Height vs. Flexibility of a DancerAn investigation on seeing if there is a relationship between the height of a dancer and

their flexibility.

Melanie Bunker

IB Mathematical Studies IA

Candidate number: XXXXXX

International School of Bangkok

Ms. Goghar

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Table of Contents

Introduction and method…………………………………………page 3

Raw Data Collection…………………………………………….page 4

Calculations: box and whisker plots…………………………..page5

Calculations: box and whisker plot and scatter plot………..page 6

Calculations: Cumulative frequency graphs…………………………………………………………....page 7-8

Calculations: Standard deviation of height, flexibility ………………………………………………….…………………..page 9

Calculations: coefficient variation, Pearson’s Correlation Coefficient ……………………………………………………….page10

Chi-Squared test: Observed Values Table…………………..page 11

Chi-Squared test: Expected Values Table…………………...page 12

Validity…………………………………………………………...page 13

Works Cited……………………………………………………...page 14

Introduction:

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Participating in dance classes has made up my extracurricular activities over the years. Every year I make personal goals to become more flexible so that my dance technique and ability will continue to grow and develop. Out of the past seven years that I have been dancing, I have noticed that some dancers are more flexible than others. Some are shorter than the average height with a wider range of flexibility while others that are around the average height (or taller) are just as flexible. To investigate this, I will focus on measuring height and flexibility. I want to see whether or not flexibility has an affect on a dancer’s height.

A measuring tape will be used to measure the height in centimeters. There are many ways in which one dancer can be flexible, and measuring the flexibility of the hamstring is one of the main ways. A Sit-n-Reach test will be used because it specifically measures the flexibility of the lower back as well as the hamstrings. To do this, a box with a board on top that extends 50 centimeters was collected; refer to the image shown in Figure 1. Each centimeter, beginning at 1 to 50 is marked off on the board. This test is typically known as the “Sit-n-Reach” test where the tester will sit on the ground putting both legs flexed on the base of the board and measure how far he or she can reach over his or her’s legs. The test will include only both feet flexed at the base of the board while the test subject reaches as far as they can on the board. The data was collected when the dancer’s muscles were not warmed up to see how flexible they are when they are not dancing.

Statement of Task:

The aim of this project is to find out whether or not the height of a teenage dancer has an affect on their flexibility.

Method: Measuring tape was used to measure the height of the dancer. A Sit-n-Reach was used to measure the flexibility of dancer’s hamstring.

1. After the materials are collected, measure the height of the dancer using the measuring tape and record it in centimeters.

2. Take the same dancer and have them place their feet at the base of the Sit-n-Reach. Have them place one hand on top of the other and reach as far as they can on the board without them bending their knees or raising their shoulders. *Note that when measuring each dancer, make sure that they are not warmed up. It is important to measure their natural flexibility.

3. Record all information onto data table. Repeat until 50 data points have been collected.

Table 1- This table shows the raw data collection from 50 dancers ranging in height and flexibility. All dancers that were tested were in between the ages of 15-18 and have all danced at least for one year.

Gender of Dancer Age Height (cm) Sit-n-Reach Both legs (cm)Male 16 171.5 30Male 16 172 41

Female 16 169 27Female 17 165 16

Figure 1: Image showing what test subjects will do to measure their flexibility. <http://www.topendsports.com/testing/tests/sit-and-reach.htm>

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Female 15 162 29Female 16 156 31Female 17 163.5 57Female 17 161.5 34Female 16 168 42.5Female 17 175 34Female 17 153 35Female 17 165 44Female 17 152 35Female 15 152 33Female 15 162 42Female 14 159 50Female 18 161 57Female 17 165 32Female 15 161 40Female 16 156 45Female 18 162 37Female 15 158 43Female 17 164 38Female 15 159 47Female 17 172 36Female 16 170 40Female 15 158 44Female 15 167 48Female 16 165 42Female 17 163 43Female 15 155 45Female 16 161 44Female 16 163 39Female 17 166 37Female 15 159 30Female 16 162 48Female 17 164 50Female 17 163 45Female 15 160 46Female 16 159 38Female 16 164 44Female 17 169 38Female 16 167 43Female 16 166 46Female 17 167 30Female 17 161 37Female 15 169 32Female 16 165 41Female 17 167 34Female 16 163 32

By calculating the average, minimum, maximum, lower quartile, median and upper quartile, it is the first step to obtain simple math processes that will be used in future calculations. These calculations help measure the spread of the data and help keep it

Average height:

∑ height

50 =

1971 .550

= 163.15 cm

Minimum height: 152 cm

Average flexibility:

∑ flexibility

50 =

1971 .550

= 39.43 cm

Minimum flexibility: 16 cm

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organized.

