Introduction

In this coursework project I will be investigating the relationship between height and weight in a sample of 60 pupils from Mayfield high school. I have chosen 30 boys and 30 girls at random from the list of pupils. The total number of pupils at the school is 1183, and the number of pupils that were randomly chosen is roughly 5% of the total. I believe that a random sample of 60 students is adequate. Even though a bigger sample might be more useful in exploring the data, if I select too many people, the data will be difficult to analyse. My hypothesis states that the taller the person, the more they will weigh. However, I have also taken into consideration that there might be some exceptions with certain individuals, eg some pupils might be shorter and heavier than others. To find evidence to investigate my hypothesis, I will create tally charts, histograms,stem and leaf diagrams and more.

Firstly, I present my random sample of 30 boys and 30 girls:

 Boys Girls Height (cm) Weight (kg) Height (cm) Weight (kg) 177 72 183 60 172 64 165 42 175 45 103 45 164 60 163 48 178 37 170 54 170 57 175 60 185 73 167 48 166 63 150 45 162 56 163 44 175 65 165 48 173 48 180 74 183 75 155 50 180 68 165 54 180 59 151 36 162 92 162 56 154 42 149 40 170 55 160 42 153 45 175 47 161 48 165 52 150 50 162 45 161 46 162 46 173 66 162 49 160 44 160 57 170 65 155 36 167 53 154 42 158 57 157 38 180 54 140 48 160 55 167 52 154 39 175 40 173 66 169 48

This data consists of a sample of 15 boys in Key Stage 3, 15 boys in Key Stage 4, 15 girls in Key Stage 3 and 15 girls in Key Stage 4.

The pupils have been randomly selected by a computer, using Excel’s =RAND formula, thus minimising bias.

The data is correct, however it is not clearly organized. I am planning on storing it in a more efficient way. This can be achieved by creating frequency tables for boys and girls, seperately.

Weight

Boys (Yr 7-11)

 Weight, w (kg) Tally Frequency 35 ≤ w < 40 2 40 ≤ w < 45 2 45 ≤ w < 50 5 50 ≤ w < 55 3 55 ≤ w < 60 6 60 ≤ w < 65 3 65 ≤ w < 70 5 70 ≤ w < 75 2 75 ≤ w < 80 1 80 ≤ w < 85 0 85 ≤ w < 90 0 90 ≤ w < 95 1

This table shows us that the spread of weight for boys in Key Stage 3+4 is greatly extended by 1 individual. This also increases the range.

The group that has the most boys is the one with weights ranging from 55-59 kg.

Girls (Yr 7-11)

 Weight, w (kg) Tally Frequency 35 ≤ w < 40 3 40 ≤ w < 45 6 45 ≤ w < 50 11 50 ≤ w < 55 5 55 ≤ w < 60 2 60 ≤ w < 65 2 65 ≤ w < 70 0 70 ≤ w < 75 1

We can clearly see from this table that the most common weight range is between

45-49 kg.

First I will be looking at the weight more closely. The data above has been sorted out into groups. Next, I am planning to draw histograms which will enable me to compare the results more easily.

 From the graphs above and below this statement I have noticed that the modal weight for the girls in my sample was lower than the modal weight for the boys.   The evidence from the sample suggests that the spread for boys’ weight is much greater than for girls. However there are a fewer boys who range between 35-44kg than girls.From the graphs I can make basic comments. The most common weight range for the boys was 55-60 kg, with 6 pupils falling into that range, while the most common weight range for the girls was 45-50 kg, with 11 (37% of the girls) falling into that  range.

This graph is a dual bar chart. It allows me to compare the 2 sets of data- weight for boys and weight for girls.

This frequency polygon will help me compare the weights between boys and girls.

Now I will attempt to find the median. To do this I will record the data in a stem and leaf diagram which will make it easier to read it off.

Boys (Yr 7-11)

 Weight (kg) (Stem) Leaf Frequency 30 7 8 2 40 5 8 2 5 8 6 4 7 50 7 6 9 5 0 3 7 4 5 9 60 4 0 3 5 8 6 5 6 8 70 2 3 5 3 80 0 90 2 1

-ordered:

 Weight (kg) (Stem) Leaf Frequency 30 7 8 2 40 2 4 5 5 6 8 8 7 50 0 3 4 5 5 6 7 7 9 9 60 0 3 4 5 5 6 6 8 8 70 2 3 5 3 80 0 90 2 1

Girls (Yr 7-11)

 Weight (kg) (Stem) Leaf Frequency 30 6 6 8 3 40 2 5 8 8 5 4 8 0 2 7 5 6 9 2 8 0 8 17 50 4 0 4 6 2 7 2 2 60 0 0 2 70 4 1 80 0 90 0

-ordered:

 Weight (kg) (Stem) Leaf Frequency 30 6 6 8 3 40 0 0 2 2 2 4 5 5 5 6 7 8 8 8 8 8 9 17 50 0 2 2 4 4 6 7 7 60 0 0 2 70 4 1 80 0 90 0

Mean Weight

Mean weight for boys = 1718 /30 = 57.26

Mean weight for girls = 1446 /30 = 48.2

Modal Weight

*From my stem and leaf diagrams I have found out that there can be several modes   for boys:

45, 48, 55, 57, 65, 66. As there are so many modes on a data that I have, I cannot state a single mode without getting more data, however, I can tell the modal class interval.

