1. Height vs. Flexibility of a Dancer
An 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
1

2. 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:
2

3. 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 Figure 1: Image showing what test subjects
height in centimeters. There are many ways in which one will do to measure their flexibility.
dancer can be flexible, and measuring the flexibility of the <http://www.topendsports.com/testing/tests/s
hamstring is one of the main ways. A Sit-n-Reach test will it-and-reach.htm>
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 30
Male 16 172 41
Female 16 169 27
Female 17 165 16
3

4. Female 15 162 29
Female 16 156 31
Female 17 163.5 57
Female 17 161.5 34
Female 16 168 42.5
Female 17 175 34
Female 17 153 35
Female 17 165 44
Average height:
Female = 17 Average flexibility:
152 =
35
Female 15 152 33
Female 15 162 42
= 163.15 cm = 39.43 cm
Female 14 159 50
Female 18 161 57
Minimum height: 152 cm
Female 17 Minimum flexibility: 16 cm
165 32
Maximum height: 175 cm
Female 15 Maximum flexibility: 57 cm
161 40
Female 16 156 45
Q1: 160Female
cm 18 Q1: 162 cm
34 37
Female 15 158 43
Q2 (Median):
Female
= 25.5th term
17
Q2 (Median):
164
= 25.5 term
th
38
Female 15 159 47
Female 17 172 36
Female = 163 cm 16 170 = 40 cm 40
Female 15 158 44
Female 15
Q3: 167cm
44 48
Q3:167 cm
Female 16 165 42
Female 17 163 43
Female 15 155 45
Female 16 161 44
Female 16 163 39
Female 17 166 37
Female 15 159 30
Female 16 162 48
Female 17 164 50
Female 17 163 45
Female 15 160 46
Female 16 159 38
Female 16 164 44
Female 17 169 38
Female 16 167 43
Female 16 166 46
Female 17 167 30
Female 17 161 37
Female 15 169 32
Female 16 165 41
Female 17 167 34
Female 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
organized.
4

5. 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.
Box & Whisker Plot: Height of 50 Dancers (cm):
Box & Whisker Plot: Flexibility of 50 Dancers (cm):
5

6. 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.
Scatter Plot of Flexibility of Dancer vs.
Their Height
180
Heighth of Dancer (cm)
175
170
165
160
155
150
0 10 20 30 40 50 60
Flexibility of Dancer (cm)
Legend:
= mean of data set; (39.43,163.15)
= each piece of data
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.
Table 3.0 Table displaying the
Table 2.0 Table displaying the intervals and frequencies of the
heights recorded from teach test flexibility measurements.
subject.
6

7. Height (cm) Frequency Cumulative Flexibility Frequency Cumulative
Interval Frequency (cm) Frequency
150-154 3 3 Interval
15-19 1 1
155-159 9 12
20-24 0 1
160-164 17 29 25-29 2 3
165-169 16 45
30-34 11 14
35-39 10 24
170-174 4 49 40-44 14 38
175-179 1 50 45-49 8 46
50-54 2 48
55-59 2 50
Figure 3-A cumulative frequency graph showing height using data from Table 2.0.
C 60
u
m
u 50
l
a
t 40
i
v
e
30
F
r
e 20
q
u
e 10
n
c
y 0
150 155 160 165 170 175 180 185
Height (cm)
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
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.
7

8. Figure 4- This cumulative frequency graph shows the length of the flexibility from Table
3.0.
60
C
u
m
50
u
l
a
t 40
i
v
e 30
F
r 20
e
q
u
e 10
n
c
y 0
0 10 20 30 40 50 60
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 2.1Calculations for the Standard Deviation of Height (cm):
Midpoint Frequency Length of Flexibility (cm)
2
Class Interval (x) (f) (f)(x) x-
150-154 152 3 456 -11.15 372.9675
8

9. 155-159 157 9 1413 -6.15 340.4025
160-164 162 17 2754 -1.15 22.4825
165-169 167 16 2672 3.85 237.16
170-174 172 4 688 8.85 313.29
174-179 177 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):
Midpoint Frequency
2
Class Interval (x) (f) (f)(x) x-
15-19 17 1 17 -22.43 503.1049
20-24 22 0 0 -17.43 0
25-29 27 2 54 -12.42 308.5128
30-34 32 11 352 -7.43 607.2539
35-39 37 10 370 -2.43 59.049
40-44 42 14 588 2.57 92.4686
45-49 47 8 376 7.57 458.4392
50-54 52 2 104 12.57 316.0098
55-59 57 2 114 17.57 617.4098
∑=2962.248
9

10. 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, fromTable 3.1.
thedenominatoris 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.
Calculating Pearson’s Correlation Coefficient:
Previously it was calculated that the mean of height ( ) is 163.56 cm and the average flexibility
( ) is 39.43 cm. With these numbers we then can plug it in to formulate an equation to find the
covariance.
= 163.56
= 39.43
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.
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 10
linear relationship at all.

11. Calculating Line of Regression:
By using the information from the calculator, we get:
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.
X2 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.
Contingency Table: Observed Values of Height vs. Flexibility
Flexibility: Flexibility: Total
15-37 cm 38-60cm
11

12. 150-165 cm 10 20 30
tall
166-179 cm 10 10 20
tall
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:
Contingency table: Calculations for Expected Values of Heights vs. Flexibility
Flexibility: Flexibility: Total
15-37 cm 38-60cm
150-165 30
cm tall
166-179 20
cm tall
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:
2
10 12 -2.0 4 0.333
12

13. 20 18 2.0 4 0.222
10 8 2.0 4 0.500
10 12 -2.0 4 0.333
∑=1.39
Degrees of freedom= 1
At a 5% significance level, the critical value is 0.004
Since the 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.
Works Cited
13

14. 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.
14