# Leaves Project

Extracts from this document...

Introduction

Mathematics

YEAR 10 – LEAVES PROJECT

Hypotheses:

- As the length increases so will the width.
- The length and width will be greater in 2002 than 2001.
- The spread of the length and width will be great in 2002 than in 2001.
- The length and width of the leaves will follow a normal to almost normal distribution.
- The length of leaf for which 10% of the leaves are longer will be greater in 2002.

From the data gathered on leaves, attached overleaf, I performed statistical techniques to attempt to prove my hypotheses.

Factors Effecting Results:

Time of year | From the data given we cannot be sure if the length and width of the leaves were collected at the same time of year each year. Leaves collected at the beginning of the year would not yet have finished growing whereas leaves collected towards the end of year (i.e. autumn) would be at their largest. |

Type of tree | We do not know if the data was collected from one specific type of tree or a mixture. |

Age of tree | The age of the tree would affect the results because a younger tree would have smaller leaves whereas a more mature tree would have larger leaves. |

Weather | We do not know if the leaves were collected during drought conditions, this would affect the size of leaf, as a tree would conserve water by not growing such large leaves. |

Location | We do not know if the leaves were collected from the same location both years i.e. in the same wood. We also do not know if the leaves were collected directly from the tree or picked up from the ground. |

The above therefore leaves my results questionable.

Before we can analyse the data gathered on leaves, I have to identify if there are any anomalies. I have highlighted these in yellow:

Length (mm) 2001 | Width (mm) 2001 | Length (mm) 2002 | Width (mm) 2002 |

72 | 44 | 45 | 27 |

105 | 56 | 62 | 41 |

61 | 37 | 80 | 47 |

66 | 39 | 82 | 39 |

33 | 11 | 64 | 35 |

62 | 41 | 85 | 40 |

55 | 30 | 76 | 40 |

85 | 47 | 75 | 43 |

74 | 36 | 66 | 32 |

36 | 20 | 64 | 42 |

90 | 55 | 67 | 32 |

13 | 6 | 78 | 45 |

57 | 30 | 70 | 38 |

79 | 43 | 69 | 37 |

65 | 41 | 52 | 26 |

47 | 32 | 64 | 29 |

55 | 31 | 71 | 45 |

55 | 30 | 76 | 44 |

56 | 27 | 79 | 45 |

61 | 34 | 69 | 37 |

55 | 32 | 67 | 39 |

53 | 25 | 77 | 43 |

57 | 32 | 76 | 44 |

55 | 31 | 83 | 47 |

50 | 28 | 83 | 44 |

59 | 35 | 85 | 46 |

72 | 45 | 89 | 45 |

71 | 39 | 87 | 46 |

60 | 38 | 95 | 53 |

66 | 44 | 84 | 48 |

58 | 32 | 85 | 42 |

71 | 38 | 86 | 45 |

67 | 35 | 64 | 38 |

55 | 37 | 87 | 46 |

65 | 41 | 77 | 43 |

80 | 45 | 75 | 39 |

78 | 44 | 71 | 38 |

98 | 50 | 64 | 30 |

85 | 43 | 60 | 26 |

93 | 46 | 82 | 50 |

90 | 42 | 80 | 45 |

31 | 18 | 91 | 52 |

30 | 15 | 84 | 51 |

27 | 14 | 93 | 54 |

44 | 22 | 88 | 51 |

41 | 20 | 95 | 54 |

42 | 22 | 102 | 56 |

90 | 50 | 76 | 40 |

82 | 41 | 79 | 42 |

66 | 34 | 104 | 52 |

I think that these numbers are anomalies because they are either too small or too big to fit in with the consistency of the data. I am going to discount these numbers from the rest of my work as including them would make my results unreliable or faulty.

Statistical Technique: Averages, Quartiles and Largest and Smallest

Numbers.

Length (mm) | Width (mm) | Length (mm) | Width (mm) | |

2001 | 2001 | 2002 | 2002 | |

## Mean | 62.57 | 35.14 | 76.71 | 43.28 |

Median | 61.00 | 36.00 | 77.00 | 44.00 |

LQ | 55.00 | 30.00 | 69.00 | 39.00 |

UQ | 72.00 | 43.00 | 85.00 | 46.50 |

IQR | 17.00 | 13.00 | 16.00 | 7.50 |

Largest | 98.00 | 56.00 | 102.00 | 56.00 |

Smallest | 27.00 | 11.00 | 45.00 | 29.00 |

Middle

The second scatter graph shows a stronger, positive correlation but there are still points being placed about the trendline. Once again the formula for the trendline shows us that the trendline crosses the y-axis at +33.474mm showing that when the length is 0mm the width is +33.474mm, once again this is impossible. The gradient of this scatter graph tells us that as the length increases by 1mm the width increases by 0.1278mm.

