The aim of the assignment is to examine the gold medalist heights for the high jump in order to find the best fitting equation to represent the heights obtained by the gold medalists from the years 1932 through until 1980.
SL Mathematics: Type II Internal Assessment
Gold Medal High Jump
High jump is an Olympic event in which the goal is to jump over a bar as high as possible. The sport changed drastically during the years that are being examined. Many factors affected the growth of this sport such as: the change and improvement in technique (the invention of the Fosbury Flop for example), World War Two, the introduction of body building in sports and the increased awareness and knowledge of human health.
The aim of the assignment is to examine the gold medalist heights for the high jump in order to find the best fitting equation to represent the heights obtained by the gold medalists from the years 1932 through until 1980. I will look into the correlation between the year and the Gold medalist’s high jump height
The following table shows the height recorded by the gold medalists at the Olympic games from 1932 – 1980. The years 1940 and 1944 have been omitted due to the Olympic games being cancelled those years as a result of World War Two.
Note: Variable x corresponds with year and variable y corresponds with height. This will be relevant in future equations.
There is no obvious trend that the points plotted on the Raw Data graph indicate, however it appears that there are two turning points. This could perhaps be an anomaly because of World War Two. I created a cubic function for my first model, and therefore must use a cubic equation: y = ax3 + bx2 + cx + d.
Judging from the results shown in the raw data, it appears that there is an increase from 1932 through to 1936. There is then a decrease in height at (1948, 198), followed by a relatively steady increase. These changes in heights illustrate two distinct turning points (at 1936, 203) and at (1948, 198). A cubic function is effective for this set of data specifically because of the concave up and down shape, although if it were for Gold Medal High Jump results in the future, a logarithmic model as it illustrates humans infinitely increasing ability. However, the years provided were focused upon. The points used for illustration are shown in bold.
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Table # 1: Raw Data
The Analytical Model
I decided to choose the points (1932, 197) and (1980, 236) so that the beginning and end points will be clear. I chose (1952, 204) and (1980, 236) because they are equidistant from the beginning of the graph and the end of the graph, respectively.
Using the results I obtained using PLYSMLT2, I input my four points in order to obtain my four variables (A, B, C, D).
y = ax3 + bx2+ cx + d
197 = a (1932)3 + b (1932)2 + c (1932) + d
204 = a (1952)3 + b (1952)2 + c (1952) + d
224 = a (1968)3 + b (1968)2 + c (1952) + d
236 = a (1980)3 + b (1980)2 + c (1980) + d
197 = 7211429568 (a) + 3732624 (b) + 1932 (c) + d
204 = 7437713408 (a) + 38103046 (b) + 1952 (c) + d
224 = 7622111232 (a) + 3873024 (b) + 1968 (c) + d
236 = 7762392000 (a) + 3920400 (b) + 1980 (c) + d
a = -19 / 26880 = -7.06845238
b = 799 / 192 = 4.161458333
c = -285788 / 38 = -7520.736842
d = 5339900
The equation for the analytical model is given below:
y = -7.06845 x3 + 4.1615 x2 + -7520.7368 x + 5339900
Table # 2: Analytical Model
This cubic model completely misses out the point (1936, 203). This anomaly can be accredited to the fact that in the years 1940 and 1944 there were no Olympic games, this is the reason for the ‘dip’. If there had not been a Second World War, then it is plausible that the height jumped in 1948 would be greater than 203cm, and even more plausible that it would be greater than 198cm; we can only speculate and make educated guesses as to the gold medal heights would be if no war occurred. The reason for this being that there was a draft of epic proportions – the United States alone had a draft of 50 million. The most athletically able were most likely drafted into the army and judging by the fall in height jumped, many of the athletes who would have participated in the 1948 Olympic games had died in the war or were injured. There is also the possibility that the athletes had not been undergoing the appropriate training for the high jump during the twelve-year period in which there was no Olympic games held. Excluding the anomalous point (1936, 203) the cubic model appears to fit the graph.
