TABLE OF CONTENTS

Part A:

INTRODUCTION

EXPECTAIONS, PREDICTIONS AND ASSUMPTIONSPAGE 3

DATA

SECONDARY DATA – STEM AND LEAF PLOTPAGE 4

SECONDARY DATA – LINE GRAPH AND OGIVESPAGE 6

SECONDARY DATA – BOX PLOT AND HISTOGRAMSPAGE 7

SECONDARY DATA – MEASURMENTS OF RANGE PAGE 12 SECONDARY DATA – BELL CURVE PAGE 12/11 INVESTIGATION PAGE 11

ANALYSIS

CONCLUSIONPAGE 13

APENDIX

PRIMARY/ RAW DATAPAGE 13/15

PART B

Simpsons paradoxPage 16/17

INTRODUCTION

The purpose of this mathematical report is to investigate a tree’s ability to capture sunlight in relation to possible correlations between the tree’s growing capacity and leaf length. A variety of mathematic techniques will be used to examine how a tree’s height relates to the length of its leaves. This data will then be used to analyse the tree’s ability to efficiently capture sunlight and move liquid around the tree.

It is predicted that this report will prove that as a tree’s growing capacity increases the average, median and mean leaf size will decrease. This is likely due providing a more efficient means of transporting fluid around the tree. However to compensate for the shorter leaves’ having less ability to capture sunlight there will be an increased quantity of them to ensure that adequate sunlight is absorbed

This study will make use of a variety of mathematical concepts, more specifically with many of them relating, or having to do with statistics. This investigation will draw on over 200 collected values for leaf length from 2 separate tree’s (100 leaves from each tree will be measured). To provide an accurate and informative way of communicating this large amount of data terms such as mean, median, mode, range, interquartile range and standard deviation will be used. The definitions of these terms are as follows:

•Mean: the average of an entire set of data

•Median: the middle value of the set of data, If n (n being the number of values) is odd then the median is the middle number. Whereas, is n is even then the median is the sum of the two middle values divided by 2.

•Mode: Is the most common number in a set of data.

•Range: The difference between the minimum and maximum values.

•Interquartile Range: The difference between the upper and lower quartile.

•Standard Deviation: The average distance in which the set of values varies from the average of the set of values.

For this report two trees will each have 100 leaves’ lengths and their height measured. 100 different leaves will be chosen from each of the selected trees (criteria for selecting a tree includes ensuring it is a healthy, well established tree and that most leaves are over 4cm in length) and then have their length measured with a ruler. To find the height of the trees an angle of elevation to the top of the tree will be measured using a protractor, string and weight. Basic Trigonometry (Tangent, Sine and Cosine rations) will then be used to calculate the height; the height of the person measuring will then be added to this value to compensate for the measurements not being taken at ground level.

To complete this investigation certain assumptions have been made.

These include:

•That the tree is not affected by any diseases (trees get diseases too). Whilst the tree was inspected and appeared healthy it is possible that some diseases may not exhibit symptoms.

•That both trees were grown under the same conditions, e.g. if one tree received fertilizer the other did also.

This suggests that both trees where...