My first hypothesis was:

As the length increases so will the width.

According to the scatter graphs (shown over the page) this hypothesis has been proved as on both of the scatter graphs, the gradient shows us that as the length increases so does the width.

Statistical Technique: Box Plots.

Length (mm) | Width (mm) | Length (mm) | Width (mm) | |

2001 | 2001 | 2002 | 2002 | |

## Mean | 62.57 | 35.14 | 76.71 | 43.28 |

Median | 61.00 | 36.00 | 77.00 | 44.00 |

LQ | 55.00 | 30.00 | 69.00 | 39.00 |

UQ | 72.00 | 43.00 | 85.00 | 46.50 |

IQR | 17.00 | 13.00 | 16.00 | 7.50 |

Largest | 98.00 | 56.00 | 102.00 | 56.00 |

Smallest | 27.00 | 11.00 | 45.00 | 29.00 |

Another way of showing this table is in the form of a “box and whisker” diagram, located overleaf.

My next hypothesis was:

The length and width will be greater in 2002 than 2001.

In order to attempt to prove this hypothesis I am going to look at the mean on the table above and also represent the data above in the form of box plots (shown overleaf).

Conclusion

5

6

7

5

0

0

5

6

Width | Freq. | Width < (mm) | Cumulative Freq. |

20-29 | 4 | 29.5 | 4 |

30-39 | 7 | 39.5 | 11 |

40-49 | 20 | 49.5 | 31 |

50-59 | 14 | 59.5 | 45 |

60-69 | 4 | 69.5 | 49 |

Total = | 49 |

Length 2002:

4 | 5 | |||||||||||||||

5 | 2 | |||||||||||||||

6 | 0 | 2 | 4 | 4 | 4 | 4 | 4 | 6 | 7 | 7 | 9 | 9 | ||||

7 | 0 | 1 | 1 | 5 | 5 | 6 | 6 | 6 | 6 | 7 | 7 | 8 | 9 | 9 | ||

8 | 0 | 0 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 6 | 7 | 7 | 8 | 9 |

9 | 1 | 3 | 5 | 5 | ||||||||||||

10 | 2 | 4 |

Length | Freq. | Length < (mm) | Cumulative Freq. |

40-49 | 1 | 49.5 | 1 |

50-59 | 1 | 59.5 | 2 |

60-69 | 12 | 69.5 | 14 |

70-79 | 14 | 79.5 | 28 |

80-89 | 16 | 89.5 | 44 |

90-99 | 4 | 99.5 | 48 |

100-109 | 2 | 109.5 | 50 |

Total = | 50 |

Width 2002:

2 | 6 | 6 | 7 | 9 | |||||||||||||||||||||

3 | 0 | 2 | 2 | 5 | 7 | 7 | 8 | 8 | 8 | 9 | 9 | 9 | |||||||||||||

4 | 0 | 0 | 0 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 7 | 7 | 8 |

5 | 0 | 1 | 1 | 2 | 2 | 3 | 4 | 4 | 6 |

Width | Freq. | Width < (mm) | Cumulative Freq. |

20-29 | 4 | 29.5 | 4 |

30-39 | 12 | 39.5 | 16 |

40-49 | 25 | 49.5 | 41 |

50-59 | 9 | 59.5 | 50 |

Total = | 50 |

N.B. Graphs are not included for width as we are only looking at lengths in 2001 and 2002.

My final hypothesis was:

The length of leaf for which 10% of the leaves are longer will be greater in 2002.

In order to prove this, I have to find out 10% of the totals on both cumulative frequency graphs.

For lengths in 2001, 4.9 and in 2002, 5.

From the two graphs I can tell that the 10th percentile in 2001 is equal to 40 and in 2002 the 10th percentile is equal to 63.

From the above I can say that my hypothesis has been proven.

In conclusion I have proven 4 out of 5 of my hypotheses and disproved the remaining one.

Francesca Tate 10H Mrs Smith

This student written piece of work is one of many that can be found in our GCSE Comparing length of words in newspapers section.

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