Although I have noticed a trend between height and weight it is not completely clear, this may be because the data is rather biased.
Although I have noticed a trend between height and weight it is not completely clear, this may be because the data is rather biased. For my first part of the investigation I used data of 30, although I did use separate girls and boys data, it is not completely reliable as it was a mixture year groups. Looking at the data now, I can see that there is not many year tens or elevens, the older people who I feel will be heavier and tall than the year groups who appear frequently in the sample, year sevens and eights, the younger ones. This shows the original samples to be a biased sample. To eliminate this bias I have taken a 10% sample for each year group and also the different sexes within the year groups, giving me ten different sets of data. I predict to see that there will be a much clear trend between height and weight between girls and boys, it will be much clearer to see that boys are taller than girls and boys also weigh more than girls. To do this I will draw cumulative frequency tables for the girl's heights and weights, and then the same for the boys for the year seven data. Then I will transfer the data on to cumulative frequency graphs and compare the data. I will repeat this for all of the year groups.
Year 7
Height
Girls
Height (cm)
Tally
Total
Cumulative Frequency
< 130
/
< 140
//
2
3
< 150
////
4
7
< 160
////
5
2
< 170
//
2
4
< 180
/
5
Boys
Height (cm)
Tally
Total
Cumulative Frequency
< 150
///
3
3
< 155
////
4
7
< 160
/
8
< 165
//// //
7
5
The cumulative graph I have drawn from the data shows both the boys and girls data. The curve for the girls is rather steep; this shows that most of the girls are about the same height. The line for the boys is even steeper; it does not curve as much as the curve for the girls it is more of a line than a curve. This shows that there is a rapid increase in the number of people of the particular height. With this line being so short and steep it shows that most people are of the same heights. With the help of the data in the tables above, ad the graph I have draw I can find out the mean, mode, median and inter quartile ranges. This will give me an idea as to what height most people are.
Mean
Girls: 149cm
Boys: 153cm
To find out the mode I will draw a stem and leaf diagram.
Girls
2
5
3
2 2
4
5 7 8
5
0 2 2 3 9
6
2 4
7
3
Boys
4
9
5
0 0 1 2 3 3 7
6
1 1 2 2 2 3
Mode
Girls: 152cm
Boys: 161cm
Now I will find out the median, upper quartile, lower quartile and inter-quartile range, to do this I will use the graph I have drawn.
Median (Q2)
Girls: 151cm
Boys: 157cm
Lower Quartile (Q1)
Girls: 142cm
Boys: 150cm
Upper Quartile (Q3)
Girls: 159cm
Boys: 169cm
Inter-quartile Range
Girls: 159-142=17
Boys: 169-150=19
By working out the mean, median and mode I can see a clear indication that boys are taller than girls in the year seven data. The mean, median and mode for the girls are all very close; the difference between the highest and lowest figure is only 3. This gives me a very clear indication of the average height of the girls in year 7. The height for boys differs slightly more though, from 153cm to 161cm. The inter-quartile range tells me the height at which most people are, this is smaller for the girls so this must show that more girls are of the same height. From this I can conclude that boys are taller than girls, therefore proving my hypothesis to be correct.
Weight
I am now going to investigate the weights of the year seven boys and girls, to prove that boys weigh more than girls. To do this I will draw cumulative frequency tables and graph, and then using them find the mean median and mode to prove my hypothesis. These will help me to prove my hypothesis as by finding the mean median and mode I have less figures to work with, but the figures were found using the original data.
Girls
Weight (kg)
Tally
Total
Cumulative Frequency
< 35
/
< 45
//// ////
9
0
< 55
////
4
4
< 65
/
5
Boys
Weight (kg)
Tally
Total
Cumulative Frequency
< 30
/
< 40
///
3
4
< 50
//// /
6
0
< 60
////
5
5
The cumulative frequency graph (see back of graph of height) I have drawn shows the weights of the boys and girls in year seven. By looking at it at a glance I can see that boys weigh more than girls. Both the curve for the boys and the one for the girls are short and steep, this shows that most of the boys are the same height as each other, and the same with the girls. I will now find out the mean and mode, then using the graph, the median upper quartile, lower quartile and the inter-quartile range. These will give me an idea as to how much the girls and the boys weight separately.
Mean
Girls: 44kg
Boys: 47kg
To find out the mode I will draw a stem and leaf diagram.
Girls
3
5 7 8 9
4
0 0 0 0 1 4
5
0 2 2 2
6
2
Boys
2
5
3
8 9
4
0 2 2 3 4 8
5
0 5 6 6 9
6
0
Mode
Girls: 40kg
Boys: 42kg
I will now use the cumulative frequency graph to find the median, upper and lower quartiles and the inter-quartile range. These will tell me between which weights most people are. The people who come within the upper and lower quartiles are the ones who weights are more significant. Those that are higher or lower than the upper or lower quartile are anomalies. The data that comes within the inter-quartile range is the information about the spread of results for the middle 50% of the populations and ignore the extremes.
