Maths Statistics Coursework Hypotheses My first hypothesis is that on average boys are taller then girls. I obviously cannot prove this by measuring every child in the world. Instead I will take a sample of the children at Mayfield School to prove this. I also cannot check every pupil of Mayfield School, so will take a random stratified sample of 200 children. This will contain the height and averages of 102 boys and 98 girls, all ages between 11 and 16.

I will first show my tally chart, which shows the ages and varying height of the children, detailing boys and girls separately. Males Grouped Height Ages 11/12 Aged 13 Aged 14 Aged 15 Aged 16 Total 1. 20 up to but not including 1. 40 0 0 0 0 0 0 1. 40 up to but not including 1. 45 3 2 0 0 0 5 1.

45 up to but not including 1. 50 7 1 1 0 0 9 1. 50 up to but not including 1. 55 7 4 2 2 2 17 1.

55 up to but not including 1. 60 3 4 1 5 0 13 1. 60 up to but not including 1. 65 6 2 2 1 2 13 1. 65 up to but not including 1. 70 0 4 3 1 2 10 1.

70 up to but not including 1. 75 0 6 6 2 4 18 1. 75 up to but not including 1. 80 0 2 3 2 1 8 1. 80 up to but not including 1. 90 0 1 2 0 2 5 1.

90 up to and including 2. 00 0 0 2 1 1 4 Total frequency 26 26 22 14 14 102 Females Grouped Height Ages 11/12 Aged 13 Aged 14 Aged 15 Aged 16+ Total 1. 20 up to but not including 1. 40 3 1 0 0 0 4 1. 40 up to but not including 1. 45 2 1 0 0 0 3 1.

45 up to but not including 1. 50 5 3 3 0 0 11 1. 50 up to but not including 1. 55 4 1 4 1 1 11 1.

55 up to but not including 1. 60 2 3 3 5 2 15 1. 60 up to but not including 1. 65 5 5 4 4 2 20 1. 65 up to but not including 1. 70 0 2 7 2 6 17 1.

70 up to but not including 1. 75 2 2 1 2 2 9 1. 75 up to but not including 1. 80 2 1 1 2 0 6 1. 80 up to but not including 1. 90 0 0 0 2 0 2 Total frequency 25 19 23 18 13 98 All grouped data: 200 From this it is noticeable that boys are taller then girls, as there are no boys, of any age, who are shorter then 1.

40 m, yet there are 4 girls who are below this mark, aged 11-13. These results obviously don't prove very much, because 4/98 is a very small fraction. I will prove my hypothesis from the year-by-year mean heights. Females Ages 11/12 Grouped height F MPV FX MEAN 1. 20 up to but not including 1. 40 3 1.

30 3. 9 1. 40 up to but not including 1. 50 7 1. 45 10. 15 1.

50 up to but not including 1. 60 6 1. 55 9. 3 1.

60 up to but not including 1. 70 5 1. 65 8. 25 1. 70 up to but not including 1. 80 4 1.

75 7 1. 80 up to but not including 2. 00 0 1. 90 0 Totals 25 34. 7 1.

388 Aged 13 Grouped height F MPV FX MEAN 1. 20 up to but not including 1. 40 1 1. 30 1. 3 1. 40 up to but not including 1.

50 4 1. 45 5. 8 1. 50 up to but not including 1. 60 4 1.

55 6. 2 1. 60 up to but not including 1. 70 7 1. 65 11. 55 1.

70 up to but not including 1. 80 3 1. 75 5. 25 1. 80 up to but not including 2.

00 0 1. 90 0 Totals 19 28. 8 1. 515789 Aged 14 Grouped height F MPV FX MEAN 1.

20 up to but not including 1. 40 0 1. 30 0 1. 40 up to but not including 1.

50 3 1. 45 4. 35 1. 50 up to but not including 1. 60 7 1. 55 10.

85 1. 60 up to but not including 1. 70 11 1. 65 18.

15 1. 70 up to but not including 1. 80 2 1. 75 3. 5 1. 80 up to but not including 2.

00 0 1. 90 0 Totals 23 36. 85 1. 602174 Aged 15 Grouped height F MPV FX MEAN 1. 20 up to but not including 1. 40 0 1.

30 0 1. 40 up to but not including 1. 50 0 1. 45 0 1. 50 up to but not including 1. 60 6 1.

55 9. 3 1. 60 up to but not including 1. 70 6 1. 65 9.

9 1. 70 up to but not including 1. 80 4 1. 75 7 1. 80 up to but not including 2.

00 2 1. 90 3. 8 Totals 18 30 1. 666667 Aged 16+ Grouped height F MPV FX MEAN 1. 20 up to but not including 1. 40 0 1.

30 0 1. 40 up to but not including 1. 50 0 1. 45 0 1.

50 up to but not including 1. 60 3 1. 55 4. 65 1.

60 up to but not including 1. 70 8 1. 65 13. 2 1. 70 up to but not including 1. 80 2 1.

75 3. 5 1. 80 up to but not including 2. 00 0 1. 90 0 Totals 13 21. 35 1.

642308 Males Ages 11/12 Grouped height F MPV FX MEAN 1. 30 up to but not including 1. 40 0 1. 35 0 1.

40 up to but not including 1. 50 10 1. 45 14. 5 1. 50 up to but not including 1. 60 10 1.

55 15. 5 1. 60 up to but not including 1. 70 6 1. 65 9.

