Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

1

INTRODUCTION Intertwined into nearly every aspect of the Korean system of martial arts known

as taekwondo is the quintessential structural tenet that expresses a method of self-defense

that allows swift, agile bodies to overcome larger, bulky opponents. This tenet asserts that

every successful student must coalesce a well-developed sense of power in the core with

the well-trained lower body musculature that enables speed. Although expressed by

taekwondo instructors in varied ways, the concept of this tenet is illustrated here by a

quote:

“The principles of Taekwondo techniques are based on the design of your body. For

power you develop the larger, powerful muscles of the torso. The speed of the techniques

comes from the fast, agile muscles of the arms and legs. As you progress in Taekwondo,

you will learn to coordinate this speed and power, and develop a concentration to focus

all of your body's strength into a small, hard striking surface like the edge of the hand or

the heel of a foot. Taekwondo allows a woman to emphasize many of her natural physical

strengths, such as power in the legs, while learning a method of self-defense efficient

against a much larger opponent” (Johnson, Nathan. Martial Arts For the Mind).

Task Statement Due to my personal experience in a taekwondo dojang, or training facility, I

witnessed firsthand how smaller, lighter students consistently had the advantage over

larger, more muscular opponents due to the development of speed and power. Therefore,

I aim to test the validity of this tenet on a larger scale by determining whether there is a

correlation between the weight of a competitive taekwondo practitioner and success

within taekwondo. The results of determining this correlation will, in a more general

sense, provide insight into how successful taekwondo is in its pursuit of an efficient

method of self defense against unevenly matched opponents.

Investigation Method Throughout my research, I will control the country and martial art system that I

will analyze by only considering taekwondo practitioners in America. The amount of

males and females will be the same with 25 practitioners of each gender. In addition, I

will determine the level of success in taekwondo based on one main variable taken from

an Internet source: the total number of competitions a practitioner has participated in. As

additional variables that may affect the ones mentioned above, I will also record each

practitioner's gender, age, and weight.

To most relevantly interpret the trends among variables and existence of a

correlation between factors, my analysis will rely on the utilization of the mean, median,

summation, and standard deviation will be employed to ascertain the measure of central

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

2

tendency as well as the characteristic of the spread of data regarding whether data points

are clustered or spread apart.

Furthermore, a box and whisper plot will portray the clearest graphical

representation of the five number summaries, consisting of the minimum value,

maximum value, upper quartile, lower quartile, and median, that will aid in the

illustration of the overall characteristics of the data. Outliers will be clearly defined and

included to demonstrate any deviations from the observed trends, determine the reliability

of the results, and guide the construction of generalizations.

Correlation will be ascertained via scatter plot of the continuous data variables of

weight and competitions in addition to weight and medals. This will then be coalesced

with a line graph including Pearson’s Product-Moment Correlation and the line of

regression. By including this graph, I will be able to represent the strength, direction, and

linearity of the correlation.

Accounting for the previously mentioned variables will provide a holistic view of

possibly influential factors on weight and success in taekwondo, while coincidingly

focusing this study on testing a cherished fundamental principle of America’s most

popular martial art system.

GENERATED MEASUREMENT

FIGURE 1: Table Showing U.S.A Taekwondo Practitioner Information

Practicioner

Name

Weight

(in kilograms)