Median (Q2)
Girls: 43kg
Boys: 45kg
Lower Quartile (Q1)
Girls: 40kg
Boys: 40kg
Upper Quartile (Q3)
Girls: 48kg
Boys: 52kg
Inter-quartile Range
Girls: 48-40=8
Boys: 52-40=12
Range
Girls: 62-35=27
Boys: 60-25=35
After taking the mean, mode and median I can see that all three measures of average in the sample were higher for the boys than girls; although the sample for the boys was more spread out, with a range of 35kg, compared to 27kg for the girls. The evidence from the sample suggests that 6 out of 15 or 40% of the boys weigh between 50kg and 60kg, whilst 4 out of 15 girls or 27% weigh the same, between 50kg and 60kg. This shows that boys are heavier than girls as most of the girl, that is 7 out 15 or 47% weigh between 40kg and 50kg.
From my investigation of year 7 girls and boys I can conclude that this shows the boys to be taller and heavier than the girls. I will continue this investigation for the other years, to see if the pattern continues.
Year 8
Height
Girls
Height (cm)
Tally
Total
Cumulative Frequency
< 145
//
2
2
< 150
0
2
< 155
//
2
4
< 160
//
2
6
< 165
////
4
0
< 170
0
0
< 175
///
3
3
Boys
Height (cm)
Tally
Total
Cumulative Frequency
< 130
//
2
2
< 140
/
3
< 150
/
4
< 160
////
4
8
< 170
///
3
1
< 180
////
4
5
The curves on the cumulative frequency graph clearly show a trend that shows as the height increases so does the number of people. This shows that many of the people; both boys and girls, are taller than about 150cm. I am now going to find out the range, mean and mode using the tables above, and also the original data.
Mean
Girls: 160cm
Boys: 157cm
To find the mode I will draw a stem and leaf diagram as this way I have the data set out in an easy to read format.
Girls
4
2 4
5
5 5 7
6
0 2 2 2 3
7
2 5 5
Boys
2
5 6
3
5
4
5
0 2 5 5 8
6
9
7
0 0 2 2 2 2
Mode
Girls: 162cm
Boys: 172cm
Range
Girls: 175-142=33
Boys: 172-125=47
I will now use the cumulative frequency graph to find the median, upper and lower quartiles and the inter-quartile range.
Median (Q2)
Girls: 164cm
Boys: 159cm
Lower Quartile (Q1)
Girls: 154cm
Boys: 146cm
Upper Quartile (Q3)
Girls: 173cm
Boys: 170cm
Inter-quartile Range
Girls: 173-154=19
Boys: 170-146=24
By using the cumulative frequency graph I can estimate how many boys or girls are of a certain height. If i select a boy at random from the school, the data suggests that the probability of him having a height between 160cm and 170cm is 0.7.
After taking the mean, mode and median I can see that two out of the three measures of average in the sample were higher for girls than boys. The mode is the only one to say that boys were taller than girls. Although the sample for the boys was more spread out, with a range of 47cm, compared to 33cm for the girls. The evidence from the sample suggests that 6 out of 13 or 46% of the girls are between 160cm and 170cm, whilst 3 out of 15 boys or 20% are of the same height, between 160cm and 170cm. This shows that girls are taller than boys as most of the boys, are smaller. This does not prove my hypothesis to be correct.
I have drawn a box and whiskers diagram to show the minimum and maximum values, the median and the upper and lower quartiles. It shows that the girls inter-quartile range is less than the boys. This suggests that the boy's heights were more spread out than the girls. The median height for the boys is 159cm. From my graph I have found out that 6 girls in the sample had height less than 159cm. Therefore 7 out of 13 girls have a height greater than the median height for the boys. This is 7/13 or 54%.
Whilst ...
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I have drawn a box and whiskers diagram to show the minimum and maximum values, the median and the upper and lower quartiles. It shows that the girls inter-quartile range is less than the boys. This suggests that the boy's heights were more spread out than the girls. The median height for the boys is 159cm. From my graph I have found out that 6 girls in the sample had height less than 159cm. Therefore 7 out of 13 girls have a height greater than the median height for the boys. This is 7/13 or 54%.
Whilst the 10% sample for the year 7 people suggests that boys are taller than girls, I have evidence to suggest other wise. From the sample of year 8 people I have found out that 54% of the girls are taller than the median for the boys. After finding this out for the year 8 people I now predict that the girls will weigh more than the boys in this sample.
Weight
Girls
Weight (kg)
Tally
Total
Cumulative Frequency
< 50
//// ///
8
8
< 60
///
3
1
< 70
0
1
<80
//
2
3
Boys
Weight (kg)
Tally
Total
Cumulative Frequency
< 30
/
< 40
0
< 50
//// //
7
8
< 60
////
4
2
< 70
///
3
5
The curves on the cumulative frequency graph clearly show a trend that shows as the weight increases so does the number of people; this then starts to even off. It shows that many of the people; both boys and girls, weight more than about 50kg. I am now going to find out the range, mean and mode using the tables above, and also the original data.
Mean
Girls: 53kg
Boys: 50kg
To find out the mode I will draw a stem and leaf diagram as this way I have the data set out in an easy to read layout.
Girls
4
2 2 5 5 6 9
5
0 0 1 3
6
0
7
2
8
0
Boys
2
9
3
4
2 2 4 6 7 9
5
2 5 7 9
6
4 4
Mode
Girls: 50kg
Boys: 42kg
Range
Girls: 80-42=38
Boys: 64-29=35
I will now use the cumulative frequency graph to find the median, upper and lower quartiles and the inter-quartile range.
