This document analyzes data on the winning heights in men's high jump in Olympic games from 1896 to 2008. It uses a quintic polynomial function to model the data from 1932 to 1980, finding the best fit equation algebraically and through a computer. The model predicts heights in 1940 and 1944 reasonably but cannot extrapolate accurately past 1980. A logistic curve may better capture the overall leveling off trend in winning heights.

1. KittitornKiatpipattanakun
20/5/11
Math IA
Gold Medal in Men’s High Jump Olympic games
Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980
Height 197 203 198 204 212 216 218 224 223 225 236
Table 1: Shows the height in centimeters, and the year of the jump heights.
Figure 1: This graph shows the winning height of each year Olympic games.
Some Functions models that had been considered was a linear equation. The
linear equation could predict that the winning height would increase every year of
the Olympic game. However, a straight line would not accurately predict the heights
since the graph of it clearly shows that it is not a direct linear relationship. Thus, a
polynomial equation will be considered. For this data, a quintic function is used
because by looking at the graph, there are 4 critical points or bends at year 1936,
1948, 1968, and 1971. A quintic function has 8 variables, a, b, c, d, e, f, and x. A
quintic function is a polynomial degree five, meaning that there are the variable x
with up to the power of 5, like as shown in the below quintic function.
Where the x variable is the year and f(x) is the height. The parameters of this
function include f, which is the y-intercept, and the coefficients a, b, c, d, and e, are
for the curving of the line. Next, a model function will be create base on the original

2. graph algebraically.Firstly, the year must be manipulated to have a y-intercept
shown where x=0. This is because to satisfy the range between 1632 and 1980 years
so that the y-intercept is at 197 cm of the year 1932 as the starting point. If the value
x would not be manipulated, then this would occur as shown below with a computer
generated quintic equation.
Figure 2: This graph shows the un-manipulated x-axis, where the y-intercept is at
approximately 460-million height cm, which shows an undesirable function.
If this model is used, then it would be harder to predict other values, so the year
number is manipulated as shown in the table below. Where every 4 years in the un-
manipulated data is equal to 1 year in the manipulated data.
Un-manipulated year Manipulated year Height (cm)
1932 0 197
1936 1 203
1940 (not played) - (2) -
1944 (not played) - (3) -
1948 4 198
1952 5 204
1956 6 212
1960 7 216
1964 8 218
1968 9 224
1972 10 223
1976 11 225
1980 12 236

3. Table 2: Notice that the year jump from 1 to 4 of the year 1936 and 1948 is because
2 years are left out 1940 and 1944. Which is the year 2 and 3 that are left out.
Secondly, to find the function model, at least 5 data points must be use to
make the model. In the following algebra work, the manipulated years 0, 4, 7, 9, 11,
and 12 will be use to generate the model.
The first thing we can get is the y-intercept, where the x or (manipulated year) is
equal to zero. Which is the height 197 cm. Thus the other 5 data points will be use to
find the model.
Manipulated Quintic Function after pluggin in the manipulated year as ‘x’
Year
4
7
9
11
12
Table 3: Shows the manipulated years and the equation with the plugged in y
(height), x (manipulated year), and f (y-intercept) of 197.
Manipulated Year Simplified Quintic Function
4
7
9
11
12
Table 4: Shows the manipulated years and the simplified quintic Function of Table 3.
Thus, from these equation, matrix is use to solve the unknown coefficient.
After using a calculator to solve the matrix, the a, b, c, d, e value is found.
Thus, a function model is found:

4. In the next step, a manipulated year data is shown, which is proportionality the
same as the original. And another graph will be shown with the original
manipulated year with the acquired quintic function model.
Figure 3: This shows the Original graph manipulated years.
Figure 4: This shows the added equation acquired earlier by the use of matrices.

5. The function model does seems to be accurate at following the data points, only 5
points that are not on the fit line does seem to be very near the line. There are
limitations however because after the last data point, the model seems to go up
infinitively, so this can only be use of the range from year 1932 to 1980, and can
predict the heights only in these years.
Figure 5: This shows a computer generated quintic function of the manipulated
years.
To compare these two functions:
Algebraically: RMSE: 2.71 cm
Using technology: RMSE: 1.848 cm
By comparing the two functions, the technology-generated function has RMSE of
1.848 cm, which has higher precision than the algebraically acquired function at
2.71 cm. However these two functions is very similar and is off by just a small
portion.
To estimate the winning heights in 1940 and 1944, plugging in the year 2 and 3 into
the computer generated formula will held a more accurate result.

6. These answer can be justified because according to the computer generated graph,
the value must be between 203 and 198 cm, since the height 203 cm is the year
1936 then the graph decreases to 198 cm in year 1948. The two values must then be
decreasing in the range of 203 and 198 cm.
Next, the computer-generated model will predict the winning height in 1984 and
2016. Where the manipulated years of these two will be 13 and 21 respectively.
From these two predicted results, it is impossible for a human being to physically
jump unassisted as high as 4790 cm. The current world record is 2.45 meters by
Javier Sotomayer in 1993. However 266 cm is still very high for human to jump.
Year 1896 1904 1908 1912 1920 1928 1984 1988 1992 1996 2000 2004 2008
Height 190 180 191 193 193 194 235 238 234 239 235 236 236
Table 5: This shows the additional data
The acquired quintic function model doesn’t seem to fit this additional data as seen
below. The line after and before each end seems to go on upwards or downwards
infinitely, leaving out the data points.

7. Figure 6: This shows the manipulated years of 1896 to be the starting point at x=0.
The line fit is only between the year 1932 and 1980 or at 8 to 20.
The overall trend from 1896 to 2008 can be described as like a logistic curve. From
the starting year to 1928, it seems to level out, except for year 1904. Then the from
year 1932 to 1980, it seems to increase steadily from height 197 cm to 236 cm. Later
on after this year, it seems to reach the human limit with around 235 cm. The
overall trend after 1980 is like a leveled line.
Some modifications that may need for the model to fit the data is to let the
computer generate again all the data point as a quintic function as shown below.
Figure 7: This shows the full generated quintic function of all the additional data.
However this function still cannot predict future years. To overcome this, a logistic
curve should be considered, since a logistic curve has a straight line incoming, an
increase in the middle, then a leveled out straight line at the end, which seems to
match the overall trend in the gold medal heights.