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