Median (Q2)
Girls: 49kg
Boys: 49kg
Lower Quartile (Q1)
Girls: 40kg
Boys: 43kg
Upper Quartile (Q3)
Girls: 74kg
Boys: 60kg
Inter-quartile Range
Girls: 60-40=20kg
Boys: 74-43=31kg
By using the cumulative frequency graph I have drawn I can pick a girl at random and by using the curve of best fit I can see that the probability of her weighing between 50 and 60kg is 0.5.
After taking the mean, mode and median I can see that two out of the three measures of average in the sample were higher for girls than boys. The median is the only one to show that boys weight the same as girls. The sample for the girls was more spread out than the boys, with a range of 38kg, compared to 35kg for the boys. The evidence from the sample suggests that 8 out of 13, or 62% of the girls weigh between 40 and 50kg. For the boys, 7 out of 15, or 47% weigh between the same figures, 40 and 50kg. It is between these figures that most of the boys weigh, it is also the same for the girls. This shows that the figures for the boys are more spread out. Although there are more boys than girls who weigh above 50kg; there being only 5 out of 13, or 38% of girls weighing above 50kg compared to 7 out of 15 boys or 47% weigh above 50kg. Many of the girls weigh between 40 and 50kg, where as the rest of the boys who weigh below 50kg, weigh less than the other girls; this causes the average weight for the girls to be higher than the boys. This proves the hypothesis i made after investigating the heights of the year 8 pupils to be correct; but it does not support the hypothesis I made at the beginning.
The box-and-whiskers diagrams show that the girls inter-quartile range is 15 kg more than the boy. This suggests that the girls height were more spread out than the boys.
The 10% sample for the year 8 pupils suggests that girls are taller than boys, and that they also weigh more than boys. This proves that there is a connection between height and weight, the taller someone is, the heavier they are. This does not support and prove my hypothesis to be true, so I will continue this investigation to see what results they give.
Year 9
Height
Girls
Height (cm)
Tally
Total
Cumulative frequency
< 150
//
2
2
< 160
//// /
6
8
< 170
////
5
3
< 180
/
4
Boys
Height (cm)
Tally
Total
Cumulative Frequency
< 150
/
< 160
///
3
4
< 170
////
4
8
< 180
///
3
1
< 190
/
2
The curves for both the boys and the girls are very similar; the girls is slightly steeper than the boys. Both of the lines are fairly straight until the last 10 or 20kg where they start to flatten out. I will now find the mean, mode and range.
Mean
Girls: 161cm
Boys: 165cm
To find the mode I will draw a stem and leaf diagram.
Girls
4
9
5
0 3 3 7 8 8
6
2 3 6 9
7
0 5 7
Boys
4
7
5
6 8
6
0 1 1 1
7
0 2 2
8
0 5
Mode
Girls: 158cm
Boys: 161cm
Range
Girls: 175-149=26cm
Boys: 185-147=38cm
I will now use the cumulative frequency graph to find the median, upper and lower quartiles and the inter-quartile range.
Median (Q2)
Girls: 158cm
Boys: 164cm
Lower quartile (Q1)
Girls: 153cm
Boys: 156cm
Upper Quartile (Q3)
Girls: 164cm
Boys: 173cm
Inter-quartile Range
Girls: 164-153=11
Boys: 173-156=17
By using the cumulative frequency graph I can estimate how many boys or girls are of a certain height. If I select a boy at random from the school, the data suggests that the probability of him having a height between 160cm and 170cm is 0.3.
After taking the mean, mode and median I can see that all three of the three measures of average in the sample were higher for boys than girls. The sample for the boys was more spread out, with a range of 38cm, compared to 26cm for the girls. The evidence from the sample suggests that's 5 out of 14 or 36% of the girls are between 160cm and 170cm, whilst 4 out of 12 boys or 33% are of the same height, between 160cm and 170cm. Most of the boys are above this height, showing that the majority of boys are taller than girls. This proves my hypothesis to be correct.
I have drawn a box and whiskers diagram to show the minimum and maximum values, the median and the upper and lower quartiles. It shows that the girls inter-quartile range is less than the boys. This suggests that the boy's heights were more spread out than the girls. The median height for the boys is 164cm. From my graph I have found out that 10 girls in the sample had height less than 159cm. Therefore 4 out of 14 girls have a height greater than the median height for the boys. This is 4/14 or 29%. This shows that most of the boys are taller than the girls, therefore proving my hypothesis, that boys are taller than girls. I will now investigate the weights of the year 9 pupils, to prove than the same will occur, boys will weigh more than girls.
Weight
Girls
Weight (kg)
Tally
Total
Cumulative Frequency
< 40
///
3
3
< 50
///
3
6
< 60
//// //
7
3
< 70
/
4
Boys
Weight (kg)
Tally
Total
Cumulative Frequency
< 40
/
< 50
//
2
3
< 60
//// //
7
0
< 70
//
2
2
The cumulative frequency graph I have drawn shows the weights of the people in year 9. The curve for the girl's data is very similar to the one for the boys data. I will now find the mean, mode and range; and then using the graph I have just drawn I will find the median and quartiles.
Mean
Girls: 51kg
Boys: 54kg
To find the mean I will draw a stem and leaf diagram.