Mathematical Process:

By using the Box & Whisker Plot it will help demonstrate the data in a way that is easier to read all of the fifty pieces of data that was collected. There is a separate Box & Whisker plot for the height of the dancers and one for their flexibility. The calculations for each Box & Whisker Plot are shown below Table 1.

Average height:

∑ height

50 =

1971 .550

= 163.15 cm

Minimum height: 152 cm

Average flexibility:

∑ flexibility

50 =

1971 .550

= 39.43 cm

Minimum flexibility: 16 cm

Box & Whisker Plot: Height of 50 Dancers (cm):

Box & Whisker Plot: Flexibility of 50 Dancers (cm):

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Next, all of the data was placed into a scatter plot to visually see the spread of data as well as the line of regression. When the data is placed into a scatter plot it is easier to see if there are any outliers. Looking at Figure 2, the dancer with a height of 165 cm has a flexibility of 16 cm. It is clear to see that this piece of data is the lowest value where dancers with shorter heights of 163.4 cm and 161 cm both have the highest value of flexibility of 57 cm.

Figure 2-This Scatter Plot shows the spread of data that was collected and as well as the line of regression. It also includes the mean of the data set.

Each variable, the height of the dancers and the flexibility of the dancers, were then placed into separate cumulative frequency tables by using the raw data that was collected. These tables make it easier to visually see the distribution of the data.

10 15 20 25 30 35 40 45 50 55 60140

145

150

155

160

165

170

175

180

Scatter Plot of Flexibility of Dancer vs. Their Height

Flexibility of Dancer (cm)

Hei

ghth

of D

ance

r (c

m)

Legend:

= mean of data set; (39.43,163.15)

= each piece of data

Table 3.0 Table displaying the intervals and frequencies of the flexibility measurements.

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Figure 3- A cumulative frequency graph showing height using data from Table 2.0.

By placing the data onto a cumulative frequency graph, it tells us the number of data items are under a certain value. In this case, the median is marked as 163 cm and from this, you know that 20 students were under the height of 163 cm. The upper quartile, which is 167 cm, tells us that 8 students were taller than the 75th percentile. And for the

Table 2.0 Table displaying the heights recorded from teach test subject.

150 155 160 165 170 175 180 1850

10

20

30

40

50

60Cumulative Frequency

Height (cm)

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Height (cm) Interval

Frequency Cumulative Frequency

150-154 3 3

155-159 9 12

160-164 17 29

165-169 16 45

170-174 4 49

175-179 1 50

Flexibility (cm)

Interval

Frequency Cumulative Frequency

15-19 1 120-24 0 125-29 2 330-34 11 1435-39 10 2440-44 14 3845-49 8 4650-54 2 4855-59 2 50

lower quartile, having a height of 160 cm, it tells us that only 8 students are shorter than 160 cm. From knowing this, we can see the heights of all the students that participated in this experiment.

Figure 4- This cumulative frequency graph shows the length of the flexibility from Table 3.0.

After placing the cumulative frequency data of the flexibility length onto a graph, we can see more clearly the number of dancers that are more flexible with the higher results and can compare it to the dancers who are not as flexible, and could not reach as far on the Sit-n-Reach test. The median for this graph is about 35 cm, telling us that 25 of the students that were tested had a flexibility of less than 35 cm. The upper quartile is about 39 cm, so this tells us that more than 12 people had a flexibility higher than 39 cm. And the lower quartile, had a flexibility of about 29 cm, so that tells us that about 38 people had a higher flexibility than 29 cm, but 12 people had a flexibility lower than 29 cm.

