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High Jump Gold Medal Heights Type 2 Maths Portfolio

Extracts from this document...

Introduction

Gold Medalist Heights

IB Mathematics SL Type 2 Portfolio

Sarah van der Post

2/4/12

Contents Page

Introduction                                                                        3

Considering Data -Heights of Olympic Games 1932 - 1980        3

Linear Model                                                                4

Natural Logarithm Model                                                7

Estimating and Predicting                                                8

Considering Additional Data - 1896 - 2008                                10

Cubic Model                                                                13

Gaussian Model                                                                        14

Conclusion                                                                        14

Introduction

The Olympic Games is a major event occurring every 4 years. International athletes come together to represent their country at a sport, first place winning a gold medal.  This portfolio aims to consider the trends of the winning men gold medalist’s high jump heights at various Olympic Games.

Considering Data - Heights of Olympic Games 1932 - 1980

The table below gives the height achieved by the men gold medalist’s high jumpers at various Olympic Games between 1932 and 1980.

Table 1: The winning heights of men’s high jumping in Olympic Games for years between 1932 and 1980. (Excluding 1940 and 1944)

 Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 Height (cm) 197 203 198 204 212 216 218 224 223 225 236

Using excel, a graph has been produced to represent this data visually.

Graph 1: The winning men’s high jump heights in Olympic Games for years between 1932 and 1980. (Excluding 1940 and 1944)

The x-axis represents the year of the Olympic Games. This begins at 1932. The y-axis represents the Gold Medalists high jump height in centimeters.

Unfortunately, there are parameters to this graph.

Middle

In order to make calculations easier, I created a graph of height versus years since 1932 as seen in graph 2. When considering the trends and possible functions this graph models, I immediately thought that the function that models the behaviour of this graph is a linear function. This can be shown by the line of best fit in Graph 3 which is linear, where the line goes through point which allows the same number of points to be above and below the line.

Graph 3: The winning heights of Olympic Games against years since 1932 showing linear line of best fit.

The overall trend of this line of best fit is a positive increase. This line of best fit passes through points E (1956) and G (1964), and touches F (1960). A (1932), B (1936), H (1968) and K (1980) are above the graph and C (1948), D (1952), I (1972) and J(1976) are below the graph. This means that there are equal point above and below, however A, B, C, D and K seem to deviate away from the line of best fit by quite a large amount.

By using the coordinates of two points whom this best fit line passes through (E and G), the equation of the line of best fit was found.

m=y1-y2x1-x2

m=218-21232-24

m=68

y-218x-32= 68

y-218=68(x-32)

y-218=68x-24

y=68x+194

y=34x+194

I have graphed this analytically found equation with the line of best fit in order to compare.

Graph 4: The winning heights of Olympic Games against years since 1932 showing best fit line (dashed)

Conclusion

Cubic Model

Graph 10: Graph of winning height against year of Olympic Games (between 1896 and 2008) showing cubic function.

*The y axis is the winning height and the x axis is the year of Olympic Games.

However the model curves upward before the year 1896 and curves downward after the year 2008 which disagrees with the trends of the data. Therefore we can assume the cubic can only be used for data between 1986 and 2008.

Gaussian Model

Considering this limitation, I sought to find another model which could fit the data and came across the Gaussian model. The Gaussian model begins with a level and horizontal line and then curves up similar to the cubic model. It also models the leveling off as the years approach 2008. But, like the cubic model slopes downwards after the 2008. It is a better representation of the data than the cubic model since it does not slope upwards in the years before 1896. Their winning height would never have begun at 0 because naturally humans can jump a certain height. Thus this level beginning fits well.

Graph 11: Graph of winning height against year of Olympic Games (between 1896 and 2008) showing Gaussian function.

*The y axis is the winning height and the x axis is the year of Olympic Games.

Conclusion

To conclude, it is very hard to find a model which fit exactly the trends of the data from Olympic Games, but this investigation has found that the Gaussian model is the model which models the data most.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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