Girls
4
0 0 0 5 8 8
5
2 2 2 6 8 8 8
6
5
Boys
3
8
4
2 5
5
5 5 5 9 9
6
0 2 6
Mode
Girls: 52kg
Boys: 55kg
Range
Girls: 65-40=25kg
Boys: 66-38=28kg
Median (Q2)
Girls: 51kg
Boys: 54kg
Lower quartile (Q1)
Girls: 42kg
Boys: 50kg
Upper quartile (Q3)
Girls: 56kg
Boys: 58kg
Inter-quartile range
Girls: 56-42=14kg
Boys: 58-50=8kg
By using the cumulative frequency graph I have drawn I can pick a girl at random and by using the curve of best fit I can see that the probability of her weighing between 50 and 60kg is 0.5.
After taking the mean, mode and median I can see that all of the three measures of average in the sample were higher for boys than the girls. The sample for the boys was slightly more spread out than the girls, with a range of 28kg, compared to 25kg for the girls. The evidence from the sample suggests that 7 out of 14, or 50% of the girls weigh between 50 and 60kg. For the boys, 7 out of 12, or 58% weigh between the same figures, 50 and 60kg. More girls weigh 50kg or below, with 6 out of 14 girls or 43% weighing 50kg or below. Compared to only 3 out of 12 of the boys or 25% weighing 50kg or below. This shows that most of the boys weigh more than most of the girls, proving my hypothesis that boys weigh more than girls to be correct.
The box and whiskers diagram I have drawn shows that the girls inter-quartile range is 6kg more than the boys. This suggests that the girl's weights were more spread out than the girls. The median for the boy's is 54kg. From my graph I found out that 9 girls in the sample had a weight less than 54kg. Therefore 5 out of 14 girls weigh more than the median weight for the boys. This is 5/14 or 36%. Whilst in general boys are taller than girls I have evidence to suggest that 36% of the girls weigh more than the median height of the boys. Overall the boys in year 9 weigh more than the girls in that year group.
From this investigation of year 9, I have found out that boys are both taller and weigh more than girls; this proves my hypothesis to be correct.
Year 10
Height
Girls
Height (cm)
Tally
Total
Cumulative Frequency
< 150
//
2
2
< 160
/
3
< 170
////
4
7
< 180
//
2
9
Boys
Height (cm)
Tally
Total
Cumulative Frequency
< 160
///
3
3
< 170
//
2
5
< 180
////
4
9
< 190
//
2
1
The cumulative frequency diagram shows the heights of the people in year 10 as shown in the tables above. The curve for the girls is more of an 's' shape than the curve for the boys, this is the type of curve expected in a cumulative frequency diagram. I will now find the mean, mode and range, and then using the graph I will find the median and quartiles.
Mean
Girls: 162cm
Boys: 156cm
To find the mode I will draw a stem and leaf diagram.
Girls
4
1
5
5
6
2 8
7
0 0 2 8
Boys
5
5 7 7
6
2 5
7
4 4 4
8
0 5 7
Mode
Girls: 170cm
Boys: 174cm
Range
Girls: 178-141=37
Boys: 187-155=32
Median (Q2)
Girls: 161cm
Boys: 171cm
Lower Quartile (Q1)
Girls: 148cm
Boys: 158cm
Upper Quartile (Q3)
Girls: 165cm
Boys: 179cm
Inter-quartile Range
Girls: 165-148=17cm
Boys: 179-158=21cm
By using the cumulative frequency graph I can estimate how many boys or girls are of a certain height. If I select a boy at random from the school, the data suggests that the probability of him having a height between 170cm and 180cm is 0.4.
After taking the mean, mode and median I can see that two out of three of the measures of average in the sample were higher for boys than girls. The only exception was for the mean, where the girls were shown to be taller than the boys. The sample for the girls was more spread out, with a range of 37cm, compared to 32cm for the boys. The evidence from the sample suggests that 2 out of 9 or 22% of the girls are between 170cm and 180cm, whilst 4 out of 11 boys or 36% are of the same height, between 170cm and 180cm. More boys than girls are of this height or above, showing that the majority of boys are taller than girls. This proves my hypothesis to be correct.
I have drawn a box and whiskers diagram to show the minimum and maximum values, the median and the upper and lower quartiles. It shows that the boy's inter-quartile range is greater than the girl's. This suggests that the boy's heights were more spread out than the girl's. The median height for the boys is 171cm. From my graph I have found out that 7 girls in the sample had height less than 171cm. Therefore 2 out of 9 girls have a height greater then the median height for boys. This is 2/9 or 22%. This shows that most of the boys are taller than the girls, therefore proving my hypothesis, that boys are taller than girls. I will now investigate the weights of the year 10 pupils, to prove my hypothesis that boy's weight more than girls.
Weight
Girls
Weight
Tally
Total
Cumulative Frequency
< 50
//
2
2
< 55
////
5
7
<60
//
2
9
Boys
Weight
Tally
Total
Cumulative Frequency
< 40
/
< 50
/
2
< 60
////
4
6
< 70
////
5
1
The cumulative frequency graph I have drawn shows the curve for the girls to be very steep and almost straight, this shows that the girls weigh about the same weights. The curve for the boys is curved at first, this shows a steady increase in the number of people at that weight; it then gets steeper. I will now find the mean, mode and range, and then using the graph I will find the median and quartiles.