The points (1972, 223) and (1976, 225) are missed by the model completely. It has been widely suggested that the high altitude and ‘thin air’ in Mexico during the 1968 Olympics was the dominant factor in the high standard of results. 1968 was also the peak of Dick Fosbury’s (winner of the 1964 high jump) high jumping career. Fosbury’s unconventional technique for jumping the high jump is the most effective way of jumping that we are aware of to this day, Dick Fosbury did not take part in any Olympic games after 1968. Fosbury can be considered an anomaly; nobody jumped higher than him until the 1980 Olympic games and the results for the 1972 and 1976 were higher than the results prior to Fosbury being a part of the Olympic games. Conceivably, without Fosbury, there would be a steady increase in gold medalist height.
There is a large jump between the point (80, 236) and the preceding point. This large increase in height can be connected with the introduction of bodybuilding to athletes in all sports. While it is true, that bodybuilding was invented in the 70s (allegedly by Bruce Lee, however this is disputed), strengthening and enlarging specific muscles through exercise was not utilized until the 1980s. the practice of bodybuilding not only lead to an increase in high jump records but it lead to an increase in records set throughout the majority of events.
The final point on the cubic model fits perfectly, which is why a cubic model is ideal for this set of data, however if the data were to continue on into the future, a logarithmic model would be ideal because a logarithmic model would illustrate an increase in high jump records – which overall is likely to happen.
The accuracy of this cubic model is shown below:
Accuracy of the Cubic Model
This table shows that the accuracy of the cubic model is not very high, having said this; we cannot tell whether this model is more effective than the technological model yet.
The Technological Model
a = -0.0004137426102040079
b = 2.43931495687917
c = -4792.80390296559
d = 3138526.4305823
In order to create the technological model, I took the original graph of the raw data with the addition of the cubic model and simply added a polynomial trend line. The trend line goes through no points on the graph, and from knowing just this, it can be derived that the cubic model represents the data more accurately than the technological model. The strength of the cubic model over the technological model will be reinforced in the table representing the accuracy of the technological model.
Accuracy of the Technological Model:
The results show that the cubic model is more accurate by a difference of 2.028. Although some of the points on the technological model have a difference that is less than that of the difference of the corresponding points on the cubic model, the overall average difference is greater, thus making the prediction that the cubic model would be more reliable is correct.
Predictions for the 1940 and 1944 Olympic Games:
If there had been no World War and/or the Olympics were held in 1940 and 1944, I estimate that the winning heights would be in very close range with the winning height of 1936 (203cm, which was set by Cornelius Johnson). I think that this is likely because Cornelius Johnson did not take part in any Olympic game after 1936 because he died in 1945. It is unlikely that an athlete’s set height would diminish in four years. It is possible that Johnson would not have hypothetically taken part in the 1944 Olympic games because of age, and because of that, the winning height may have been lower than those of the 1936 and 1940 games because of the fact that Johnson may not be participating.
Because my initial, analytical model proved to be the more accurate one, I will use it to estimate the winning heights for the years 1940 and 1944.
Predicted 1940 and 1944 Results
My initial prediction was incorrect. The predicted winning heights are more similar to those of 1932 and 1948 and are steadily increasing. However, the model does not take into account the factor of having an anomalous athlete such as Cornelius Johnson, or any other athlete for that matter.
Predictions for the 1984 and 2016 Olympic Games:
For the 1984 Olympic games, I estimate that the winning height will be very similar to that of the 1980 winning height. It should be slightly more than that of the previous winning height. As for the winning height of the 2016 games, it should be significantly higher than the winning height of the 1984 height. However, we know that the cubic model used only illustrates the original raw data well, and other models (logarithmic) would better predict the 2016 Olympic games winning height. Because I am using a cubic model, the prediction given using the model will be significantly lower than the 1980 winning height and the prediction will be very unreliable. The results are as follows:
Predicted 1984 and 2016 Results:
My prediction for the winning 1984 Olympic height was similar to what the model generated. This result is plausible because the height is not drastically larger than that of the 1980 Olympics. As for the prediction for the 2016 height, it is 57.6cm less than the 1980 high jump height. This statistic is very unlikely because history as shown that in the field of athletics, over time statistics improve for a various number of reasons.
Table # 3: Raw Data # 2:
The results shown on the graph show that the 1904 winning high jump height was far less than every other height. This could be due to the fact that there were only six participants in the high jump. Why so few athletes took part in the event in 1904, I cannot say.
After 1980, a great increase in height is shown, the reason for this is the execution of bodybuilding for athletes and the increased knowledge on the topic of bodybuilding.