1. IB Mathematical Studies Internal Assessment:
Shoe Size versus Length of Forearms
Exam Session: May 2012
School Name: International School Bangkok
IB Number: 000307-161
Teacher: Ms. Goghar
Date: January 20, 2012
Course: IB Math Studies SL
Word Count: 2,338
Name: Jennifer Purgill

2. Purgill 1
Introduction
Growing up, children always hear all kinds of urban myths about many different topics.
Some of these myths include the idea that the length of a person’s arm span is equal to their
exact height, the length of a person’s thumb is always about an inch long no matter who the
person is, the length of your waist from hip to hip is equal to the circumference of your neck, and
that the size of a person’s foot is the exact same as the length of their forearm. As a child who
heard many of these myths while growing up, I was always curious to find out whether or not
they were really true. The myth that was heard most often, though, was the one stating that a
person’s shoe size is extremely close, if not exactly the same, as the length of their forearm. This
myth will be analyzed using various mathematical processes to test whether it is actually true or
not.
Statement of Task
The main purpose of this investigation is to deduce whether or not the size of a person’s
shoe has a direct correlation with the length of their forearm.Information for the project will be
collected from randomly selected students attending the International School of Bangkok. It is
necessary to have a varied group of students tested to ensure that the correlation happens in all
people, not only a certain race or gender. After data is collected and recorded, a graph will be
used as a visual aid to show exactly what the correlation is between the shoe size of a person and
their forearm length. If the hypothesis is correct, then the results should support the idea that as
the length of a person’s forearm increases, then so does their shoe size. If results are exact
enough, it should be proven that the size of a person’s shoe is equal, or very close to equal, to
their forearm length. The main reason I chose this topic for my internal assessment is because I

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have always heard rumors that the length of people’s forearms and their shoe size are exactly the
same but it will be interesting to see if it can be mathematically proven with proper calculations.
Hypothesis: The length of a person’s forearm is directly correlational to the size of their
shoe.
Ho null hypothesis: The size of a person’s shoe is independent of the length of
their forearm.
H1 alternative hypothesis: The size of a person’s shoe is not independent of the
length of their forearm.
Plan of Investigation
The data used for this internal assessment was gathered from various students attending
the International School of Bangkok. The reason data was collected from randomly selected
students is to ensure the fact that this occurrence is present in all people, not only people of a
certain gender, race, or height. There will be 50 students measured overall, 25 males and 25
females. The even number of both males and females measured will ensure that this data does
not only apply to one gender. They will first have their forearms measured from the point where
the inside of the elbow ends to the bend of the wrist using centimeters. This information will then
be recorded on paper. The students will then be asked for their shoe size and that data will be
recorded as well. After all 50 students’ information is collected, the shoe sizes will be converted
into centimeters to see if they are directly correlational, if not exactly equal, to the length of their
forearms. This raw data will be displayed in tables 1 and 2 of the internal assessment.

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Mathematical Processes/Interpretation of Results
Collected Data
Table 1: Table displaying50 students’ shoes sizes in centimeters and their forearm lengths
Student Size of Shoe (cm) Forearm Length (cm)
1 25.7 25.5
2 22.8 22.7
3 25.4 25.5
4 24.1 24.0
5 24.1 24.3
6 24.8 24.7
7 23.5 23.4
8 25.1 25.0
9 24.8 24.6
10 22.8 23.0
11 26.0 25.8
12 25.4 25.5
13 24.8 25.0
14 23.5 23.5
15 24.5 24.5
16 23.5 23.4
17 25.4 25.5
18 23.5 23.4
19 24.1 24.0
20 24.1 24.3
21 24.8 25.0
22 22.8 23.0
23 24.1 24.0
24 23.5 23.0
25 23.8 24.0
26 27.6 27.5
27 26.7 26.5
28 27.0 27.2
29 26.3 26.2
30 26.3 26.1
31 27.9 28.0
32 27.9 27.7
33 28.3 28.2
34 26.7 26.5
35 28.3 28.1
36 28.3 28.5
37 27.0 27.2
38 27.6 27.5
39 27.6 27.5