Modal weight Group for boys = 50-59 kg

Modal weight Group for girls = 40-49 kg

Median Weight

Median weight for boys =

(30 +1) /2 = 15.5 = (15th + 16th) / 2 =  (56 + 57) / 2  =  56.5

Median weight for girls =

(30 +1) / 2 = 15.5 = (15th + 16th) / 2 = (48 + 48) / 2 = 48

Range for weight

Weight range for boys =  92  - 37 = 55 kg

Weight range for girls =  74 -  36 =  38 kg

To summarize those results here I have created a table:

 Weight (kg) Mean Median Modal class interval Range Boys 57.26 56.5 50-59 55 Girls 48.2 48 40-49 38

The two measures of average (mean and median) are greater for boys than for girls. The range for weight for boys is greater than for girls. In conclusion, although there are a few boys who weigh low and a few girls that weigh high, the evidence suggests that in general, the weights for boys are greater than the weights for girls.

Evidence from the sample suggests that 6 out of 30, or 20% of boys weigh in the range of 55-60 kg and that 11 out of 30, or 37% of girls weigh in the range of          45-50 kg

Height

Boys (Yr 7-11)

 Height, h (cm) Tally Frequency 150 ≤ h < 160 5 160 ≤ h < 170 9 170 ≤ h < 180 11 180 ≤ h < 190 5

Girls (Yr 7-11)

 Height, h (cm) Tally Frequency 100 ≤ h < 110 1 110 ≤ h <  120 0 120 ≤ h < 130 0 130 ≤ h < 140 0 140 ≤ h < 150 2 150 ≤ h < 160 6 160 ≤ h < 170 15 170 ≤ h < 180 4 180 ≤ h < 190 2

This table shows us that the range is extended by a single individual. This will affect the the average by a certain amount. Still, we can see that 15 out of the 30 girls (50%) weigh in the range from

160-169 kg.

Now I will draw histograms to present my data more clearly:

If we  compare the 2 bar charts, we can clearly see thatthe spread for girls is much greater than the spread for boys. The girls also have the highest frequency in a single group, 15, while boys’ highest frequency is 11.

This graph is a dual bar chart. It allows me to compare the 2 sets of data- weight for boys and weight for girls.

Next, I will do the stem and leaf diagrams to find out the median for the height.

Boys (Yr 7-11)

 Height  (cm) (Stem) Leaf Frequency 150 4 3 0 8 4 5 160 4 6 2 2 1 6 0 7 0 9 170 7 2 5 8 0 5 3 0 3 0 3 11 180 5 3 0 0 0 5

-ordered:

Height  (cm) (Stem)

Middle

-ordered:

 Height (cm) (Stem) Leaf Frequency 100 3 1 110 0 120 0 130 0 140 0 9 2 150 0 1 4 5 5 7 6 160 0 0 2 2 2 2 3 3 5 5 5 5 7 7 9 15 170 0 5 5 5 5 5 180 0 3 2

Mean Height

Mean height for boys = 5051 /30 = 168.36

Mean height for girls = 4829 /30 = 160.96

Modal Height

Modal height Group for boys =  170- 179

Modal height Group for girls = 160- 169

Median Height

Median height for boys =

(30 +1) / 2 = 15.5 = (15th + 16th) / 2 =  (170 + 170) /2 = 170

Median height for girls =

(30 + 1) / 2 = 15.5 = (15th + 16th) / 2 = (162 + 163) /2 = 162.5

Range for height

Height range for boys = 185 – 150 = 30 cm

Height range for girls =  183 – 103 = 80 cm

To summarize those results here I have created a table:

 Height (cm) Mean Median Modal class interval Range Boys 168.36 170 170- 179 30 Girls 160.96 162.5 160- 169 80

The three measures of average (mean, median and the mode) are greater for boys than girls. However, the spread for girls’ height is significantly greater than for boys because of a particular individual, which is 80cm, compared to the boys which is 30cm In general the height for boys is greater than the height for girls.

The evidence from the sample suggests that 11 out of 30,  37% of the boys’ heights are between 170-179 cm, whilst 15 out of 30 girls, 50% of the girls’ heights are between 160-169 cm.