Table 2.1 Calculations for the Standard Deviation of Height (cm):

10 15 20 25 30 35 40 45 50 55 600

10

20

30

40

50

60Cumulative Frequency

Length of Flexibility (cm)

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Class IntervalMidpoint

(x)Frequency

(f) (f)(x) x- 2

150-154 152 3 456 -11.15 372.9675155-159 157 9 1413 -6.15 340.4025160-164 162 17 2754 -1.15 22.4825165-169 167 16 2672 3.85 237.16170-174 172 4 688 8.85 313.29174-179 177 1 177 13.85 191.8225

∑=1478.15425

= 5.44cm

The data collection for the height of the dancers can be expressed in a range as follows;

152 h 175 cm. The standard deviation that was calculated can tell us that the spread of the height data is ±5.44 cm away from the , therefore it is a wide range. These values tells us that for the heights of the dancers that there is a wide range of data away from the mean, and how far off from the mean the data is.

Table 3.1 Calculations for the Standard Deviation of Flexibility Measurements (cm):

Class IntervalMidpoint

(x)Frequency

(f) (f)(x) x- 2

15-19 17 1 17 -22.43 503.104920-24 22 0 0 -17.43 025-29 27 2 54 -12.42 308.512830-34 32 11 352 -7.43 607.253935-39 37 10 370 -2.43 59.04940-44 42 14 588 2.57 92.468645-49 47 8 376 7.57 458.439250-54 52 2 104 12.57 316.009855-59 57 2 114 17.57 617.4098

∑=2962.248

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cm

The number from the numerator in the equation was obtained from the sum of all the

numbers that were in the column with using the equation, from Table 3.1. the denominator is the total number of dancers that participated in gathering the data. The data collection for the flexibility measurements of the dancers can be expressed in a range as follows; 15 m 57 cm. The standard deviation that was calculated can tell us that the spread of the height data is ±7.70 cm away from the , a wide range. These values tells us that for the flexibility there is a wide range of data away from the mean, and how far off from the mean the data is.

By calculating the standard deviation of both variables, height of the dancer and flexibility of the dancer, we can now compare them by using the coefficient variation to make a comparison between the variables.

Flexibility of Dancer Height of Dancer

The results show that the measurement of the dancers flexibility has a greater relative dispersal than the height of the dancers. Since 19.5% is a greater percentage than 3.33% it is conclusive to say that the flexibility of the dancers has greater dispersion.

Previously it was calculated that the mean of height (y

) is 163.56 cm and the average flexibility

(x

) is 39.43 cm. With these numbers we then can plug it in to formulate an equation to find the

covariance.

y = 163.56

= 39.43

Calculating Pearson’s Correlation Coefficient:

Sxy=∑ xy

n−x∗ y

=321936 .550

−(39 .43 )(163 .56 )

=-5 .5369 10

With the calculation of the covariance, plugging it into Pearson’s correlation coefficient formula along with the standard deviation of both the height and the flexibility can help tell if the data has a linear relationship.

Calculating Line of Regression:

By using the information from the calculator, we get:

y−163 . 56=(−5 . 5369) 2

(7 .70)( x−39 .43 )

y−163 .56=3.9815( x−39.43 )y=167 .54 ( x−39. 43)y=167 .54 x−6606 .16

This shows a negative correlation between the dancer’s height and their flexibility. This can be predicted that there is a extrapolation of the data, meaning that there are predictions outside the rand of data used to derive the line of regression.

X 2 Test of Independence

Lastly, with the collected data, the Chi-Square Test is used to determine if there is a significant differenced between the observed frequencies and the expected frequencies. We will test if one of them affects the occurrence of the other. Is there a relationship between the height of the dancers and their flexibility that exists? By using this test we will be able to conclude the answer.

Hypothesis: The dancer who is closer to the average height will be more flexible than those dancers who are taller.

Ho null Hypothesis: Height and flexibility are independent.

HI alternative hypothesis: Height and flexibility are dependent.

A Pearson’s Correlation Coefficient with a -0.132 indicates that the relationship between the data has a weak negative linear relationship, which is close to having no linear relationship at all.