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40 27.3 26.9
41 26.7 26.5
42 26.3 26.2
43 27.0 27.0
44 27.9 28.0
45 27.3 27.2
46 27.6 27.8
47 27.6 27.9
48 26.3 26.5
49 27.0 27.4
50 26.7 26.5
Sum 1288.1 1287.2
Average 25.8 25.7
The averages of the x and y columns displayed above were calculated by first finding the sum of
all 50 data points per column then diving that total by 50. A sample calculation to show finding
the average of x for this is shown below:
The average of the y column was found using the same equation but substitutes y in place of x:
The modes of the shoe size for the 50 students (x) are 23.5, 24.1, and 27.6
The mode of the forearm length for the 50 students (y) is 26.5

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A t-test was performed to check if the difference between a person’s shoe size and their forearm
length are statistically significantly different in the case of women.
Figure 1: Results of t-test for all 50 students comparing shoe size and forearm length
It was found that the size difference between a student’s shoe size and the length of their forearm
is not statistically significant. This means that the size of a student’s shoe and the length of their
forearm are very similar if not exactly the same in most of the measurements presented.
Figure 2: Graph displaying the comparison between students’ shoe size compared to the
corresponding students’ forearm lengths
It can be seen in the graph above that as the size of the students’ shoes increase, the lengths of
their forearms increase as well. There is a high positive correlation that can be seen between the
two variables. A line of best fit was used to analyze the data, and it is stated that the correlation
between the two variables is 0.9976. The linear fit for this graph was calculated using the
formula:

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Table 2: Table displaying chi-squared information where is the students’ shoe sizes (cm) and
is the length of the respective student’s forearm
22.8 22.7 517.6 519.8 515.3 9.00 9.00
22.8 23.0 524.4 519.8 529.0 9.00 7.29
22.8 23.0 524.4 519.8 529.0 9.00 7.29
23.5 23.0 540.5 552.3 529.0 5.29 7.29
23.5 23.4 549.9 552.3 547.6 5.29 5.29
23.5 23.4 549.9 552.3 547.6 5.29 5.29
23.5 23.4 549.9 552.3 547.6 5.29 5.29
23.5 23.5 552.3 552.3 552.3 5.29 4.84
23.8 24.0 571.2 566.4 576.0 4.00 2.89
24.1 24.0 578.4 580.8 576.0 2.89 2.89
24.1 24.0 578.4 580.8 576.0 2.89 2.89
24.1 24.0 578.4 580.8 576.0 2.89 2.89
24.1 24.3 585.6 580.8 590.5 2.89 1.96
24.1 24.3 585.6 580.8 590.5 2.89 1.96
24.5 24.5 600.3 600.3 600.3 1.69 1.44
24.8 24.6 610.1 615.1 605.2 1.00 1.21
24.8 24.7 612.6 615.1 610.1 1.00 1.00
24.8 25.0 620.0 615.1 625.0 1.00 0.49
24.8 25.0 620.0 615.1 625.0 1.00 0.49
25.1 25.0 627.5 630.0 625.0 0.49 0.49
25.4 25.5 647.7 645.2 650.3 0.16 0.04
25.4 25.5 647.7 645.2 650.3 0.16 0.04
25.4 25.5 647.7 645.2 650.3 0.16 0.04
25.7 25.5 655.4 660.5 650.3 0.01 0.04
26.0 25.8 670.8 676.0 665.5 0.04 0.01
26.3 26.1 686.4 691.7 681.2 0.25 0.16
26.3 26.2 689.1 691.7 686.4 0.25 0.25
26.3 26.2 689.1 691.7 686.4 0.25 0.25
26.3 26.5 697.0 691.7 702.3 0.25 0.64
26.7 26.5 707.6 712.9 702.3 0.81 0.64
26.7 26.5 707.6 712.9 702.3 0.81 0.64
26.7 26.5 707.6 712.9 702.3 0.81 0.64
26.7 26.5 707.6 712.9 702.3 0.81 0.64
27.0 26.9 726.3 729.0 723.6 1.44 1.44
27.0 27.0 729.0 729.0 729.0 1.44 1.69
27.0 27.2 734.4 729.0 739.8 1.44 2.25
27.0 27.2 734.4 729.0 739.8 1.44 2.25
27.3 27.2 742.6 745.3 739.8 2.25 2.25
27.3 27.4 748.1 745.3 750.8 2.25 2.89
27.6 27.5 759.0 761.8 756.3 3.24 3.24
27.6 27.5 759.0 761.8 756.3 3.24 3.24
27.6 27.5 759.0 761.8 756.3 3.24 3.24