To compare the data graphically, I have drawn a frequency polygon with 2 lines: 1 for boys and 1 for girls.

These conclusions are based on a sample of only 30 girls and 30 boys. I could extend the sample or repeat the whole exercise to confirm my results, and make them more informative.

To sum up and see whether there is relationship between height and weight, I have chosen a sample of 30 students from Key Stage 3 and from those results I have created a scatter diagram:

 Height (cm) Weight (kg) 149 40 160 42 175 47 165 52 162 45 162 46 162 49 160 57 155 36 154 42 147 38 140 48 167 52 175 40 169 48 154 42 170 55 153 45 161 48 150 50 166 46 173 66 160 44 170 65 167 53 158 57 180 54 160 55 154 38 173 66

There is a positive correlation between height and weight. This tells us, as I predicted in  my introduction, that the taller the person, the more they will weigh. This is a qualititive statement.

To support my hypothesis, I will use Spearman’s rank corellation

later on in this coursework to quantify this result.

Further Investigation

I reckon that the correllation between height and weight will be better if I consider boys and girls seperately.

I have created the scatter diagrams (the last 3 pages) and the evidence shows that there is a stronger correllation between shoe size and height if boys and girls are considered seperately.

I have drawn by hand the lines of best fit. Those lines predict that a girl who was 172cm tall would weigh 52 kg, whereas a boy of the same height would weigh 64 kg.

The above values have been read from the graph. To be more precise I can use the formula  y= mx + c.

If y represents the height in cm, and x represent weight in kg, the equations of the lines of best fit for my data set are:

Boys only:  y =mx + c

Using the triangle from the boys scatter diagram, the change in y = 10, the

change in x = 12. Gradient = change in y / change in x , therfore, 10/12 =

5/6.

 5 6

y=     x + c

To find c, I’m using the point (144,40) which lies on the best fit line. I am

substituting these values for x and y

 5 6

40 =x 144 + c

40 = 120 + c

40 – 120  = c

 5 6

c = -80

The formula is y = x - 80

Where x = height (cm)

Where y = weight (kg)

Girls only: y =mx + c

Using the triangle from the boys scatter diagram, the change in y = 8, the

change in x = 18. Gradient = change in y / change in x , therefore, 8/18 =

4/9.

 4 9

y= x + c

To find c, I’m using the point (150,42) which lies on the best fit line. I am

substituting these values for x and y

 4 9

52 = x 150 + c

 2 3

52 =  66 +  c

 2 3

52 – 66  = c

 2 3

c= -14

 4 9 2 3

The formula is y =  x   -      14

Combined sample: y = mx + c

Using the triangle from the boys scatter diagram, the change in y = 18, the

change in x = 18. Gradient = change in y / change in x , therefore, 18/18 =

1.

y =1 x + c

To find c, I’m using the point (166,52) which lies on the best fit line. I am

substituting these values for x and y

52 = 1 x 166 + c

52 = 166 + c

52 - 166 = c

c =  -114

The formula is y = x – 114

Where x = Height (cm)

Where y = Weight (kg)

The equations above are to make a prediction of weight when I  know height, and vice versa. I can predict a boy’s weight who is 166cm tall using the formula

 5 6

y  =  x  - 80

 5 6

y  = x 166 – 80

 1 3

y   =   58

so the weight of the boy who is 166cm tall would be roughly 58kg to the nearest whole number.

The line of best fit is the best estimation of relationship between height and shoe size. There are exceptional values in my data, for example the girl who is 103cm tall and weighs 45 kg which fall outside the genereal trend. The line of best fit is a continous relationship between height and weight in this case, however both of them are rounded to the nearest whole numbers.

A good idea would be to draw a cumulative frequency graph, as it is going to help me compare different data sets. The table below shows  the cumulative frequency for weight for boys, girls, and for the combined sample.

 Weight (kg) Cumulative Frequency Boys Girls Combined <40 2 3 5 <45 4 9 13 <50 9 20 29 <55 12 25 37 <60 18 27 45 <65 21 29 50 <70 26 29 55 <75 28 30 58 <80 29 30 59 <85 29 30 59 <90 29 30 59 <95 30 30 60

 Weight (kg) Median Lower Quartile Upper Quartile Interquartile Range Boys 58 48 66 18 Girls 48 44 52 8 Combined 51 46 60 14

Next, I will create a table for height, and draw a graph just like I did with weight.

 Height (cm) Cumulative Frequency Boys Girls Combined <110 0 1 1 <120 0 1 1 <130 0 1 1 <140 0 1 1 <150 0 3 3 <160 5 9 14 <170 14 24 38 <180 15 28 43 <190 30 30 60

 Height (cm) Median Lower Quartile Upper Quartile Interquartile Range Boys 181 163 185 22 Girls 164 157 168 11 Combined 167 161 182 21

I can now estimate, thanks to the cumulative frequency graphs, how many boys in the school are between 160cm and 170cm tall.