Gender Number of

Competition

Nia Abdallah 56.7 female 55

Natalie Aguliar 54.9 female 9

Alexander

Ahlstrom

72.6 male 17

Nathanael

Ahlstrom

62.6 male 14

Kacey Anderson 54.4 female 13

Lilian Angel 52.2 female 19

Ryan Arcaina 66.2 male 8

Jesus Armendariz 60.8 male 14

Sean Aston 58.1 male 26

Dalia Avivi 72.6 female 29

Curtis Barnett 84.0 male 26

Antony Graf 83.5 male 64

Christine Beckel 68.0 female 5

Stephanie Beckel 58.5 female 50

Anthony Bell 70.8 male 16

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

3

Kristina Bowen 61.2 female 12

Barbara Brand 46.7 female 32

Caroline Bui 59.0 female 7

Christi Burke-

Maves

51.3 female 40

Lauren Cahoon

Hamon

77.1 female 36

Kenneth Carter Jr. 61.2 male 32

Danielle Harrison 72.3 female 20

Kristi George 58.1 female 26

Aziza Chambers 54.0 female 37

Kevin Chavez 54.9 male 2

Rene Chen 52.2 female 19

Muhammad

Chishti

69.9 male 19

Kelly Chu 50.8 female 18

Alexandra Cinque 58.5 female 25

Josh Coleman 77.1 male 28

Garth A. Cooley 81.6 male 54

Jon Marc Cortez 57.6 male 24

Daniel Cosmillo 77.1 male 16

Nikolau

Gottschalk

62.1 male 15

Samantha Cravens 57.6 female 8

Ally Creel 57.6 female 27

TJ Curry 79.4 male 11

Whitney Daniel 57.6 female 20

Jennifer Daye 61.2 female 16

Medhi Dehghani 86.2 male 43

Brandon DeSouza 76.2 male 15

Simone DeVito 49.9 female 41

Faith Dortch 66.7 female 14

Phil Drake 86.2 male 14

John Drath 78.0 male 9

Matt Dunlap 66.7 male 51

John Duhart II 53.5 male 31

Daniel Elkowitz 53.5 male 23

Candy Esparza 72.6 female 6

Luke Ford 79.8 male 21

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

4

All USA Taekwondo Athletes. United States Olympic Committee. 2011.< http://usa-

taekwondo.us/athletes>

DESCRIPTION: In the table above, the athletes chosen were listed in the order they

were found on the online database. The weights were converted from pounds to

kilometers using the conversion factor:1 kilogram = 2.20 pounds in order to ascertain

that the measurements were in accepted metric units. These kilogram values were

rounded to one decimal place in order to keep the data within three significant figures. No

outlier points have been omitted or singled out of the data in this particular table. The

numbers of medals include gold, medal, and bronze. The numbers of competitions

include all the competitions the practitioner participated in regardless of where or when

the competition was held or whether or not a medal was obtained.

MATHEMATICAL PROCESSES

FIGURE 2: Scatter Plot of Weights Versus Competitions For Both Genders

DESCRIPTION: The following graph was created through the aid of the website

program, nces.ed.gov/nceskids/createagraph/. Through this program, adding data points,

naming labels and legends, selecting graph style, choosing colors and symbols, and

identifying the starting and ending points of the scales on each axis were all selected or

manually inputed. Only the image of the graph and intervals of the scale were

electronically created by the program. In order to increase the clarity of the graph, all

weights were rounded to the nearest whole number. This graph was not used to do the

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

5

find pearson’s coefficient or the regression line, but to visually demonstrate the differing

spreads of data between the male and female practitioners.

FIGURE 3: Scatterplot Showing Weight Versus Competitions For Males

DESCRIPTION: Also facilitated through the utilization of

nces.ed.gov/nceskids/createagraph/, this graph only shows the number of competitions

and weight in kilograms for the male practicioners. The information on this graph was

used to find both the r value in pearson’s product moment correlation and the equation of

the regression line. As in the first joint scatterplot, the weights were rounded to the

nearest whole number for clarity.

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

6

FIGURE 4: Scatterplot Showing Weight Versus Competitions For Females

DESCRIPTION: Just as with Figure 1, this graph was created through the utilization of

nces.ed.gov/nceskids/createagraph/. However, this graph only shows the number of

competitions and weight in kilograms for the female practitioners. The information on

this graph was used to find both the r value in pearson’s product moment correlation and

the equation of the regression line. As in the first joint scatterplot, the weights were

rounded to the nearest whole number for clarity.