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27.6 27.7 764.5 761.8 767.3 3.24 4.00
27.6 27.8 767.3 761.8 772.8 3.24 4.41
27.9 27.9 778.4 778.4 778.4 4.41 4.84
27.9 28.0 781.2 778.4 784.0 4.41 5.29
27.9 28.0 781.2 778.4 784.0 4.41 5.29
28.3 28.1 795.2 800.9 789.6 6.25 5.76
28.3 28.2 798.1 800.9 795.2 6.25 6.25
28.3 28.5 806.6 800.9 812.3 6.25 7.84
Sum 1288.1 1287.2 33302.6 33326 33280.5 141.3 142.4
Average 25.8 25.7 666.1 666.5 665.6 2.83 2.85
The average for the x column displayed in the table above was found by using the formula:
This same formula was used to find the averages for the rest of the columns as well, but
substitute for what data the column contains and finding the sum of the respective column
rather than the column of x.
The correlation coefficient ) will be used to test is there is a correlation between students’ shoe
sizes and their forearm length. The formula for this test is:
In this formula, Sxyis the covariance.

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In this formula for the correlation coefficient ( ), the work for is shown above. In the
formula, the work to find and are shown below:
When the data from the table is inserted into the calculator, the correlation coefficient (r) for the
data is equal to 0.994.This means that there is a positive direct correlation between the length of
the students’ forearms and their shoe sizes. Because r=0.994 this means that r2=0.988, which
indicates that there is a strongcorrelation between the length of students’ forearm lengths and the
size of shoe they wear.
Chi-squared is a test used to determine whether two factors are independent or dependent of each
other. When using chi-squared, a table of observed and expected data is shown and calculated to
check if there is a significant difference between the two factors. The formula for this is:

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Observed values:
Table 3: Sample calculation for chi-squared table of observed values
A1 A2 Total
B1 A B A+B
B2 C D C+D
Total A+C B+D n
Expected Values:
Table 4: Sample calculation for chi-squared table of expected values
A1 A2 Total
B1 A+B
B2 C+D
Total A+C B+D n
Table 5: Table displaying the relationship between the shoe size (cm) and length of the forearms
(cm) of the students
Size of Shoe (cm)
Length of Forearm 22.8-26.0 26.1-28.3 Total
(cm)
22.7-25.7 24 0 24
25.8-28.5 1 25 26
Total 25 25 50

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Figure 3: Graph displaying shoe size (cm) versus forearm length (cm) for the 50 students
It can be seen in the graph shown above that it is not mathematically possible to have at least five
data points for each section of the chi-squared table (see table 5) due to the data points being in
very close proximity to each other. This supports the alternate hypothesis that a person’s shoe
size is not independent of their forearm length.
Degrees of freedom measure the number of values in the final calculation that are free to vary.
The formula for degrees of freedom is:

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Discussion/Validity
Limitations
Throughout the investigation, there have been a few limitations present which may affect the
reliability of the outcomes found. One limitation of the experiment is that all of the data was
collected from students of the International School of Bangkok. Because of this, the participants
do not represent a random sample but instead an opportunity sample and there was not a very
wide variety of types of participants although the students were from many different cultures and
backgrounds. All of the students measured were in a similar age group (ages 14 to 18) and it is
possible that not all of the students had grown to their full potential (i.e. their arms still have
room to grow and or their shoe size is not as large as it will be when they have stopped growing).
Another limitation to the data was that all measurements were done using simple rulers and
therefore the numbers recorded may not have been 100% accurate as well as the fact that it is
difficult to determine the exact location where a person’s forearm begins and ends. Because of
this, some of the measurements may have been off by a few millimeters or centimeters for the
column of forearm length in both the males and females measured.

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Conclusion
Despite the limitations mentioned in the previous section, the data overall supports the
alternative hypothesis and rejects the null hypothesis that the size of a person’s shoe is
independent of the forearm length of the person. This means that in fact a person’s shoe size is
dependent on a person’s forearm length. Furthermore, the data not only supports the alternative
hypothesis, but it also shows that there is not an extreme significance between a person’s shoe
size and the length of their forearm. The data is also not biased towards one gender or race over
another because there was a wide variety of people with different backgrounds and ancestry
measured as well as equal amounts of both genders tested. Because of this and the results found
it can be determined that the shoe size is in fact directly correlational to the length of a person’s
forearm.