14 – 5 = 9

9/30 = 0.3

This means that 30% of the boys in Mayfield high school will be between 160 and 170 cm tall. This also can tell us that the probability of choosing a boy at random from the sample who’s height is in the range between 160 and 170 cm tall is 0.3

The median that I have worked out is slightly different than the median that I have recorded from the graphs. I will use the median from the working out because it is more precise. The median height for boys is 170cm. If I draw a line going upwards from the x axis to the curve for girls’ height, then I will read off how many girls in the sample had less than 170cm. 24 out of 30 girls i.e.  80% of the girls are shorter than the median height for boys. This means that 6 girls out of 30 i.e.  20% of the girls are taller than the median height for boys. In general, boys are taller than girls but as  this evidence shows, 20% of girls have a height greater than the median height of the boys.

I have analysed my data and I have made quite a lot of findings about my data. Firstly, in all circumstances (boys, girls and mixed) there is a positive corellation between height and weight. In general, the taller the person is, no matter if its a girl or a boy, the more they will weigh. If we look on the scatter diagrams we can notice that the points on the one for girls are more dispersed than for boys. This means that the corellation for boys is better than for girls. The points  on the scatter diagrams for boys and girls are less dispersed than the points on the scatter graph for the mixed sample of boys and girls. This suggests that the correllation between height and weight is better when boys and girls are considered seperately. The overall relationship between height and weight is linear, however there are a couple of points that do not fit the trend. We can predict a person’s height or weight just by looking at a scatter diagram. The line of best fit will  give a rough estimate or the equation of a line to give a more precise result, using  y=mx +c formula. Taking a look at the cumulative frequency curves, it is clearly shown that boys are heavier than girls. The median height and weight are also higher than the median height and weight for boys. Analysing this data is limited for the reason that my data is limited, and I could only choose a random sample of 30 boys and 30 girls. In a few cases there have been odd results, for example a girl that is 103 cm tall. Individuals like that fall out of the trend.

Considering Age

The sample that  I have chosen at the beginning of this coursework was partly stratified. However, this data was bias and unprecise. I have taken boys from year     7-11 into 1 group, and I done the same for girls. If I consider age then this will reduce bias, because height and weight are affected by age. Corellation is better when age is taken into consideration. This is why I will stratify my data so that I will reduce bias as much as possible. The table below shows the spread of pupils across each year.

 Boys Girls Total Year 7 151 131 282 Year 8 145 125 270 Year 9 118 143 261 Year 10 106 94 200 Year 11 84 86 170

To take a stratified sample, I must divide the total numer of boys and girls by the total number of pupils in the school and multiply it by 60 (number of pupils in the  sample).

 BoysYear 7 = 151/1183 x 60 = 7.6 ~  8Year 8 = 145/1183 x 60 = 7.3 ~ 7Year 9 = 118/1183 x 60 = 5.98 ~ 6Year 10 = 106/1183 x 60 = 5.3 ~ 5Year 11 = 84/1183 x 60 = 4.2 ~ 4 GirlsYear 7 = 131/1183 x 60 = 6.6 ~ 7Year 8 = 125/1183 x 60 = 6.33 ~ 6Year 9 = 143/1183 x 60 = 7.2 ~ 7Year 10 = 94/1183 x 60 = 4.7 ~ 5Year 11 = 86/1183 x 60 = 4.36 ~ 5

Conclusion

In the year 11’s case, we  can see that the mean deviation for boys is enormous. This is due to a certain individual who is short, as well as another individual who is tall. When we look at girls however, their spread isn’t that great, yet it is still greater than the spreads in year 7.

Concluding from this I can say that by the time pupils get to year 11 they might grow at different rates. Also, when they reach this stage, boys tend to be significantly taller than girls.

Conclusion

• A sample of 60 students (30 from each sex) stratified over age and gender showed a mean height of 168 cm for boys and 161cm for the girls. Although the mean height for boys was greater, the range of boys’ height was also greater, which suggests that there would be boys who would be below the girls’ average, thus shorter than some girls.
• We can see that there is a positive corellation between the height and the weight, however, there are some points which do not exactly fit the pattern. This is due to some exceptions, for example a person might not be as tall, or they might not be very heavy. These are just some of the reasons.
• At a 10% sample of boys and girls, both in year 7 and 11 suggests that boys and girls in year 7 are different with a range of 8cm between. In year 11 however, I noticed this spread to increase up to 20cm, which suggests that boys tend to grow at a faster rate and are more taller than they used to be.
• The taller the person, the more they will weigh.

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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