Math IA
- 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