Mean
Girls: 54kg
Boys: 58kg
To find the mode I will draw a stem and leaf diagram, as this will set out the data in an easier to use layout.
Girls
4
8
5
0 1 2 5 5 5 8
6
0
Boys
4
0
5
0 2 6 8
6
0 2 4 4 4
7
0
Mode
Girls: 55kg
Boys: 64kg
Range
Girls: 60-48=12kg
Boys: 70-40=30kg
Median (Q2)
Girls: 52kg
Boys: 60kg
Lower Quartile (Q1)
Girls: 50kg
Boys: 52kg
Upper Quartile (Q3)
Girls: 54kg
Boys: 65kg
Inter Quartile Range
Girls: 54-50=4kg
Boys: 65-52=13kg
By using the cumulative frequency graph I have drawn I can pick a girl at random and by using the curve of best fit I can see that the probability of her weighing between 50kg and 60kg is 0.8.
After taking the mean, mode and median I can see that all of the three measures of average in the sample were higher for boys than the girls. The sample for the boys was spread out more than the girls, with a range of 12kg, compared to 3kg for the girls. The evidence from the sample suggests that 7 out of 9, or 78% of the girls weigh between 50kg and 60kg. For the boys, 4 out of 11, or 36% weigh between the same figures, 50kg and 60kg. Although, more boys weigh above 60kg, 6 out of 11 or 55%. This shows that most of the boys weigh more than most of the girls, proving my hypothesis, that boys weigh more than girls to be correct.
The box and whiskers diagram I have drawn shows that the girls inter-quartile range is 9kg less than the boys. This suggests that the boy's weights were more spread out than the girls. The median for the boy's is 60kg. From my graph I found out that 5 girls in the sample had a weight less than 60kg. Therefore 4 out of 9 girls have a weight greater than the median height for the boys. This is 4/9 or 44%. Whilst in general boys are taller than girls I have evidence to suggest that 44% of the girls weigh more than he median height of the boys. Overall the boys in year 10 weigh more than the girls in that year group.
From this investigation of year 10, I have found out that boys are both taller and weigh more than girls; this proves my hypothesis to be correct.
Year 11
Height
Girls
Height
Tally
Total
Cumulative Frequency
< 150
/
< 160
///
3
4
< 170
///
3
7
< 180
//
2
9
Boys
Height
Tally
Total
Cumulative Frequency
< 160
///
3
3
< 170
///
3
6
< 180
//
2
8
The curves on the cumulative frequency graph are both very straight; they only differ in length, as the curve for the boys is very short compared to the girls. Also the curve for the boys is situated below the curve for the girls. I will now find the mean, mode and range; then using the graph, the median and quartiles.
Mean
Girls: 161cm
Boys: 165cm
To find the mode I will draw stem and leaf diagrams as this will set the data out in an easy to read layout.
Girls
5
0 2 2 5
6
3 3 5
7
2 4
Boys
5
1
6
0 2 4 8
7
8
0 0
Mode
Girls: 163cm
Boys: 180cm
Range
Girls: 150-174=24cm
Boys: 180-151=29cm
Median (Q2)
Girls: 162cm
Boys: 163cm
Lower Quartile (Q1)
Girls: 155cm
Boys: 160
Upper Quartile (Q3)
Girls: 170cm
Boys: 170cm
Inter-quartile Range
Girls: 170- 155=15cm
Boys: 170-160=10cm
By using the cumulative frequency graph I can estimate how many boys or girls are of a certain height. If I select a boy at random from the school, the data suggests that the probability of him having a height between 160cm and 170cm is 0.5.
After taking the mean, mode and median I can see that all three of the measures of average in the sample were higher for boys than girls. The sample for the boys was more spread out, with a range of 29cm, compared to 24cm for the girls. The evidence from the sample suggests that 3 out of 9 or 33% of the girls are between 160cm and 170cm, whilst 4 out of 9 boys or 44% are of the same height, between 160cm and 170cm. A higher percent of boys are of this height, showing that the majority of boys are taller than girls. Also most of the rest of the girls are below 160, where the rest of the boys are actually above 170. This proves my hypothesis, that boys are taller than girls to be correct.
I have drawn a box and whiskers diagram to show the minimum and maximum values, the median and the upper and lower quartiles. It shows that the girl's inter-quartile range is greater than the boy's. This suggests that the girl's heights were more spread out than the boy's. The median height for the boys is 163cm. From my graph I have found out that 5 girls in the sample had height less than 171cm. Therefore 4 out of 9 girls have a height greater then the median height for boys. This is 4/9 or 44%. This shows that most of the boys are taller than the girls, therefore proving my hypothesis, that boys are taller than girls. I will now investigate the weights of the year 11 pupils, to prove my hypothesis that boy's weigh more than girls.
Weight
Girls
Weight
Tally
Total
Cumulative frequency
< 40
/
< 50
////
5
6
< 60
//
2
8
< 70
/
9
Boys
Weight
Tally
Total
Cumulative Frequency
< 40
///
3
3
< 50
0
3
< 60
0
3
< 70
////
4
7
< 80
/
8
The cumulative frequency graph I have drawn shows the weight of the year 11 boys and girls separately. The curve for the girls data is steep at first, it then evens off, this shows a slow down in the original rapid increase in the number of people. The curve for the boys is a definite 's' shape; it starts off level, and then becomes very steep, before evening off again. I will now find the mean, mode and range; then using the graph, the median and quartiles.
