I will use the fictional database from Mayfield High School to do my investigation. I would like to investigate the relationship between:

- Boy’s height and girl’s height
- Boy’s shoe size and girl’s shoe size
- Boy’s height to shoe size ratio
- Girl’s height to shoe size ratio
- The differences between the two ratios

This will show me the gender variation. I will use a sample of 30 boys and 30 girls from the original 1183 students in the database. I will take a fair sample by using random sampling. I will take a stratified sample of 120, and from this I will use a random number table to get the desired 60 students, which I will base my investigation on.

From these 60 samples I will work out the mean, median, mode and range to research the topic. I will create graphs and pie charts to analyse the information I will collect. I feel this will assure the observations I make are successful. I will be able to compare them successfully and make a valid conclusion.

### SAMPLE

Below is the data I have collected:

7

1.63

5

1.61

11

1.57

11

1.55

9

1.72

7

1.65

11

1.67

10

1.74

12

1.88

9

1.70

4

1.80

7

1.73

12

1.67

3

1.33

13

2.06

2

1.65

From this desired sample, I will create tally charts, using groups of 10cm for height, and normal groups for shoe size.

TALLY CHARTS(SHOE SIZE)

Girls Shoe size

Size

Tally

Frequency

1

2

2

2

3

1

4

5

5

3

6

6

7

3

8

4

9

1

10

2

11

1

30

Boys Shoe Size

Size

Tally

Frequency

4

5

5

3

6

2

7

4

8

5

9

3

10

2

11

2

12

2

13

1

14

1

30

I have calculated the median by halving thirty to get fifteen. I then found the fifteenth result when the whole data was put in chronological order.

I have worked out the mean by multiplying each shoe size with its frequency. Once all these results were obtained I added them together, and divided by thirty.

The range is the largest shoe size minus the smallest shoe size, which happens to be ten in each case.

The modal group is simply the most common data value.

TALLY CHARTS (HEIGHT)

Girls Height

Height

Tally

Frequency

1.30-1.39

2

1.40-1.49

2

1.50-1.59

9

1.60-1.69

11

1.70-1.79

5

1.80-1.89

1

30

Boys Height

Height

Tally

Frequency

1.50-1.59

10

1.60-1.69

9

1.70-1.79

6

1.80-1.89

4

1.90-1.99

0

2.00-2.09

1

30

I have worked out the mean, median, mode and range for the height in similar ways as for the shoe size. I worked out the mean by multiplying the frequency by the mid point of each group. I added up all of these results and then divided by thirty. The median, mode and range are the same as for the shoe sizes.

The figures I have worked out from the data seem quiet promising, and already I have observed:

* The differences between the boy’s shoe size mean and the girl’s shoe size mean,

* The differences between the boy’s height mean and the girl’s height mean,

* The difference between the range in girl’s height and boy’s height,

* And many more observations can be made.

From this data I will make a stem and leaf diagram.

STEM AND LEAF DIAGRAMS

Boys Height

Stem

Leaf

Frequency

Cumulative frequency

1.50

2,4,4,5,5,7,7,8,9,9

10

10

1.60

0,2,3,3,5,6,7,7,7

9

19

1.70

2,2,2,3,3,5

6

25

1.80

0,0,2,8

4

29

1.90

0

29

2.00

6,

1

30

Girls Height

Stem

Leaf

Frequency

Cumulative frequency

1.30

2,3

2

2

1.40

3,5

2

4

1.50

1,1,2,2,3,4,5,5,7

9

13

1.60

0,0,1,1,2,3,5,5,5,5,5

11

24

1.70

0,0,3,3,4

5

29

1.80

0,

1

30

A stem and leaf wouldn’t be appropriate for shoe size so I only mad them for height. They are very useful, as they will help me make a cumulative frequency graph

### CUMULATIVE FREQUENCY

VARIENCE

Although range is a good and simple way of working out the spread of results, it is not as effective as variance. Variance is a complex but highly accurate way of determining the spread of data around the mean. I have only worked out the variance for boy’s and girl’s height due to the fact that it is so complex and time consuming.