Pearson’s Product-Moment Correlation With this process, the direction and strength of the correlation between two

variables can be ascertained. If the sign of r is -1, then the two variables are said to be in a

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

7

perfect negative correlation. However, if the sign of r is +1, then the two variables are

said to be in a perfect positive correlation. When the value of r is between +1 and -1, the

correlation is considered strong. When the value of r is close to 0, then the correlation is

considered weak. Depending on the how close the points of the scatterplot are to the

regression line, the correlation can be strongly, moderately, or weakly linear. The formula

is as follows:

1) where Sxy, or covariance, is the sum of all x values minus the mean of all x values

multiplied by the sum of all y values minus the mean of all y values

2) where SxSy is the square root of the covariance of the x values squared multiplied by

the covariance of the y values squared

FIGURE 5: Table of Covariance, Mean, and Sum Values for Male Practitioners

Male Weight and Competition Data

X 73 63 66 61 58 84 71 61 55 70 77 82 55 77 79 86 76 86 78 67 54 54 80 62 84 70

∑1759

Y 17 14 8 14 26 26 16 32 2 19 28 54 24 16 11 43 15 14 9 51 31 23 21 15 64 24

∑593

(x-x) 3 -7 -4 -9 -

12

14 -1 -9 -

15

0 7 12 15 7 2 16 6 16 8 -3 -

16

-

16

10 -8 14 ∑29

(y-y) -7 -

10

-

16

-

10

2 2 -8 8 -

22

-5 4 30 0 -8 -

13

19 -9 -

10

-

15

27 7 -1 -3 -9 40 ∑-7

(x-x)² 9 49 16 81 14

4

19

6

1 81 22

5

0 49 14

4

22

5

49 4 25

6

36 25

6

64 9 25

6

25

6

10

0

64 19

6

∑2766

(y-y)² 49 10

0

25

6

10

0

4 4 64 64 48

4

25 16 90

0

0 64 16

9

36

1

81 10

0

22

5

72

9

49 1 9 81 16

00

∑5535

DESCRIPTION:In order to keep the values consistent with the scatterplot, the x and y

values were rounded to the nearest whole number as opposed to having one decimal

point. Both the means and the summation were included. The table was presented

horizontally in order to increase clarity and save space.

Final Calculation

Sxy = (29)(-7) SxSy = √ (2766)(5535)

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

8

= -203 = √(15309810)

= 3912.775

r = Sxy/SxSy = (-203/ 3912.775) = -0.05188 = -0.0519 so r = -0.0519

FIGURE 6: Table of Covariance, Mean, and Sum Values for Female Practitioners

Female Weight and Competition Data

X 57 55 54 52 73 68 59 61 48 59 51 77 72 54 52 51 59 58 58 58 61 50 67 73 52 59

∑147

9

Y 55 9 13 19 29 5 50 12 32 7 40 36 20 37 19 18 25 8 27 20 16 41 14 6 26 23

∑584

(x-x) -3 -5 -6 -8 13 8 -1 1 -

12

-1 -9 17 13 -6 -8 -9 -1 -2 -2 -2 1 -

10

7 13 -8 ∑-20

(y-y) 31 -

15

-

11

-5 5 -

19

26 -

12

8 -

17

16 12 -3 13 -5 -6 1 -

16

3 -4 -8 17 -

10

-

18

2 ∑-15

(x-x)² 9 25 36 64 16

9

64 1 1 14

4

1 81 28

9

16

9

36 64 81 1 4 4 4 1 10

0

49 32

4

4 ∑172

8

(y-y)² 96

1

22

5

12

1

25 25 36

1

67

6

14

4

64 28

9

25

6

14

4

9 16

9

25 36 1 25

6

9 16 64 28

9

10

0

32

4

4 ∑459

3

DESCRIPTION: In order to keep the values consistent with the scatterplot, the x and y

values were rounded to the nearest whole number as opposed to having one decimal

point. Both the means and the summation were included. The table was presented

horizontally in order to increase clarity and save space.

