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Phase II
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Statistics 1510
Research Project;
Is there a Correlation Between Circumference and Diameter of
Circular Objects?
Laura Imbelloni
Professor Brian Jean
Taft College
Abstract
This experiment looked at the relationship between two variables of interest in
order to determine if there is a correlation between the two. The two variables of interest
were circumference and diameter. Summary statistics as well as various forms of statistical
analyses were done including a regression analysis. This study was able to conclude that
there is a strong correlation between circumference and diameter of circular objects.
Introduction
The purpose of this experiment is to gain experience in collecting measurement data
and being able to apply inferential and graphical techniques to analyze the collected data
and report it in a professional and clear manner (Jean, Brian). Our variables of interest are
circumference and diameter which are both quantitative with a measurement scale being
ratio as well as well PI () which represents the correlation between circumference and
diameter of circular objects, also a quantitative variable. Rules of data collection were
followed in which the objects were defined, along with the population, sampling technique,
measuring devices and variables of interest with the use of randomization.
This study looked at the relationship between circumference and diameter. Our
hypothesis was that there is a positive, direct correlation between the two variables of
interest between circular objects.
Materials & Methods
The circumference of each object was found by wrapping a string around the object
once until the ends were touching as seen in the examples in Figure 1 & 2. The amount of
string between the two points was then measured with a ruler in order to find the value of
the circumference of the specific object as seen in the examples, Figure 3 & 4. Objects
included; CD, Foldgers Coffee lid, Men’s After Shave, Ben & Jerry’s Ice Cream lid, Bracelet,
Lifewater Bottle lid, Tuna can, bottom of drinking glass, Wine Bottle base, Peanut Butter jar
lid, Tea Light Candle, paper towel roll, paper plate, drum, clock, nickel, quarter, can of soup,
Yo-yo, Hat Box, Hockey puck, Frosting can, Soda can, face cream, Vitamin Bottle.
Diameter of objects were then measured with a ruler by measuring one end of the
object to the opposite end of the specific object and then recorded as seen in the examples
from Figures 5 & 6. Objects or samples were chosen at random from various circular objects
found throughout the household as well as randomly chosen objects from the class data set.
All objects or samples were chosen at random to ensure the sample is reflective of the
population.
Figure 1: First part of measurement of
Circumference of Quarter.
Figure 2: First part of measurement of
Circumference of CD.
Results:
Circumference Diameter PI ()
Average 13.3132 4.168 3.194146
Sx 9.849772 3.116083 3.160946
N 25 25 25
Min X 2.6 0.8 2.857143
Figure 3: Second part of measurement of
the circumference of Quarter.
Figure 4: Second part of measurement of
the circumference of CD.
Figure 5: Measurement of the diameter of
Quarter.
Figure 6: Measurement of the diameter of
CD.
Table 1: Summary Statistics
Q1 7.4 2.4 3.1371
M 9.5 3 3.166667
Q3 17.64 5.125 3.2668
Max X 38 12 3.789474
0
1
2
3
4
CD
Fold
gers
Co
ffee
Lid
Men
s A
fter
shav
e Li
d
Ben
& J
erry
's I
ce …
Bra
cele
t
Life
Wat
er B
ott
le L
id
Tun
a C
an
Bo
tto
m o
f D
rin
kin
g …
Win
e B
ott
le b
ase
Pea
nu
t B
utt
er J
ar L
id
Tea
ligh
t C
and
le
Pap
er T
ow
l Ro
ll
Pap
er P
late
Dru
m
Clo
ck
Nic
kel
Qu
arte
r
Sou
p C
an
Yo-y
o
Hat
bo
x
Ho
ckey
pu
ck
Fro
stin
g C
an
Sod
a ca
n
Face
cre
am
Vit
amin
Co
nta
iner
V
lau
es
Objects
- Values for Objects
Figure 8: Histogram displaying (gray) of Sample objects.
Figure 7: Linear association between
Diameter and Circumference.
Discussion:
Shown in Table 1 by the descriptive data of , displaying little difference in the mean
and median as well as very similar variability within the sample allows us to conclude that
there is a relationship between circumference and diameter of circular objects. This also
shows that the sample is symmetrical and an appropriate measure of center and spread is
the sample mean as well as the sample standard deviation.
Figure 9 & 10: Box and
Whisker Plot of
circumference,
diameter and . Figure
10 is rescaled.
Circumference
Diameter
The results also show that there is correlation between circumference and diameter
of circular objects. The calculated correlation coefficient or “r” is .997748. This is a strong
correlation due it being extremely close to 1. The data met the necessary assumptions for
Pearson’s correlation. The data set also met the assumptions for a simple linear regression.
A complete regression analysis using diameter to predict circumference was then done to
predict and check the sample data. The regression further concluded that there is a
correlation between circumference and diameter due to the small values of epsilon or error
between the observed value and the expected value. The regression coefficients (0)
corresponds with the Y-intercept which is equal to .1680 and (1) corresponds with the
slope of the line which is equal to 3.154. The equation for the sample regression line would
then be y-hat= b(o) + b(1)(x) or y-hat= .1680 + 3.154(x). The linear association can be seen
in Figure 7.
One may also observe in Figure 8 that the values for the sample is uniform, further
supporting the direct correlation between the circular objects’ circumference and diameter.
Figure 9 and Figure 10 also support the conclusion that there is a correlation between the
circumference and diameter of circular objects by displaying in both the circumference and
diameter skewed left and in Figure 10, a relative uniform and symmetrical shape.
Conclusion:
In conclusion, this study was able to show a correlation between circumference and
diameter of circular objects. This was done by several statistical analyses that were done
on the sample data. All rules and assumptions to ensure a fair test or experiment were
followed in order for this sample to reflect the population.
Appendix:
Jean, Brian, David Meyers, and Rene Sporer. Data Analysis: An Applied Approach to
Statistics with Technology. 3rd ed. N.p.: 3RingPublishing, 2004-2008. Print.
Raw Data:
Objects
Circumference Diameter PI (
CD
18 4.75 3.789474
Foldgers Coffee Lid
12.5 4 3.125
Mens Aftershave Lid
6 1.75 3.428571 Ben & Jerry's Ice Cream Lid
12 3.75 3.2
Bracelet
9 2.75 3.272727
Life Water Bottle Lid
5 1.5 3.333333
Tuna Can
10.625 3.375 3.148148 Bottom of Drinking Glass
7.5 2.4 3.125
Wine Bottle base
9.5 3 3.166667
Peanut Butter Jar Lid
11.625 3.625 3.206897
Tea light Candle
5 1.5 3.333333
Paper Towl Roll
18.75 5.75 3.26087
Paper Plate
27 8.5 3.176471
Drum
33 10.5 3.142857
Clock
34.5 11 3.136364
Nickel
2.6 0.8 3.25
Quarter
3.1 0.9 3.444444
Soup Can
8.5 2.625 3.238095
Yo-yo
8 2.8 2.857143
Hat box
38 12 3.166667
Hockey puck
9.1 2.9 3.137931
Frosting Can
10.7 3.5 3.057143
Soda can
8.25 2.625 3.142857
Face cream
7.3 2.4 3.041667
Vitamin Container
17.28 5.5 3.141818
Average
13.3132 4.168 3.194146
x
332.83 104.2 80.32348
x²
6759.465 667.345 258.7758
Sx
9.849772 3.116083 3.160946
σx
9.6508 3.0531 0.1675
N
25 25 25
Min X
2.6 0.8 2.857143
Q1
7.4 2.4 3.1371
M
9.5 3 3.166667
Q3
17.64 5.125 3.2668
Max X
38 12 3.789474