GOLD MEDAL HEIGHTS.docx

8/10/2019 GOLD MEDAL HEIGHTS.docx

1/14

1

Ahmet Ulusoy College

IB Mathematics SL Internal Assessment

Type II

Gold Medal Heights

Candidate Name: evval Beli

Candidate Number: 006615-006/dxm 499


2/14

evval Beli 006615-006

2

GOLD MEDAL HEIGHTS

Aim: The aim of this task is to consider the winning height for the mens high jump in theOlympic Games.

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

Table 1: The height (in centimeters) achieved by the gold medalists at various OlympicGames.

Graph 1: Height (in centimeters) achieved by the gold medalists at various Olympic Games. i

Variables, Parameters and Constraints

In this graph, x axis represents the year that the Olympic Game has taken place and y axis

expresses the gold medal winning height in each year. Domain of the function in the graph is

the x axis, years 1932-1980 and the range is from 197 cm to 236 cm. The year of the Olympic

Game is independent variable and the height is the dependent variable. The graph shows an

increase in height achieved over years, with an exception of the games in 1948 and 1972. This


3/14


3

situation makes it harder to determine a function that models the behaviour of the graph, since

there is not a monotonic increase or decrease in height between years. This situation is most

probably caused by the differences in capabilities of each competitor in the games. Only gold

medal winning mens high jump competitors have been taken into account and Olympic

Games were not held in 1940 and 1944. All of these are parameters and constraints for this

task.

Linear Function

I have noticed that the function in the graph is in an increasing trend, hence I have first tested

the linear function to see how it models the graph.

Graph 2: Line of best fit for the linear function in the Gold Medal Heights graph.


4/14


4

As seen in Graph 2, the line of best fit passes through two points and near to some points.

Even though the points for the years 1936, 1948 and 1952 are not close to the line of best fit,

the lines correla tion coefficient is close to one (0.9398); which indicates a good linear fit.

To be able to examine the graph analytically, we need to operate on the linear equation. Since

data points for years 1956 and 1964 are closest to the line, I thought it would be most suitable

to work with those values.

Linear equations can be written in this form:

y=mx+b

where m represents gradient of the line and b stands for the y-intercept.

m=

m= = =

y-intercept for data points of 1956 and 1964:

212+218= [ (1956+1964)] +2b

430=2940+2b

Average b value for data points of years 1956 and 1964= -1255

Using these calculations, I have drawn a new linear function, adjusting m as (0.75) and b as

-1255.


5/14


6/14


6

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

Table 2: The height (in centimeters) achieved by the gold medalists at various OlympicGames. Blue shaded column shows the median of the given data and grey shaded ones show

the data which deviates the most from the median value.

To refine my model, I have omitted the data points for the years 1932, 1948 and 1980 (grey

shaded columns in Table 2) and redrew the graph.

Graph 4: Linear line of best fit for the Gold Medal Heights graph (data points for the years

1932, 1948 and 1980 are omitted).

As seen in Graph 4, line of best fit passes through two data points, 1964 and 1972. Most data

points are very close to the line and the RMSE value decreased from 4.523 to 3.207,

indicating that this line is more accurate than the previous line of best fit. I have used two data

Median


7/14


7

points which the line of best fit passed through in Graph 2 to calculate the gradient of the line.

Hence, I have used data points of 1964 and 1972 to create a new equation.

m= =

y-intercept for data points of 1964 and 1972:

218+223= [ (1964+1972)]+2b

441= 2460+2b

Average b value for data points of years 1964 and 1972= -1009.5

Using these calculations, I have drawn a new linear function, adjusting m as (0.625) and b

as -1009.5.

Graph 5: Linear line of the derived equation on Gold Medal Heights graph (data points for

the years 1932, 1948 and 1980 are omitted).


8/14


8

Two missing data points in the graph, 1940 and 1944, makes it harder to find an accurate

model to explain the behaviour of the graph. The function model is a straight line, however

the winning heights do not increase rapidly, so this is a limitation for this model. Also, the

continuous increase in the linear function causes this model to be inapplicable to the real life

situation in this task. Human body has limits and because of this, assuming the gold medal

winning height will infinitely increase is unrealistic. Thus, a new model, which would be

maintainable under real life conditions, needs to be found.

Natural Logarithm Function

In the long run, it is reasonable to expect a stabilization rather than a continuous increase in

the graph. For this reason, I have thought that the natural logarithm function could model the

behaviour of the graph accurately and help me to visualize the possible gold winning heights

in the next Olympic Games.

Graph 6: Auto fit for the natural logarithm curve on the Gold Medal Heights graph.