Final Calculation

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

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Sxy = (-20)(-15) SxSy = √(1728)(4593)

= (300) = √(7936704)

= 2817.216

r = Sxy/SxySy = (300/2817.216) = .10649 so r = .106

Least Squares Method of Finding the Regression Line Equation Through this process, I can ascertain the equation of the line of regression and

whether it is an appropriate fit for the particular data based on the principle that most of

the distances between a specific data point and the regression line will be small. As a

result, the sum of their squares will be small as well. The formula, which is in the

gradient-slope form is as follows:

y - = Sxy/ Sx² (x - )

FIGURE 7: Table of Male Values for Regression Equation

Sxy Sx²

70 24 -203 841

DESCRIPTION: The values in this table were taken from Figure 5 and the calculation

of the r value below it. Although Sx² was not a part of that previous calculation or the

table, it was easily determined from the value of Sx in the calculation, which was 29.

Therefore (29)² = 841.

Final Calculation

y - = Sxy/ Sx² (x - )

y - 24 = (-203) /(841)(x - 70) = y -24 = (-.2414)(x - 70) = y - 24 = -.2414x +16.898

y = -.241x + 40.898

FIGURE 8: Table of Female Values for Regression Equation

Sxy Sx²

59 23 300 400

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

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DESCRIPTION: The values in this taken were taken from Figure 6 and the calculation

of the r value below it. Although Sx² was not a part of that previous calculation or the

table, it was easily determined from the value of Sx in the calculation, which was -20.

Therefore (-20)² = 400.

Final Calculation

y - = Sxy/ Sx² (x - )

y - 23 = (300) /(400)(x-59) = y-23 = .75(x -59) = y -23 = .75x -4425

y = .75x - 4402

Quartiles Calculation The quartiles serve as percentages that, when displayed on a box and whisker

plot, allow conclusions to be made concerning whether the spread of data is more

clustered together or more spread apart. The lower quartile is the 25% mark of the data,

the median is the 50% mark, and the upper quartile is the 75% mark. The median is an

effective measure of central tendency in this case because it gives the halfway point of

the data by only considering middle values and it is not affected by extreme values. The

formulas for finding each quartile are listed below:

upper quartile = Total x (¼) lower quartile = Total x ( ¾) median = {(n+1)/2}

FIGURE 9: Table Of Quartiles, Minimum Value, and Maximum Value for Males

Minimum X Maximum X Lower Quartile Median Upper Quartile

54 84 62 71 79

Minimum Y Maximum Y Lower Quartile Median Upper Quartile

2 64 15 19 28

(n + 1) x (¼) = 26 x (¼) = 6.5th value

(n + 1) x (¾) = 26 x (¾)= 19.5th value

(n+1/2) = (25+1/2) = 13th value

X Values:

54, 54,55,55, 58, 61, 61, 62,63, 66, 67, 70, 71, 73, 76, 77, 77, 78, 79, 80, 82, 84, 84, 86,86

Y Values:

2, 8, 9, 11, 14, 14, 14, 15, 15,16, 16,17, 19, 21, 23, 24, 26, 26, 28, 31, 32, 43, 51, 54, 64

FIGURE 10: Table of Quartiles, Minimum Value, and Maximum Value for Females

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

11

Minimum X Maximum X Lower Quartile Median Upper Quartile

47 77 53 58 61

Minimum Y Maximum Y Lower Quartile Median Upper Quartile

5 55 13 20 32

(n + 1) x (¼) = 26 x (¼) = 6.5th value

(n + 1) x (¾) = 26 x (¾)= 19.5th value

(n+1/2) = (25+1/2) = 13th value

X Values:

48, 50,51, 51, 52, 52, 52, 54, 54, 55, 57, 58, 58, 58, 59, 59, 59, 61, 61, 67,68,72,73,73,77

Y Values:

5, 6, 7,8,9, 12,13,14,16, 19, 18,19, 20, 20,25,26,27,29,32, 36,37, 40, 41, 50, 55

FIGURE 11: Box Plot for Male X (Weight) Values

DESCRIPTION: Due to the inability of both the GDC and graph-making programs to

show labels on their generated box plots, I opted for the traditional method of hand-

drawn graphs and proceeded to take a picture of each. The same is true for the following

box plots. In this particular box and whisker plot, the distribution imitates that of a

perfect bell curve with symmetrical distribution, with the values being more clustered

together.