Mean
Girls: 49kg
Boys: 54kg
To find the mode I will draw a stem and leaf diagram, as this will set it out in a layout which is easier to read.
Girls
3
9
4
4 4 5 5 8
5
6
0 6
Boys
3
8 8
4
0
5
6
0 0 3 3
7
2
Mode
Girls: 45kg
Boys: 60kg
Range
Girls: 66-39=27kg
Boys: 72-38=34kg
Median (Q2)
Girls: 47kg
Boys: 62kg
Lower Quartile (Q1)
Girls: 43kg
Boys: 40kg
Upper quartile (Q3)
Girls: 54kg
Boys: 68kg
Inter-quartile Range
Girls: 54-43=11kg
Boys: 68-40=28kg
By using the cumulative frequency graph I have drawn I can pick a girl at random and by using the curve of best fit I can see that the probability of her weighing between 50kg and 60kg is 0.1.
After taking the mean, mode and median I can see that all of the three measures of average in the sample were higher for boys than the girls. The sample for the boys was spread out more than the girls, with a range of 34kg, compared to 27kg for the girls. The evidence from the sample suggests that 2 out of 9, or 22% of the girls weigh between 60kg and 70kg. For the boys, 4 out of 8, or 50% weigh between the same figures, 60kg and 70kg. Most of the boys weigh 50kg or above, this is 5 out of 8 or 63%. Where as most of the girls weigh below 50kg, that is 6 out of 9 or 67%. This shows that most of the boys weigh more than most of the girls, proving my hypothesis, that boys weigh more than girls to be correct.
The box and whiskers diagram I have drawn shows that the girls inter-quartile range is 17kg less than the boys. This suggests that the boy's weights were more spread out than the girls. The median for the boy's is 62kg. From my graph I found out that 8 girls in the sample had a weight less than 62kg. Therefore 1 out of 9 girls have a weight greater than the median height for the boys. This is 1/9 or 11%. Whilst in general boys are taller than girls I have evidence to suggest that 11% of the girls weigh more than the median height of the boys. Overall the boys in year 11 weigh more than the girls in that year group.
From this investigation of year 11, I have found out that boys are both taller and weigh more than girls; this proves my hypothesis to be correct.
From this investigation of the 10% samples of the year groups separately, I can conclude that boys weigh more than girls, and they also are taller than girls, this is what I said in my hypothesis. Four out of the five year groups or 80% of the year groups prove my hypothesis to be correct, and they support the statement I originally made. It is only the year 8 year group which differs from the rest; I found that the girls weighed more than the boys and were also taller. The reason for this result may be due to the limitation I had. I could have had greater confidence in my results if I had taken larger samples, such as instead of taking a 10% sample of each year group, a 20% or 30% sample as that way I would have more data to work with, and these would account for any anomalies. Also my predictions were made on general trends observed in the data. In all of the samples there were exceptional individuals whose results fell outside the general trend. These could be seen on the box and whiskers diagram where one of the whiskers may have extended further away from the box than expected. Overall the majority of the data support my hypothesis, and could be used to back it up.
In my pre-test I sampled 30 students at random from the school. However because the school is growing there are more students in year 7 than in year 11, therefore the sample is biased. To ensure that students from all year groups are equally represented I will take a stratified sample. This is a sample where the numbers from each group are in proportion to that group's size within the whole population. By taking a stratified sample I can be as sure as possible that my sample is a repetitive of the whole school. As far as possible my sample is free from bias caused by gender or age diversion.
Using the 10% sample I took for the year groups individually I will find another measure of spread. I have already used the range and inter-quartile range as a measure of spread, but another measure of spread which is useful is the mean deviations from the mean (the average distance of the data values from the mean). Once I have found the mean deviation for the year 7 boys 10% sample I will compare it to the stratified sample for the year 7 boys who represent the whole of the school.
Year 7
Boys (10% sample)
Height
Difference from mean
Deviation squared
53
0
0
51
2
4
62
9
81
53
0
0
50
3
9
52
57
4
6
62
9
81
61
8
64
63
0
00
61
8
64
62
9
81
61
8
64
49
4
6
50
3
9
Total
78
590
Mean height - 153cm
Mean deviation = 78 = 5.2cm
15
Standard deviation = 590 = 39.33 = 6.27cm
15
Boys (stratified sample)
Height
Difference from Mean
Deviation squared
42
1
21
55
2
4
52
42
1
21
65
2
44
52
65
2
44
72
9
361
66
3
69
55
2
4
46
7
49
75
22
484
90
37
369
57
4
6
63
0
00
52
Total
65
3089
Mean deviation = 165 = 16.5cm
15
Standard deviation = 3089 = 205.93 = 14.35cm
15
In the stratified sample there was a large spread between the smallest and tallest boy; there were two who were 142cm tall this is a 11cm difference from the mean. Compared to the tallest being 190, which is 37cm greater than the mean height. Where as in the 10% sample most people are of the same height the tallest being 163cm, which is 10cm from the mean. This may be the reason why the mean deviation is greater for the stratified sample.