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=(−5 . 5369)(7 .70 )(5 . 44 )

=-0 .132

Contingency Table: Observed Values of Height vs. Flexibility

Flexibility: 15-37 cm

Flexibility: 38-60cm

Total

150-165 cm tall

10 20 30

166-179 cm tall

10 10 20

Total 20 30 50

This data was organized in such a manor so that we can easily find the Chi-Squared later. A 2 x 2 contingency table was created to sort out the data into intervals of both the height and the flexibility length of all of the 50 dancers.

Calculating degrees of freedom:

df =(r−1)(c−1 )df =(2−1)(2−1 )df =(1 )(1 )df =1

Contingency table: Calculations for Expected Values of Heights vs. Flexibility

Flexibility: 15-37 cm

Flexibility: 38-60cm

Total

150-165 cm tall

30∗2050

=1230∗3050

=18 30

166-179 cm tall

20∗2050

=830∗2050

=12 20

Total 20 30 50

As you can see, the calculations were calculated within the expected values table. To get the numbers used in the expected values table, we had to use the values from the contingency table. An expected values table was also created to sort out the data from the contingency table. When comparing the values from the contingency table to the expected values table we can see that the expected values are not the same as the values from the contingency table. The values in the expected values table have either plus two or minus two difference from the contingency table. Since the values are not the same, it is possible that there could be an influencing factor between the height and the flexibility length the dancers.

Calculating the chi-squared value for heights of dancers vs. their flexibility:

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f o f e f o−f e ( f o− f e) 2 ( fo−fe )2

fe10 12 -2.0 4 0.33320 18 2.0 4 0.22210 8 2.0 4 0.50010 12 -2.0 4 0.333

∑=1.39Degrees of freedom= 1

At a 5% significance level, the critical value is 0.004

Since the c 2calculations of 1.39 > critical value of 0.004, we must reject the null hypothesis and accept the alternate hypothesis that the dancer’s height is independent of their flexibility. With the results, there is no relationship, the classifications are therefore independent.

Validity:

The investigation I chose to do helped me to determine whether or not height makes a difference on someone’s flexibility, which is something that I have often wondered over the years as a dancer. After doing several mathematical tests, it can be concluded that both the dancer’s height and flexibility are entirely independent of each other. I went into this investigation with the idea that these variables are independent of each other. As I was collecting data I noticed that some of the taller dancers had less flexibility in their hamstrings. The tallest height recorded was 175 cm with a flexibility of 34 cm whereas a dancer that is 165 cm had the lowest recorded flexibility of 16 cm. Even a dancer with a height of 161.5 cm had the highest flexibility of 57 cm, and that dancer is shorter than the dancer who had the lowest flexibility measurement. The shortest dancer that was 152 cm measured their flexibility to be 35 cm. Before I calculated the statistics I could see that there was a wide range of height and their capacity of their flexibility, so I wasn’t sure if the variables would have an affect on each other. After the different tests were calculated, each result supported another in saying that the height of the dancer has no relationship with their flexibility.

Reflecting upon my method, I noticed several factors that could have been improved. I wanted to keep my investigation as controlled as possible. I tried my best to keep the age of the dancer between 15 years old and 17 years old so that I can focus on a certain age group where the dancers have been dancing for a year or longer. I think that I should’ve narrowed my experimental group down even further by having all of my test subjects dance for the same amount of years. Some dancers are either naturally flexible from their genetics or it can come from the number of years they dance and how often they work on their flexibility. I think I got a substantial amount of data, however having more than 50 data pieces can always improve and support the results. I also limited my data in a way that I only measured one type of flexibility. Even by using the Sit-n-Reach board, there are at least three ways one can measure flexibility but I choose only one. By choosing only one way, measuring both of their feet against the board, is the simplest way but to be more accurate with the results other methods of measuring could have been taken into account.

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Works Cited

Coad, Mal, et al. Mathematics for the International Student:IB Mathematical studies

course. Adelaide:Haese and Harris Publications, 2004

Wood, Rob. "Sit and Reach Flexibility Test." Www.Topendsports.com. Rob Wood of

Topend Sports, 27 Oct. 2011. Web. 28 Oct. 2011.

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