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Math Portfolio Type II Gold Medal heights

Extracts from this document...

Introduction

The Olympic games were created thousands of years ago with the purpose to determine who could push himself the hardest to jump higher or run faster. High jump has always been part of the revived Olympics and has been carried out ever since the 1896's Olympics.

The achievements of athletes almost put itself forward for analysis, as the data is already precisely recorded and therefore it is rather easy to make predictions. This data is useful for trainer or scouts when they have to estimate whether an athlete is worth their while or not.

As in all of the other programs a steady improvement of the athletes’ performances have been recorded as the games advanced and almost all skilled sportsmen were taken under supervision of trainers and were made professionals.

The recorded heights of the gold medal winners starting from the 1932's Olympics until 1980 are as follow:

Year

1932

1936

1948

1952

1956

1960

1964

1968

1972

1976

1980

Height in cm

197

203

198

204

212

216

218

224

223

225

236

Table 1.0 Data table for the  height of the gold medallists in high jump from 1932 until 1980.

Using these raw values to plot a graph without any manipulation results in a picture which is both hard to read and not well suited to convey the wanted information.

 If the x-Axis was to correspond with the years starting from year zero, the graph would be unnecessarily broad. The data of Table 1.0 has to be processed in order to provide a readable graph. As well no years before 1896 are relevant for this task as the first modern Olympic games were held in that year. Therefore zero on the x-Axis will represent the year 1896 on the graph and the new data is listed in the following table.

Year with year 1896 as year 0

36

40

52

56

60

64

68

72

76

80

84

Height in cm

197

203

198

204

212

216

218

224

223

225

236

Table 1.1 Processed data from Table 1.0 for the height of the gold medallist in high jump from 1932 until 1980

...read more.

Middle

image13.png

image15.png

image18.pngimage17.pngimage16.pngimage59.png

image60.png

One should not at this point that these points were not randomly chosen. They both represent anomalies and therefore do not fit the general pattern, however their effect on the curve of the graph must also be taken into consideration. The graph attained by this calculation does fit these points very well as the  graph on the following page proves.

The axis' have again been shifted to provide a better visualization of the graph. The x-axis begins at 30(1926).image01.pngimage62.png

One can see that the graph fits really well with (36, 197),

(40, 203) and

(84, 236). However the other points are not really included on the graph.

Therefore it is fit to create a graph which suits the other points and then average the image00.png

value of the parameters in order to find the best fitting graph.image19.png

One can see from Figure 3.0 that the points from (52, 198) to (80, 225) follow a rather regular pattern and so following the same steps as before using the system of equations one can attain a suitable graph for these points.

image21.pngimage50.pngimage20.png

image04.pngimage05.png

image22.pngimage23.pngimage07.pngimage10.pngimage09.png

image11.pngimage64.pngimage63.png

image65.pngimage24.png

image66.png

image15.png

image26.pngimage25.png

image27.pngimage17.pngimage16.pngimage68.pngimage67.png

Now using the just attained new equation                                   the functions graph will added to the graph already present Figure 3.0's draw a comparison between the two and prove that an averaged function would provide the best fit.image69.png

A graph that represents the average function can be calculated by averaging the b and k of Equation 1 and Equation 2. To distinguish between the two function’s parameters let b1  and k1 be the parameters of Equation 1 with b2  and k2 belonging toimage01.pngimage70.png

Equation 2.

image72.png

image73.png

image74.png

b now represents the averaged value.

image00.pngimage75.png

Forming a new equation using the newly acquainted values one gets the following functionimage28.pngimage76.png

To test if the averaged function is really better suited to visualise the progress of high jump athletes one has to take Figure 3.1and add Equation 3(                                ).image78.pngimage77.pngimage77.png

image01.png

image00.png

image29.png

...read more.

Conclusion

The general trend of the data points show an over all increase of height as the years progress.

However there are a few random fluctuations throughout the graph.

The first notable swing can be seen from year 1896 to 1908. During this period there were foreshadows of the Great War, where the nations main concern were not athletic competition, but preparations for the on coming war. The First World is also responsible for the missing value at x=20 which represents the year 1916, as with the Olympics there is also the ancient tradition that no games can be held if there is war.

The same explanation is there for the missing values of the years 1940 (44) and 1944 (48) as the Second World War was raging.

No value is available for the 1900 Olympics which were actually held in Paris. However they were part of the Parisian World Exhibition, the same goes for the 1924 Olympics also held in Paris, however they were not part of a World Exhibition.

One can see a pattern that whenever a year's value is missing the next years height drops below the height of the previous years.

The graph starting from 1960 (64) seems to be reaching a plateau however this trend is broken in 1968 (72). This is the same year as the Fosbury Flop was introduced and started to become the prominent high jump technique.

Starting from that year again a steady increase can be seen until again a plateau is reached 1988 (92).

From that point on the height mostly stayed the same.

In this portfolio for the display of equations the software  MathType Light (Version 6.7a) was used.

The graphs were displayed using Autograph (Version 3.3.0010) and the same software was also used to generate the Lines of best Fit.

The data tables were taken from Microsoft Excel Starter (Version 14.0.5128.5000).

...read more.

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

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