_____________________________________________________________________17
= 144+149+150+152+152+154+154+155+155+155+160+165+165+166+170+170+170
_____________________________________________________________________17
= 2686÷17
= 158.00 (2 d.p.)
So the mean height of a year 7 girl is 158 Cm
Now I am going to find the standard deviation for the girls in year 7 using the formula:
= √∑fx²÷∑f – (x) ² Where f-frequency
x-height (Cm)
x-mean height (Cm)
= √∑fx²f–(x) ²
= √425458÷17 – (158)²
= √62.94
= 7.93 Cm (2 d.p.)
Year 7 Boys
Mean = (137×1) (138×0) (139×0) (140×0) (141×0) (142×0) (143×0) (144×0) (145×0) (146×1) (147×0) (148×1) (149×0) (150×0) (151×0) (152×2) (153×0) (154×0) (156×0) (157×0) (158×0) (159×0) (160×0) (161×0) (162×0) (163×0) (164×1) (165×1) (166×1) (167×1) (168×0) (169×1) (170×2) (171×0) (172×1)
_____________________________________________________________________13
= 137+146+148+152+152+164+165+166+167+169+170+170+172÷13
= 159.85 Cm (2 d.p.)
So the mean height for a year 7 boy is 159.85 Cm
Now I am going to find the standard deviation for the boys in year 7 using the standard deviation formula.
= √∑fx²÷∑f – (x)²
= √333708÷13 – (160)²
= √25669.8 – 25600
= √69.8
= 8.36 Cm (2 d.p.)
Year 9s
Next I am going to look at pupils in year 9. I will start by finding a stratified random sample of 30 pupils.
№ of girls 62÷100×30=18.6=19
№ of boys 38÷100×30=11.4=11
I will label the girls 00→61 and the boys 62→99
I have used my calculator and have generated the following numbers:
598 338 782 124 862 846 474 911 975 432 947 803 491 305 019 438 087 207 761 504 597 733 871 053 138 194 478 014
I want to obtain two digit random numbers from these random numbers, so I pair the digits together.
59|8 3|38 |78|2 1|24 |86|2 8|46 |47|4 9|11 |97|5 4|32 |94|7 8|03 |49|1 3|05 |01|9 4|38 |08|7 2|07 |76|1 5|04 |59|7 7|33 |87|1 0|53 |13|8 1|94 |47|8 0|14|
59 Pick Girl
83 Pick Boy
38 Pick Girl
78 Pick Boy
21 Pick Girl
24 Pick Girl
86 Pick Boy
28 Pick Girl
46 Pick Girl
47 Pick Girl
49 Pick Girl
11 Pick Girl
97 Pick Boy
54 Pick Girl
32 Pick Girl
94 Pick Boy
78 Ignore
03 Pick Girl
49 Ignore
13 Pick Girl
05 Pick Girl
01 Pick Girl
94 Ignore
38 Pick Girl
08 Pick Girl
72 Pick Boy
07 Pick Girl
76 Pick Boy
15 Pick Girl
04 Ignore
59 Ignore
77 Pick Boy
33 Ignore
87 Pick Boy
10 Ignore
53 Ignore
81 Pick Boy
47 Ignore
80 Pick Boy
14 Ignore
My sample of girls are: 59 38 21 24 28 46 47 49 11 54 32 03 13 05 01 38 08 07 15
My sample of boys are: 80 81 87 77 76 72 83 78 86 97 94
I will use the data shown above to calculate the mean height and standard deviation for both boys and girls in year 9.
Year 9 Girls
First I am going to find the mean height of a girl in year 9.
Mean = (154×1) (155×1) (156×0) (157×2) (158×3) (159×0) (160×3) (161×1) (162×1) (163×1) (164×0) (165×1) (166×1) (167×0) (168×1) (169×0) (170×2) (171×1)
_____________________________________________________________________19
=154+155+157+157+158+158+158+160+160+160+161+162+163+165+166+168+170+170+171
_____________________________________________________________________19
= 3073÷19
= 161.74 Cm (2 d.p.)
The mean height for girls in year 9 is 161.74 Cm
Now I am going to find the standard deviation for the girls in year 9.
= √∑fx²÷∑f−(x)²
= √497511÷19−26160
= √24.96
= 5.10 (2 d.p.)
Year 9 Boys
First I am going to find the mean height of a boy in year 9.
Mean= (169×1) (170×0) (171×0) (172×0) (173×0) (174×2) (175×1) (176×0) (177×0) (178×0) (179×1) (180×0) (181×2) (182×0) (183×0) (184×1) (185×2) (186×0) (187×0) (188×0) (189×0) (190×0) (191×0) (192×0) (193×0) (194×0) (195×1)
_____________________________________________________________________11
= 169+174+174+175+179+181+181+184+185+185+195÷11
= 1982÷11
= 180.18 (2 d.p.)
The mean height of a boy in year 9 is 180.18 Cm
Now I am going to find the standard deviation for the boys in year 9.
= √∑fx²÷∑f−(x)²
= √357412÷11−32465
= √27
= 5.21 (2 d.p.)
Year 11
Now I am going to look at pupils in year 11. I will start by finding the stratified random sample of 30 pupils.
№ of Females 50÷100×30=15=15 Females needed
№ of Males 50÷100×30=15=15 males needed
I will label the girls 00→49 and the boys 50→99.
I have used my calculator and have generated the following random numbers:
421 684 739 016 378 709 308 106 448 318 329 572 169 899 698 074 710 339 797 348 600 149 431 918 135 118
I want to obtain two digit random numbers from these random numbers, so I pair the digits together.
