Table 2 shows all data with the Year row showing the number of years since 1932.
Table 2:
The graph below (Figure 2) shows the points given (Table 2) modeled by a linear equation using Geogebra to produce the graph. The graph is made with years on the x axis measured in years since 1932 and the height of the winning high jump shown in cm on the y axis.
Figure 2: Points from Table 2 plotted on a linear graph with line of best fit generated by Graphical Analysis
Gradient: Using points E (24,212) and G (32,218)
m= y2-y1x2-x1 = 218-21232-24=68=34
Linear Equation Shown on Graph:
y-212x-24=34 → Cross multiply
y-212=34(x-24) → Simplify
y-212= 34x-18 →Put into form y=mx+c
y=34x+194
y=0.75x+194
Refined Linear Equation: y=0.76x+194
Figure 3: The linear function y=0.75x+194 (as shown in Figure 2) compared to the ‘line of best fit’ generated by graphical analysisLinear Equation: y=0.75x+194
I used technology to refine the equation y=0.75 x+194 by finding the line of best fit using Geogebra which generated the equation in the form
-1267x+1678y=325764
To get the equation in the form y=mx+c:
-1267x+1678y=325764 Rearrange equation into form y=mx+c
1678y=325764+1267x Divide whole equation by 1678
y=0.76x+194 (All figures rounded to 3sf)
Table 3 shows the data excluding that gathered in 1932 and 1936 with the Year row showing years since 1948
Table 3:
Figure 4: Points from Table 3 plotted on a linear graph with line of best fit generated by Graphical Analysis
Gradient: Using points B (4,204) and E (16,218)
m= y2-y1x2-x1 = 218-20416-4=1412=76
Linear Equation Shown on Graph:
y-218x-16=76 → Cross multiply
y-218=76(x-16) → Simplify
y-218= 76x-16 →Put into form y=mx+c
y=76x+202
y=1.16x+202
Figure 5: The linear function y=1.16x+202n (as shown in Figure 4) compared to the ‘line of best fit’ generated by graphical analysis
Linear Equation generated using technology: y=0.76x+194
Linear Equation: y=1.16x+202
I used technology to refine the equation y=1.16 x+202 by finding the line of best fit using Geogebra which generated the equation in the form
-49x+48y=9648
To get the equation in the form y=mx+c:
-49x+48y=9648 Rearrange equation into form y=mx+c
48y=9648+49x Divide whole equation by 48
y=1.02x+201 (All figures rounded to 3sf)
In comparison the linear model does not fit the Table 3 data (as shown in Figure 4 and 5) as well as it fits the data shown in Table 2 (as shown in Figure 2 and 3). The line of best fit generated using technology passes through two points in Figure 3 where as it passes through only one point in Figure 4. In Figure 5 , the line showing the manually generated gradient is quite different from the line of best fit generated using technology as this passes through one point only. It is for these reasons that I used the linear equation found for the graph that incorporated all data points.
The linear equation shown in Figure 3 and refined using Graphical Analysis fits the data well, however it is unrealistic to presume that the model of the equation would continue to be valid for data gathered after 1980 as there are natural limitations on human beings ability to overcome gravity and continue to improve following this trend. For this reason I wanted to find a graph that would fit the steep positive slope formed by the data but would gradually plateau as humans began to be held back by the limitation of natural forces. I thought that a logarithmic graph would best fit this.
Figure 6: This graph shows a log10graph generated using Graphical Analysis to fit the data points given (Table 2)
I didn’t think this graph properly modeled the equation so I decided to remove the first two pieces of data gathered, those from 1932 and 1936 because there was a 12 year gap in the games between 1932 and 1948 when they were cancelled due to WWI i.e. to use the data shown in Table 3.
Figure 7: This graph shows a log10graph generated using Graphical Analysis to fit the data points given (Table 3)
This graph is a much better model for the data. It passes close to all points and follows the curve of the data. The equation of the log10 graph was generated using Graphical Analysis.
y=A*log(Bx)
y=28.75 log(2.82*10^6x)
Figure : A comparison of the refined log10 function and the linear function on the same set of data points (data from Table 3)
The linear function passes through more points than the logarithmic function however it is unrealistic to expect the linear function to be a good indicator of future trends as humans have a limited capacity for improvement. For this reason the logarithmic graph is a better choice to use as a model for the data and to indicate future trends as the gradient of the graph (height of winning jump/year) showing rate of improvement of the winning height jumped gradually becomes less steep over time. Visually the shape of the logarithmic function is similar to the shape made by the data points while the shape of the linear function is quite obviously different.