FIGURE 12: Box Plot for Male Y (Competition) Values

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

12

DESCRIPTION: The distribution of this graph is positively skewed, with the majority

of the number of competitions tending toward the higher values. However, the mean is

much nearer to the lower quartile than the upper quartile.

FIGURE 13: Box Plot for Female X (Weight) Values

DESCRIPTION: Similar to the male weights, the female weights remained clustered

together, however a slight positive skew is evident from the mean’s proximity to the

upper quartile and the larger amount of weights tending toward the higher values.

FIGURE 14: Box Plot for Female Y (Competition) Values

DESCRIPTION: This quartiles on this graph were more spread apart than on the others.

The positive skew is again evident and the scale in increments of 5 applies well as in

Figure 12.

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

13

INTERPRETATION AND VALIDITY

Conclusions In the results of pearsons's product moment correlation, the correlation between

the weights and number of competitions for the male practitioners was a weak, negative

correlation, as shown by the value, r = -0.0519. This means that as the weights of the

male practitioners increases, the number of competitions they are able to participate in

decreases. One point of evidence for this is in the scatterplot in Figure 3, which

demonstrates how the points that are clustered near 70-80 kilograms do not reach an

amount of competitions higher than 30. However, since the correlation is weak, there are

points which lie in contradiction to this overall conclusion. As would be expected from

such a conclusion, the practitioners with weights between 50-60, should have the highest

number of competitions, yet none of the points that fell within those weights had an

amount of competitions higher than 36. Therefore, they lighter males do not have a

significant advantage over the heavier males. In fact, the male with the lowest number of

competitions, two, had a weight of 55 kilograms, which does not conform to the meaning

of the overall negative correlation. In addition, all except one of the four males with the

highest number of competitions had weights over 80 kilograms.

Concerning the quartiles for the males, both the upper and lower quartiles for the

X and Y values were particularly close to the mean, with no difference being greater than

10. However, the large variability of the data can be seen in that the range between the

maximum and minimum X values was 30, and for the Y values was 62. This relationship

is more clearly understood by the visual layout provided by the box and whisker plots.

For the X values, or weight of the male practitioners, shown in Figure 11, the distribution

of the data conforms almost perfectly to a normal bell curve, demonstrating that despite

the large range between the minimum and maximum values, the points are clustered

together. As for the box plot of the Y values, or number of competitions, in Figure 12, the

variability of the data can be seen as the graph is positively skewed. Overall, for the male

data, according to the weak, negative r value, conforms to the taekwondo size tenet that

smaller opponents have a greater advantage than larger ones.

As for the results of the female practitioners, pearson's product moment

correlation resulting in a weak, positive correlation due to the fact that r = .106. This

leads to the conclusion that as the weights of female practitioners increases, the number

of competitions they are enabled to participate in increases as well. After analyzing the

scatterplot in Figure Four, it can be noted that there are contradictions with this

conclusion as well. For instance, most of the women have weights that are between 50-60

kilograms. In fact, the two women with the lowest number of competitions have weights

of 68 kilograms and 73 kilograms. Although one female of 77 kilograms participated in

36 competitions, the five women with the highest number of competitions had weights

below 60 kilograms, which does not conform with the meaning of the positive

correlation.

In the quartile calculations for the female practitioners, the differences between

the median and the upper and lower quartiles are below 10, with one exception of 12

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

14

between the median and upper quartile for the Y values, or number of competitions. In

fact, the quartiles and the median coincided so closely that the difference between the

upper quartile and median for the X values, or weights, was only 3. The ranges between

the maximum and minimum values, like the males, was considerably large, with the

range of minimum and maximum X values being 30 and the range for the Y values being

50. As portrayed by the box plot in Figure 13, the X values were more spread apart near

the higher weights. This same trait is apparent with the Y values in Figure 14, however,

the distance between the median and upper quartile is significantly larger. Overall, the

females stood in dissident with the taekwondo size principle due to the weak, positive

value of r.

Limitations Concerning the topic itself, the difficulty arose in attempting to define “success”

in taekwondo, when so many factors contribute to such a broad, inspecific term.