The heights for the boys for the whole school appear to be far more spread out than the heights for the 10% sample of boys. (The mean of the deviations from the mean was 5.2cm, when only year 7 boys were sampled and 16.5cm with the stratified sample. That is a difference of 11.3.)
I have also found the standard deviation; this is 14.35cm for the stratified sample for the whole school, compared to only 6.27cm for the 10% sample. This shows that the stratified sample is very spread out , when compared to the 10% sample.
I have drawn a scatter diagram with the heights and weights of the boys in year 7. I have noticed an untypical point at (25,152), so I have drawn three lines of best fit, one which excludes this point, one that includes all of the points and finally a curve of best fit that takes all the points into account. I have analysed the lines of best fit to see which is most suitable by measuring the average distance or vertical dispersion of the datum points from the line or curve. These can be seen in the tables below.
Line excluding point (25, 152)
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
40
51
2
39
50
44
54
9
48
56
3
50
58
8
25
40
2
55
61
4
56
62
60
65
2
43
53
8
42
52
0, 10
38
49
0
59
64
4
Mean of vertical dispersion
5.6cm
Line including all points
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
40
53
0
39
52
44
55
7
48
56
3
50
57
7
25
47
5
55
59
2
42
54
8, 8
56
60
60
62
43
54
7
38
52
9
59
61
1
Mean of vertical dispersion
4.67cm
Curve including all points
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
40
53
0
39
52
44
55
7
48
58
5
50
59
9
25
34
8
55
60
3
42
54
8, 8
56
61
0
60
63
0
43
55
6
38
51
0
59
62
2
Mean of vertical dispersion
5.8cm
The means of the vertical dispersions suggest that when all points are considered a line is a better approximation to the relationship than a curve (4.67cm instead of 5.8cm). However a strong correlation can also be found by excluding the points at (25, 152) and drawing a line of best fit, this gave me a mean of 5.6cm. I have drawn the stratified sample results for the boys from the whole of the school on to the graph and drawn a line of best fit to find the vertical dispersions.
Line of best fit for stratified sample
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
26
42
0, 0
38
50
5
45
51
, 14, 5, 24, 1
35
48
4
56
64
48
58
4
54
63
2
50
60
5
70
76
4
40
52
5
59
67
4
Mean of vertical dispersions
6.6
I have evidence to suggest that the correlation between height and weight may be greater when I restrict myself to students from a single year group. The mean of the vertical dispersion for a sample of 15 year 7 boys was 4.67cm when all of the points were taken into account. The mean for a stratified sample of 15 boys from the whole school was 6.6cm. Therefore I am going to find the mean height, mean of deviations from mean height and mean of vertical dispersions on the line of best fit using the 10% sample from each year group as I feel this is more accurate. I will then summarise it in a table.
Girls
Height (cm)
Difference from mean
Deviation squared
25
24
576
73
24
576
59
0
00
47
2
4
32
7
289
45
4
6
52
3
9
52
3
9
50
41
8
64
64
5
225
48
62
3
69
32
7
89
53
4
6
Total
46
2244
Mean height =149
Mean deviation = 146 = 9.73cm
15
Mean deviation excluding unrealistic points = 98 = 7.54
13
Standard deviation = 2244 =149.6 = 12.23cm
15
The heights of the girls in year 7 in the 10% sample appear to be fairly spread out with a mean deviation of 9.73cm; this may be because there are two figures which stand out from the others, they are the smallest and the tallest people. They both differ from the mean height by 24. This would cause the mean deviation to increase quite a bit, if I recalculate the mean deviation without those points then I would get the result 7.54, this s more realistic.
The standard deviation for the 10% sample is 12.23cm, this shows there to be a large spread, showing that not many people are of the same height.
I will now draw a scatter diagram to find the vertical dispersions.