9/14


9

(I have subtracted 1900 from the years of the Olympic Games in the x-axis of the graph (ex.

32 instead of 1932) in order to show the natural logarithm curve more clearly.)

As seen in the Graph 6, the natural logarithm curve is very close to or passing through data points 60, 64, 72 and 76. Apart from the years 1948, 1952 and 1980, the natural logarithm

function seems to model the behaviour of the graph better than the linear function.

Graph 7: Natural logarithm and linear functions on the Gold Medal Heights graph.

Most of the data points are very close to the natural logarithm curve, causing it to be a more

accurate model than the linear line. Also, predictions made with the natural logarithm

equation will most probably be closer to the actual future values than the ones made with the

linear equation, since the natural logarithm curve models the behaviour of the data more

realistically. Hence, I have estimated the results for the years 1940 and 1944 using natural

logarithm equation. I chose to work with data points for years 1960 (60) and 1976 (76)


10/14


10

because the natural logarithm curve passes through them and so I thought they would

represent the general behaviour of the curve more accurately than other points.

Natural logarithm equations can be written in this form:

y= a+ b

Data point for year 1960 1:

216=a+ b

216=a+ 7.580699752b

Data point for year 1976:

225=a + b

225=a+ 7.588829878b

Solving equations simultaneously:

225=a+ 7.588829878b

216=a+ 7.580699752b

9=0.008130126b

b=1106.993914

a=-8175.788487

y=-8176+ 1107

1940:

1 Calculations have been done using Texas Instruments TI-84 Plus Silver Edition graphing calculator.


11/14


11

y=-8176 + 1107

y=204.48068

1944:

y=-8176 + 1107

y=206.7608044

Calculations showed that the winning heights for 1940 and 1944 Olympic Games would have

been 204 cm and 207 cm, respectively. Although these values are realistic and within the

range of the given data, I believe the winning heights in the 1940 and 1944 Olympic Games

would have been lower. Considering the reasons behind the cancellation of the Olympic

Games in these years, such as World War II, I presume that the athletes could not have been

able to train as much as they could have for other Games. In my opinion, the actual heights

would have been around 190 cm, taking the winning heights in 1936 and 1948 into account.

My predictions for the winning height in 1984 and in 2016 are as follows:

1984:

y= -8176 + 1107

y= 229.3074086

2016:

y= -8176 + 1107

y= 247.0197865

Calculations showed that the winning heights for 1984 and 2016 Olympic Games would be

229 cm and 247 cm respectively. Considering the winning heights in 1980 and 2008, these


12/14


12

values are reasonable and realistic. The actual mens high jump gold medal winner in 1984

Games was Dietmar Mgenburg with 235 centimeters. This is a higher value than my

prediction but it is not different enough to invalidate my model.

Gold Medal Heights 1896-2008

Graph 8: Natural logarithmic curve on the Gold Medal Heights 1896-2008 graph.

With additional data, natural logarithmic function still fit the Gold Medal Heights graph

reasonably well. Even though there are some deviations, the function seems to reflect the

overall trend of growth in the graph.


13/14


13

Graph 9: Natural logarithmic curve on the Gold Medal Heights 1896-2008 graph. (Zoomed

out)

Although there are some inconsistencies in the graph, the overall trend from 1896 to 2008 is a

positive increase. Because of the differences in the abilities of human beings, a number of

unstable results can be expected, however there are some significant fluctuations in the Gold

Medal Heights graph. Especially the data points for winning heights in the 1904, 1948 and

1980 Games remarkably deviates from the natural logarithm curve. When I have looked for

the reasons behind these variations, I came across some interesting facts. Firstly, 1904 was the

year of the third Olympic Games, thus I assume the inexperience of athletes and trainings

without a good technique caused the deviation in this year. 1948 Games had been held three

years after the World War II, so the athletes may not have been able to train as much as they

could have in those years. I could not find any information about the 1980 Olympics that

could be related to the variation in the graph; in my opinion the winner athlete worked harder


14/14


14

and/or used a different technique that helped him to stand out amongst others and break the

previous world record.

For my part, no matter which function will be used to model the behaviour of the Gold MedalHeights graph, there will always be some amount of variation present because each human

being has different capabilities and there are many factors affecting each athletes success;

motivation, stress, diet, technique and so on. The winning heights expected to rise in the

future Olympic Games until the physical limits of the human body are reached. Natural

logatihm function was appropriate to show the potential stabilization of the graph in the

future, but a new model that could also fit the fluctuating data would represent the overall

trend more accurately.

i I have used Logger Pro 3.8.6.1 Demo to draw all graphs in this assignment.

Documents

GOLD MEDAL HEIGHTS.docx