Although I decided to focus on the number of competitions as the definition of “success,”

this is a far from accurate portrayal for several reasons. For one, although a certain level

of skill is required to be able to enter into competitions, having entered a huge quantity of

competitions does not necessarily mean that a person is exceptionally skilled at

taekwondo. This is because a practitioner could have entered 50 competitions, but have

won no medals, whereas another practitioner may have entered 10 competitions and

walked away with 25 medals. However, even looking at the number of medals alone is

not sufficient because there are gold, silver, and bronze medals which indicate differing

skill levels.

In addition, although gender is considered as an additional factor, height and age

are neglected. However, even if age is considered, a practitioner who is 30 years old may

not necessarily have more experience and skill than a 15 year old practitioner, depending

upon when each person began to seriously practice taekwondo. In this investigation, only

competitive taekwondo practitioners were considered, which limits the reliability of this

investigation to mirror the larger population. Furthermore, the athletes were all taken

from the official United States Olympic team. Certainly smaller dojangs across the nation

have skilled taekwondo athletes, but may not have had the money or transportation to

send their practitioners to the competitions, but this by no means should suggest that they

are less skilled or “successful” in the martial art.

As for the mathematical processes used, the use of least squares linear regression

was suitable for the data being dealt with, however the trend line equations did not offer

any really valuable information to the investigation. This is due to the fact that the

equations were not included on the scatterplots, so that the visual aspect was not fulfilled.

However the slopes and y intercept in the equation would not have coincided well with

the scale on the scatterplots. In addition, the coalescence of the r value and visual

scatterplots sufficiently describe the variability and nature of the correlation so that the

linear regression equation is largely unnecessary.

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

15

Although a chi-squared test was originally attempted, the type of data being used

was not appropriate for such as sophisticated math process. In addition, testing the data

for independency largely would distract from the primary goal of determining correlation

between weight and the number of competitions.

The utilization of pearson’s product moment correlation, statistical quartiles,

scatterplots, and box and whisker plots were appropriate, effective choices of

mathematical processes for this data. With pearson’s correlation, the existence, strength,

and direction of a correlation could be deduced all from one compact equation. As for the

scatterplots, they were a prudent choice of graph because they are designed to handle

continuous data, such as weights, and are usually employed when the alteration of the

two variables is not possible, as they were in the case of this investigation since the data

was generated from an online source. In addition, the visual layout of the scatterplot

allows quick, clarified observation of the clustering of the data, the presence of outliers,

and the existence of a positive or negative skew within the data. Lastly, the quartiles and

their presentation in box and whisker plots allows for the relationship between the

measure of central tendency, which in this case is the median, and upper and lower

percentages of the data to be readily visible along with an alternative visual of the

variablity of the data.

However, it could be arguable that the median was not the best measure of central

tendency because the mode would have given the most usual value by only taking the

most common values into account and it also would not have been affected by extreme

values. Likewise, the mean could have been a useful measure of central tendency due to

its most frequent usage and easy comprehension among people. However, all values

would have been taken into account and the mean would have been affected by extreme

values.

Possible Modifications In light of the previously mentioned limitations and evaluation of the applicability

of the mathematical processes employed, the following improvements could be made in

order to increase the reliability of the investigation and make is more exemplary of the

larger population.

One possible alteration would be to consider other variables such as height and

age, which can easily be found from a database such as the one I used. The way in which

these variables could be “considered” is by creating a double column graph showing age

and height of each practitioner and then analyzing to see if any trends exist between

extremes of heights and ages with the success in taekwondo.

On the subject of success, this could be redefined in terms of medals won, hours

spent training, or years spent studying taekwondo formally. However, this information

would need to be gathered by survey, which would make the accumulation of data more

difficult and time consuming. Lastly, the investigation could be shifted to focus on non-

competitive practitioners or even employ a comparison between competitive and non-

competitive practitioners.

Does Size Matter?

A Correlative Study in the Martial Arts

Andrea Mendez

16

Even so, no combination of modifications can be conjured that will completely

evaluate a practitioner’s success. However, these modifications can serve to evaluate the

proficiency of a practitioner in certain, specific areas of the martial art.