42|1 6|84 |73|9 0|16 |37|8 7|09 |30|8 1|06 |44|8 3|18 |32|9 5|72 |16|9 8|99 |69|8 0|74 |71|0 3|39 |79|7 3|48 |60|0 1|49 |43|1 9|18 |13|5 1|18|
42 Pick Girl
16 Pick Girl
84 Pick Boy
73 Pick Boy
90 Pick Boy
16 Ignore
37 Pick Girl
87 Pick Boy
09 Pick Girl
30 Pick Girl
81 Pick Boy
06 Pick Girl
44 Pick Girl
83 Pick Boy
18 Pick Girl
32 Pick Girl
95 Pick Boy
72 Ignore
16 Ignore
98 Pick Boy
99 Pick Boy
69 Pick Boy
80 Pick Boy
74 Pick Boy
71 Pick Boy
03 Pick Girl
39 Pick Girl
79 Pick Boy
73 Ignore
48 Pick Girl
60 Pick Boy
01 Pick Girl
49 Pick Girl
43 Pick Girl
19 Ignore
18 Ignore
13 Ignore
51 Ignore
18 Ignore
I will use the above data to calculate the mean height and the standard deviation for both boys and girls in year 11.
Year 11 Girls
First I am going to find the mean height of a girl in year 11.
Mean = (156×1) (157×1) (158×1) (159×0) (160×2) (161×1) (162×0) (163×1) (164×2) (165×3) (166×0) (167×0) (168×1) (169×1) (170×1)
_____________________________________________________________________15
= 156+157+158+160+160+161+163+164+164+165+165+165+168+169+170
_____________________________________________________________________15
2445÷15
= 163.00 Cm (2 d.p.)
The mean height for a year 11 girl is 163 Cm
Now I am going to find the standard deviation for the girls in year 11.
= √∑fx² ÷ f − (x)²
= √398791 ÷ 15 − 26569
= √17
= 4.13 (2 d.p.)
Year 11 Boys
First I am going to find the mean height of a boy in year 11.
Mean= (162×1) (163×0) (164×0) (165×1) (166×0) (167×1) (168×1) (169×1) (170×3) (171×1) (172×0) (173×0) (174×1) (175×3) (176×0) (177×0) (178×0) (179×0) (180×0) (181×2)
_____________________________________________________________________15
= 162+165+167+168+169+170+170+170+171+174+175+175+175+181+181
_____________________________________________________________________15
= 2573÷15
= 171.53 Cm (2 d.p.)
The mean height for a boy in year 11 is 171.53 Cm
Now I am going to find the standard deviation for the boys in year 11.
= √∑fx² ÷ ∑f − (x)²
= √441757 ÷ 15 − 29422.54
= √27.93
= 5.28 (2 d.p.)
Summary of results table
I have put all of my results into the table shown below:
My results show me that the mean height increases from year 7 up to year 11 for both boys and girls.
In year 7 the boys are on average taller than the girls, this is also the same in years 9 and 11.
The largest spread between the heights is the year 7 boys who have a standard deviation of 8.36.
Conclusion
Year 7
From my results I can see that year 7 boys are on average taller than girls in year 7 because the mean height of a year 7 boy is 159.85 Cm and the mean height of a year 7 girl is 158 Cm. The boys heights are also more dispersed in year 7 than the girls heights, since the standard deviation for the year 7 boys is 8.36 and for a year 7 girl the standard deviation is 7.93, so my hypothesis is supported for year 7 pupils.
Year 9
From my results I can see that year 9 boys are on average taller than girls in year 9 because the mean height of a year 9 boy is 180.18 Cm and the mean height of a year 9 girl is 161.74 Cm. The boys heights are also more dispersed in year 9 than the girls heights, since the standard deviation for the year 9 boys is 5.21 and for a year 9 girl the standard deviation is 5.10, so my hypothesis is supported for year 9 pupils.
Year 11
From my results I can see that year 11 boys are on average taller than girls in year 11because the mean height of a year 11 boy is 171.53 Cm and the mean height of a year 11 girl is 163 Cm. The boys heights are also more dispersed in year 11 than the girls heights, since the standard deviation for the year 11 boys is 4.28 and for a year 11 girl the standard deviation is 4.13, so my hypothesis is supported for year 11 pupils.
I will now represent my results using histograms.
Year 7 Girls
Year 7 Boys
Year 9 Girls
Year 9 Boys
Year 11 Girls
Year 11 Boys
The histograms I have drawn further illustrate my results. The histogram for year 7 girls showed that their heights were not very dispersed. However, the histograms for year 7 boys showed a greater dispersion, this is also the same for years 9 and 11.
The boys in year 7 showed the greatest dispersion of heights.
Conclusion and evaluation
My hypothesis stated that “On average boys are taller than girls and boy’s heights are more dispersed than girl’s heights.” My hypothesis is supported by my year 7, 9 and 11 results as can be seen from my summary of results table and my histograms. These results are only valid for the school I have investigated. If I were to look at another sample from another school then my results may be different. If I were to attempt this investigation again, I would:-
- Look at a number of different schools, including primary schools and colleges and I would include all of the year groups.
- Look at school in different parts of the country.
- Take a large sample of the pupils.
In this way, my results would be more representative of the population as a whole, and I would be more likely to eliminate any anomalous results from my investigation. I could further extend this investigation by looking at other measurements such as hand span, head circumference etc to see whether these measurements fall in line with my hypothesis i.e. that on average boys have larger measurements than girls and that their measurements are more dispersed than girls measurements.