Girls Variance

Height

Mid point (X)

FX

(X-X) 2

F (X-X)

1.30-1.39

1.345

2.69

0.070

0.14

1.40-1.49

1.445

2.89

0.030

0.06

1.50-1.59

1.545

13.9

0.040

0.36

1.60-1.69

1.645

18.09

0.002

0.02

1.70-1.79

1.745

8.57

0.020

0.10

1.80-1.89

1.845

1.85

0.060

0.06

47.99

0.74

Mean= 47.99/30=1.5996

Variance= 0.742/30=0.025

This is a very low variance shows that the results are very close to the mean.

Boys Variance

Height

Mid point (X)

FX

(X-X) 2

F (X-X)

1.50-1.59

1.545

15.450

0.0200

0.200

1.60-1.69

1.645

14.805

0.0007

0.006

1.70-1.79

1.745

10.470

0.0050

0.030

1.80-1.89

1.845

7.3800

0.0300

0.120

1.90-1.99

1.945

0.0000

0.0700

0.000

2.00-2.09

2.045

2.0450

0.1390

0.139

50.150

0.495

Mean= 50.15/30=1.672

Variance= 0.495/30=0.0165

This Variance is even lower, so the results must be even closer together. From these observations, I can tell that data in the boy’s height is packed much closer around the mean than the girl’s height data.

GRAPHS

I have investigated a lot of the data so far, and I think all I need to complete my collecting and representing of data is some simple graphs and pie charts.

From these pie charts and histograms it is easy to compare the gender variation, but I feel a scatter graph comparing boy’s height with boy’s shoe size or girl’s height with girl’s shoe size would also be useful.

### OBSERVATIONS

After completing my investigation with graphs and my calculations of various averages, I feel there are a lot of observations for me to make. I can now analyse my results.

I feel my investigation successfully proved that girls are not as tall as boys. The height mean for girls is 160.5cm whereas for boys it is 167.2cm. This may only be a minor change but when we look at the histogram of height we can clearly see the difference. There is a high percentage change of 4.1%

The investigation also proved that girl’s feet are smaller than boys. The mean shoe size for girls is 5.76, whereas for boys it is 7.83, which is a large percentage change of 35.9%. The histogram shows well the clear difference between boys and girls. In fact, not a single boy from my sample had size 1,2 or 3 feet, which shows clearly that they have larger feet. Not a single girl from my survey had size 12,13,or 14 feet, again, they had smaller feet.

I hoped my scatter diagram would show a strong correlation, but I don’t feel it was very successful. I presumed that with boys, their shoe size would increase as their height did; I predicted the same for girls, but perhaps in a different ratio. Unfortunately the positive correlation is so slight, that a line of best fit can hardly be drawn. Obviously this relationship is very slight. When analysing the scatter graph closely, a slight trend can be seen.

My results I collected after calculating the variance show the accuracy of the mean in each case. The variance for girl’s height was 0.025, and for boy’s height was 0.0165. Although both are extremely low figures, it shows the mean for the girl’s is not as accurate as for the mean of boy’s height. It also shows that possibly boys have more variety of shoe sizes than girls. This is quiet interesting as it shows the range is not that accurate. The range of both groups is 10, but obviously, as the variance proves, they are not the same.

The cumulative frequency graph I have drawn is useful too. It shows the differences between boy’s height and girl’s height, of which there are lots. They both follow the same pattern, but boys on average are clearly taller than girls.

### CONCLUSION

After my successful collected sample, I feel I presented the data well, allowing myself to make some good observations. Now I feel I am able to conclude my results.

- I have clearly proven that on average, boy’s feet are larger than girl’s feet.
- I have also proven boys on average are taller than girls
- From the graphs the slight positive correlation shows that with girls and boys, when height increases so does shoe size.

If I wished, I could research this subject further, perhaps by using a larger sample of boys and girls, or perhaps by using more accurate measurement, (perhaps heights to one decimal place). If I had measured the boys and girls myself, it may have been a more accurate investigation. But overall, considering the data I had to complete the investigation, I feel it turned out well. I am happy with the results I have collected and feel I have made a valid conclusion.