9 1. 70 up to but not including 1. 80 0 1. 75 0 1. 80 up to but not including 2.

00 0 1. 85 0 Totals 26 39. 9 1. 534615 Aged 13 Grouped height F MPV FX MEAN 1. 30 up to but not including 1.

40 0 1. 35 0 1. 40 up to but not including 1. 50 3 1. 45 4.

35 1. 50 up to but not including 1. 60 8 1. 55 12. 4 1. 60 up to but not including 1.

70 6 1. 65 9. 9 1. 70 up to but not including 1. 80 8 1. 75 14 1.

80 up to but not including 2. 00 1 1. 85 1. 85 Totals 26 42. 5 1. 634615 Aged 14 Grouped height F MPV FX MEAN 1.

30 up to but not including 1. 40 0 1. 35 0 1. 40 up to but not including 1. 50 1 1. 45 1.

45 1. 50 up to but not including 1. 60 3 1. 55 4. 65 1. 60 up to but not including 1.

70 5 1. 65 8. 25 1. 70 up to but not including 1. 80 9 1. 75 15.

75 1. 80 up to but not including 2. 00 4 1. 85 7. 4 Totals 22 37. 5 1.

704545 Aged 15 Grouped height F MPV FX MEAN 1. 30 up to but not including 1. 40 0 1. 35 0 1.

40 up to but not including 1. 50 0 1. 45 0 1. 50 up to but not including 1. 60 7 1. 55 10.

85 1. 60 up to but not including 1. 70 2 1. 65 3. 3 1. 70 up to but not including 1.

80 4 1. 75 7 1. 80 up to but not including 2. 00 1 1. 85 1. 85 Totals 14 23 1.

642857 Aged 16 Grouped height F MPV FX MEAN 1. 30 up to but not including 1. 40 0 1. 35 0 1. 40 up to but not including 1. 50 0 1.

45 0 1. 50 up to but not including 1. 60 2 1. 55 3.

1 1. 60 up to but not including 1. 70 4 1. 65 6.

6 1. 70 up to but not including 1. 80 5 1. 75 8. 75 1. 80 up to but not including 2.

00 3 1. 85 5. 55 Totals 14 24 1. 714286 In these tables I have calculated a number of things: The frequency is the amount of times that height occurs. The MPV (Mid-point value) is simply the middle number in the grouped heights.

The FX is the frequency column, multiplied by the mid-point value column. The totals in the F and FX are all the amounts added up. I have calculated the mean by dividing the FX column total, by the Frequency column total. Total means 11/12 13 14 15 16 Average means Males 1. 5346 1.

6346 1. 7045 1. 6429 1. 7143 1. 6462 Females 1. 3880 1.

5158 1. 6022 1. 6667 1. 6423 1. 5630 As you can see from the year-by-year means every male age group has a higher mean height, then there female equivalents. It is also interesting to note, as you go through the age groups that year-by-year the mean height is higher.

The only exception to this is that 15 year old girls have a higher mean height then the 16 year old girls. I have also calculated the cumulative frequency's, for both boys and girls, separately. Cumulative Frequency: Boys Frequency Cumulative Frequency Coordinates 1. 30 up to but not including 1. 40 0 0 (1. 4, 0) 1.

40 up to but not including 1. 50 14 14 (1. 5, 14) 1. 50 up to but not including 1.

60 30 44 (1. 6, 44) 1. 60 up to but not including 1. 70 23 67 (1. 7, 67) 1. 70 up to but not including 1.

80 26 93 (1. 8, 93) 1. 80 up to but not including 2. 00 9 102 (2. 0, 102) Totals 102 102 I have put all this information onto a graph, with the above coordinates. I have labelled the X-axis Height, and the Y-axis Cumulative Frequency.

I have also calculated: Lower Quartile = 1/4 n = 1/4 (102) = 25. 5 th value = 1. 55 m Upper Quartile = 3/4 n = 3/4 (102) = 76. 5 th value = 1. 765 m Median = 1/2 n = 1/2 (102) = 51 st value = 1. 64 m These are also marked out on the graph.

Cumulative Frequency: Girls Frequency Cumulative Frequency Coordinates 1. 25 up to but not including 1. 40 4 4 (1. 4, 4) 1. 40 up to but not including 1. 50 14 18 (1.

5, 18) 1. 50 up to but not including 1. 60 26 44 (1. 6, 44) 1. 60 up to but not including 1. 70 37 81 (1.

7, 81) 1. 70 up to but not including 1. 80 15 96 (1. 8, 96) 1. 80 up to but not including 2. 00 2 98 (2.

0, 98) Totals 98 98 I have put all this information onto a graph, with the above coordinates. I have labelled the X-axis Height, and the Y-axis Cumulative Frequency. I have also calculated: Lower Quartile = 1/4 n = 1/4 (98) = 26. 5 th value = 1. 525 m Upper Quartile = 3/4 n = 3/4 (98) = 79. 5 th value = 1.

695 m Median = 1/2 n = 1/2 (98) = 49 th value = 1. 629 m These are also marked out on the graph. The Lower Quartile, Upper Quartile and Median strengthen my point, as in each case the boys Lower Quartile, Upper Quartile and Median are higher then the girls equivalent. Boys Girls Lower Quartile 1. 55 m 1. 525 m Upper Quartile 1.

765 m 1. 695 m Median 1. 64 m 1. 629 m.