Correlation of height and shoe size

This is an analysis to see if height and shoe sizes are correlated among people. To see if whether the bigger feet you have the taller you will be or otherwise.

1. Correlation of Height and Shoe Size
Introduction:
There have been claims that you can tell how tall a person will be just by looking
at their shoe size. On average, a tall person is figured to have bigger feet than a shorter
person. The height of a person is measured from the top of the head to the bottom of the
feet. Depending on a person’s nationality height can vary due to the different
environmental and genetic factors. The shoe size is measured from the tip of the longest
toe to the ankle. There are several scales in which a person’s shoe size is recorded. Some
scales start measuring on 0 and others on 1. Shoe size also varies depending on age
(adult or child) and gender (male or female).
Statement of Task:
The main purpose of this investigation is to determine whether there is a
correlation between the height of a person and their shoe size.
Plan of investigation:
Data will be collected from at least 40 students and/or teachers and see if their
height correlates with their shoe size. In order to get the most accurate calculation for
this investigation, only people who are over the age of sixteen will be asked because if
younger students still have time to grow taller causing a fault in the calculation. The
independent variable is the shoe size seeing that it is said that the bigger your foot is the
taller you will get and the dependent variable is the height.

2. Collected Data
Table 1: Height and Shoe Size of 40 students between the ages of 16-19
# of Students Height Shoe Size # of Students Height Shoe Size
1 153 5 21 170 8.5
2 154 6 22 171 9
3 154 6 23 173 10
4 155 6 24 174 8
5 158 5 25 174 10
6 159 7 26 174 9
7 160 6 27 175 12
8 161 5 28 175 11
9 163 6 29 176 9
10 164 7 30 177 10
11 165 7 31 178 11
12 165 6 32 178 11
13 165 7 33 178 12
14 166 10 34 179 10.5
15 167 9.5 35 179 11.5
16 167 10 36 179 11
17 168 10 37 180 13
18 168 9 38 180 12
19 170 10.5 39 183 12.5
20 170 9.5 40 185 13
Table 1: Table 1 displays the data that was collected from each number of students for
their height and shoe size.

3. Data Analysis/Mathematical Processes
First off we will look at a logger pro generated scatter plot of the collected data

Graph 1: Graph 1 shows the height vs. shoe size on a scatter plot. So far it seems to have a
positive correlation, and its strength is moderate.
Standard Deviation Calculations
Standard deviation measures the variability/ dispersion of the particular variable, in
this case the height and shoe sizes. The formula for the standard deviation is:

4. Standard Deviation for:

= 8.613495672
8.81 is the standard deviation for x, the height. This indicates a condensed range of data
yet sufficient for a statistical analysis.
Standard Deviation for:

= 2.3648
2.36 is the standard deviation for y, the shoe size. This is somewhat the same as the
height’s standard deviation; this indicates a condensed range of data although it
remains sufficient for statistical analysis.

5. Least Square Regression
Least square regression calculations identify the relationship between the independent
variable, x, and the dependent variable, y. the least square regression is given by the
following equation:
where = -


Therefore:




is the least squares regression equation for this
particular set of data. As can be seen later on in graph 2
The following is the previously displayed logger pro generated graph with the
programmes calculation of the least square regression line.

6. Graph 2: Scatter Plot with Line of best fit

Graph 2: this graph indicates that there is a strong correlation. this is also indicated
through the value of the correlation coefficient, 0.89913.
Line of Best Fit Calculations:

7. Parson’s correlation coefficient:
Parson’s correlation coefficient indicates the strength of the relationship between the
two variables (shoe size vs. height). The formula for the correlation coefficient is:
where ,

To which


The correlation of determination is used to predict the future outcome based on the
data collected, it is given in terms of.
To which
With this being said it is safe to say that the strength of the two variable’s (height and
shoe size) correlation is strong.

8. Chi-Square Test
Chi-squared test measures the independence of two variables and whether they are
related or not. The following equation is used:
Observed Values:
B1 B2 Total
A1 a b a+b
A2 c d c+d
Total a+c b+d N

Calculations of Expected Values:
B1 B2 Total
A1

a+b
A2

c+d
Total a+c b+d N


Degree of freedom measures the number of values in the calculation that can vary:

r = row c = column
Null Hypothesis: Height and shoe size are independent
Alternative Hypothesis: Height and shoe size are not independent

9. Table 2: Observed Values
5 -> 9 10 -> 13 Total
153 -> 169 14 4 18
170 -> 185 5 17 22
Total 19 21 40
Table 2: This table shows the observed values of the Height vs. Shoe Size. Both data sets
has been grouped in order to be put in a 2 by 2 table.
Table 3: Calculations for the Expected Values:
5 -> 9 10 -> 13 Total
153 -> 169

18
170 -> 185

22
Total 19 21 40
Table 3: This table shows the individual calculations for each of the expected values.
Table 4: Expected Values
5 -> 9 10 -> 13 Total
153 -> 169 8.55 9.45 18
170 -> 185 10.45 11.55 22
Total 19 21 40
Table 4 shows the expected values, retrieved by calculations in table 2.



The x^2 critical value at 5% significance with 1 degree of freedom is 3.841. As the chi
square value ( ) > than the critical value, the null hypothesis has been rejected.
Hence, the alternative is supported and we can conclude that the height and shoe size
are NOT independent.

10. Discussion
Data Interpretation
Graph 2 clearly shows that the two variables, height and shoe size, not only have
a positive linear but they also have a strong correlation. This is supported by the value
of r^2 =0.808, which was generated along with the logger pro. Furthermore, it is also
supported by the Pearson’s correlation coefficient, which in its turn shows r^2 as 0.808.
Thus, all these calculations support the notion that a person’s height is affected by the
size of their shoe size.
When you look at the chi squared value, x^211.031, you can clearly see that the
null hypothesis is not supported, meaning that the alternate hypothesis takes into affect
showing that the height is dependent on the shoe size.
This should be taken into note that with the height and shoe size being
investigated to see if there is a correlation between them, you can clearly see that on
normal circumstances the height can be determined by the size of your shoes. Being that
the bigger your shoes size the taller you will become.
Limitations
The limitations of this investigation are that the data that was collected cannot
be account for all the people in the world. The data collected was a mixture of all races,
ranging from Asians to Europeans, etc. Due to the different ethnic races and ancestry
line there could be different outcomes. There are also anomalies where a person that is
short can have bigger feet that a person who is taller than them. The data was also
collected from both men and women, which have different ranges of shoe sizes. The
shoe sizes for women are normally smaller than men’s shoe sizes. Therefore causing the
data collected to have a high uncertainty.
Another limitation is that seeing that the growth in a person doesn’t stop still a
certain age (men 21-25, women 18-21) thus altering their final shoe sizes and height.

11. Appendix
Table 5: Coefficient of determination table
value Strength of correlation
r
2
=0 No correlation
Very weak correlation
Weak correlation
Moderate correlation
Strong correlation
Very strong correlation
r
2
=1 Perfect correlation
Table 5: This table shoes the correlation of determintaion in order to find the strength
of the data that was collected and put on the scatter plot.

12. Work Citied
Coad, Mal, Whiffen, Owen, Haese, Haese, Bruce, Mathematics for the
internationalstudent Mathematical Students SL, Adelaide, 2004