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
35
35
0
62
72
38
39
20
44
47
0
40
42
0, 22, 20, 10
41
43
2
52
58
6, 8, 5
50
55
3
39
41
0
37
38
0
Mean of vertical dispersions
8.47
Year 8
Boys
Height (cm)
Difference from the mean
Deviation Squared
55
2
4
69
2
44
58
35
22
484
55
2
4
70
3
69
26
31
961
52
5
25
50
7
49
25
32
024
72
5
225
70
3
69
72
5
225
72
5
225
72
5
225
Total
200
3934
Mean height - 157
Mean deviation = 200 = 13.33
15
Standard deviation = 3934 = 262.27 = 16.19
15
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
64
70
5, 15
59
65
4
55
62
4
29
37
2
47
53
7
44
51
25
52
53
41
43
7
61
66
31
42
49
23, 23
49
55
5
46
53
9
57
62
0
Mean of vertical dispersions
4.07
Girls
Height (cm)
Difference from the mean
Deviation Squared
55
5
25
55
5
25
57
3
9
62
2
4
60
0
0
75
5
225
72
2
44
62
2
4
42
8
324
62
2
4
75
5
225
44
6
256
63
3
9
Total
98
254
Mean height - 160
Mean deviation = 98 = 7.54cm
13
Standard deviation = 1254 = 96.46 = 9.82cm
13
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
60
67
2
42
54
, 6, 8
53
62
5
80
81
8
45
57
6, 18
50
60
8, 12
46
57
5
51
61
72
75
0
49
59
5
Mean of vertical dispersions
9.62
Year 9
Boys
Height (cm)
Difference from the mean
Deviation Squared
61
4
6
56
9
81
58
7
49
72
7
49
47
8
324
61
4
6
85
20
400
60
5
25
72
7
49
61
4
6
70
5
25
80
5
225
Total
05
275
Mean height - 165
Mean deviation = 105 = 8.75cm
12
Standard deviation = 1275 = 106.25 = 10.31cm
12
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
45
57
4
60
73
7
51
64
8
57
70
2, 2
42
54
7
62
75
4
55
63
22, 3, 7
38
50
1
66
80
0
Mean of vertical dispersions
8.08cm
Girls
Height (cm)
Difference from the mean
Deviation Squared
69
8
64
58
3
9
53
8
64
62
49
2
44
53
8
64
57
4
6
66
5
25
58
3
9
75
4
96
70
9
81
70
9
81
50
1
21
63
2
4
Total
97
879
Mean height - 161
Mean deviation = 97 = 6.93cm
14
Standard deviation = 879 = 62.79 = 7.92cm
14
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
48
60
9, 7
40
55
3, 6, 3
58
67
4, 5, 13
52
63
6, 7, 0
45
58
8
56
65
0
65
72
22
Mean of vertical dispersions
8.07cm
Year 10
Boys
Height (cm)
Difference from the mean
Deviation Squared
74
8
64
87
31
961
57
65
9
81
85
29
841
57
80
24
576
62
6
36
55
74
8
64
74
8
64
Total
26
2690
Mean height - 156
Mean deviation = 126 =11.45cm
11
Standard deviation = 2690 = 244.55 =15.64cm
11
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
64
79
5, 22, 5
70
84
3
58
73
8
62
77
8
60
75
5
52
68
6
50
66
1
56
71
3
Mean of vertical dispersions
6.91cm
Girls
Height (cm)
Difference from the mean
Deviation Squared
70
8
64
55
7
49
78
6
256
68
6
36
41
21
441
72
0
00
70
8
64
41
21
441
62
0
0
Total
97
451
Mean height - 162cm
Mean deviation = 97 = 10.78cm
9
Standard deviation = 1451 = 161.22 = 12.7cm
9
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
55
65
5, 24, 24
50
61
6
52
63
5
58
68
0
51
62
0
60
70
0
48
59
3
Mean of vertical dispersions
9.67
Year 11
Boys
Height (cm)
Difference from the mean
Deviation Squared
51
4
96
68
4
6
51
4
96
80
5
225
60
5
25
80
5
225
62
3
9
64
Total
71
893
Mean height - 165
Mean deviation = 71 = 8.88
8
Standard deviation = 893 = 111.63 = 10.57
8
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
38
49
2, 11
63
72
4, 10
40
50
72
80
0
60
69
1, 5
Mean of vertical dispersions
5.5
Girls
Height (cm)
Difference from the mean
Deviation Squared
63
2
4
72
1
21
52
9
81
55
6
36
74
3
69
65
4
6
63
2
4
50
1
21
52
9
81
Total
67
633
Mean height - 161
Mean deviation = 67 = 7.44cm
9
Standard deviation = 633 = 70.33 = 8.39cm
9
Weight (kg)
Height of line (cm)
Vertical dispersions (cm)
45
60
3, 10
51
64
8
44
59
7
60
69
4, 6
39
56
8
66
73
8
48
62
0
Mean of vertical dispersions
9.33
Results Summarised
Year
Gender
Mean height (cm)
Mean of deviations from mean height (cm)
Mean of vertical dispersions on line of best fit
7
Boys
53
5.2
4.67
Girls
49
9.73
8.47
8
Boys
57
3.33
4.07
Girls
60
7.54
9.62
9
Boys
65
8.75
8.08
Girls
61
6.93
8.07
0
Boys
56
1.45
6.91
Girls
62
0.78
9.67
1
Boys
65
8.88
5.5
Girls
61
7.44
9.33
Final Summery
A sample of 30 students stratified over age and gender shows a mean height of 169.93cm for the boys and 149.13cm for the girls. However the range of heights for the boys was considerably greater than that for the girls. This suggests that there will be many boys who are shorted than 149cm, the girls mean height.
A 10% sample of boys in year 7 suggests that this year group of boys has a mean height of 153cm with a mean deviation about that mean of 5.2cm. My evidence suggests that the girls in this year group have a mean height of 149cm with a mean deviation about this mean of 9.73cm. The deviations for these two only differ by 4.53cm, so I can fairly conclude that the boys in year 7 are in general taller than the girls.
This conclusion is supported by the evidence gathered from 10% samples in other year groups. The 10% sample means that the sample sizes for years 9, 10 and 11 were particularly small, especially when compared to the 10% samples for years 7 and 8.
There is a positive correlation between height and weight both across the school as a whole and within each year group. The correlation appears to be stronger when individual year groups and separate genders are considers, as opposed to when a stratified sample is used to represent the school as a whole.
Over the full range of heights, there is evidence to suggest that a line of best fit is not the most suitable model to describe the relationship between height and weight. Further sampling and analysis could determine whether a curve of best fit would more accurately describe the relationship.
However I can fairly conclude that there is a connection between height and weight; the taller the person is, the more they weigh. I have also found out that age and gender are connected to how height and weight are linked.