Had the Olympic Games occurred in 1940 and 1944 I believe that the data would have fit the logarithmic function y=28.75 log(2.82*10^6x) with years being counted from 1932. I believe that without the hindrance of WWII there would be no need to exclude the first two data points and the trend would continue, modeled accurately by the logarithmic equation. This is indicated by the fact that the rate of improvement between the first and second data points (collected in 1932 and 1936) is the same as that of the third and fourth pieces of data (collected 1948 and 1952), i.e. m= 34 . This indicates that had the data continued to be collected in four year increments without interruption that the data collected in 1940 and 1944 would be similar to the third and fourth data points given in Table 1, (1948, 198) (1952,204).
y=28.75 log(2.82*106x)
For the year 1940, let x = 8
y=28.75log(2.82*106* 8)=211 (3sf)
For the year 1944 let x = 12
y=28.75log(2.82*106*12)=216 (3sf)
To test whether this model would work for values outside of those given the model is tested for years 1984 and 2016
For 1984:
1984-1948=36
y=28.75 log(2.82*106x)
y=28.75 log(2.82*106*36 ) = 230cm (3sf)
For 2016:
2016-1948= 68
y=28.75 log(2.82*106x)
y=28.75 log(2.82*106*68) = 238cm (3sf)
This model is limited in its use as it does not account for the fluctuation in data, particularly that caused by the development of the Fosbury Flop technique which was developed in 1968 which caused a leap in the winning heights from the 1980s onward as more athletes adopted his new method of high jump. This is the most likely reason the actual value for the winning height for the Men’s’ high jump event in 1984 was 5cm higher than predicted using the model. While the actual data for the 2016 Olympics is not yet available it is likely that this would also be higher than the height predicted using the model. However the difference between the heights of the winning jumps predicted using the model and the actual winning heights is likely to become less significant over time as the natural limitations of man’s capacity to overcome gravity causes the rate of improvement to lessen as the model predicts.
Table 4 shows all data collected 1896-2008. The ‘Year’ row shows the number of years since 1896.
Table 4
Table 4 continued
Figure 9: Scatter plot produced using Graphical Analysis showing data from Table 4
It is important to note the more significant fluctuations on the graph such as from 1896 – 1908. This could be because of WWI or the relative newness of the Olympic Games at an international scale. Other fluctuations such as that caused by WWII (1936-1948) and that caused by the development of the Fosbury Flop technique, both mentioned previously. The effect these fluctuations have had on the winning heights can be seen most clearly on the scatter plot using Table 4 data (Figure 9) as it gives a more complete picture of the Games and the effect fluctuations have on the trend of the data.
I tried my logarithmic model against the data for the entire games (Figure 10, below) but it did not give an accurate trend for the data. The logarithmic model suggest rapid improvement that gradually levels out with time however this model does not account for the rapid decrease in the rate of improvement of winning high jump heights that begins to occur during the 1960’s. Neither does it account for the small gradient that shows a much smaller rate of improvement at the end of the 19th century and beginning of the 20th century. It is likely that the steep gradient of the portion of the data I first analyzed (1932-1980) marked a period of rapid technological advances in sports technology and technique as many technological advances were made during this time. This means that using a logarithmic function is not an appropriate model to use to indicate the winning heights of high jumpers over the entire course of the games.
Figure 10: Logarithmic model y=28.75 log(2.82*106x) shown against data from Table 4 produced using Geogebra
The graph above (Figure 10) shows the points given (Table 4) modeled by the logarithmic equation: y=28.75 log(2.82*106x) Geogebra was used to produce the graph.
Figure 10: Gaussian function shown against data from Table 4 produced using Geogebra
Because of my limited knowledge of Gaussian functions I used technology to produce the graph. I wanted a function that would show the decrease in the slope reflecting the lessening rate of improvement. The Gaussian function has the steady slope similar to that of the linear function that accurately reflects the trend of the data over the middle of the 20th century and has a decreasing gradient similar to that of a logarithmic function near the end of the 20th century as humans approach the limit imposed on them by natural forces such as gravity. This function is limited in its use however as it would show a gradual decrease in winning high jump heights if projected into the future and there is no current evidence to suggest that this would be the case. This graph has the equation:
y=7.503*105 -1152x+0.5895x2 -1.005*10-4x3
In conclusion the equation y=7.503*105 -1152x+0.5895x2 -1.005*10-4x3 seems best to illustrate the trend of winning high jump heights of men in the Olympic Games between 1932-1980 while fitting the data both before and after this period with accuracy.
