NAME:  Hat measurements, including hat size
TYPE:  Observational
SIZE:  26 observations, 6 variables

DESCRIPTIVE ABSTRACT:
 
The dataset contains hat size as well as circumference, length of major axis 
and length of minor axis of the inner hat band for 26 hats. The manufacturer 
and the country of manufacture are also included.

SOURCE:

Katherine Brady and Kari Cornelius, Carleton College ('96) students, gathered 
the observations from hats at a store at the Mall of America in Bloomington, 
Minnesota.

VARIABLE DESCRIPTIONS:

Columns 
 1- 5	  Hat size
 7-11	  Circumference (inches)
13-16	  Length of major axis (inches)
18-21	  Length of minor axis (inches)
23	  Where made? Italy = 1, U.S.A. = 2
25	  Manufacturer? Beaver = 1, Borsalino = 2, Dobbs = 3, Stetson = 4

Measurements recorded in inches are rounded to the nearest quarter inch.

Values are aligned and delimited by blanks.

STORY BEHIND THE DATA:

When shopping for a bicycle helmet for my son, I found a "sizing tape" 
included in the packaging. Such a tape is used to correctly identify the 
correct helmet size for an individual. The tape I found, when wrapped 
around the rider's head, gives head circumference as well as a "helmet 
size" ranging from 5 to 8 1/4 in increments of 1/8. Since the divisions 
between consecutive sizes were equally spaced (that is, the spacing 
between sizes 5 5/8 and 5 3/4 is the same as that between 6 3/8 and 6 1/2 
on the tape) helmet size must be linear in circumference. If helmet sizing 
is identical to men's hat sizing, men's hat size must be linear in the 
circumference of the inner hat band.

A plot of hat size versus circumference does indeed reveal a linear 
pattern. If the origin is included in such a plot, it is clear that a line 
through the origin will fit the data quite well. Lord (1995) actually 
indicates that men's hat size is the inner band circumference divided by 
pi (and then, presumably, rounded to the nearest eighth).

PEDAGOGICAL NOTES:

Fitting the no-intercept model hat size = c * circumference by least squares 
with the given data gives an estimated value of c of about .3285 compared to 
about .3183 for the reciprocal of pi. Students may also try fitting hat size 
to a constant times either the length of the major axis or the length of the 
minor axis. Interestingly enough, fitting hat size by a constant times the 
length of the major axis gives a standard error which is less than that when 
using the correct model (about .08 versus about .10). This may be largely 
due to the fact that axis length is much easier to measure than 
circumference; one needs to carefully press the measuring tape against the 
inner hat band, without slippage, to accurately measure circumference. 
Fitting hat size to a linear combination of the two axes through the origin 
very nearly gives the pleasing least squares model of 
hat size = (2/3)*(major axes) + (1/3)*(minor axes) with a standard error of 
about .07. It's not surprising that such a model fits well since if the hat 
band is an ellipse its perimeter can be expressed exactly in terms of (an 
elliptic integral involving) these two axes. See Beyer (1981), for example, 
for details. As far as determining an individual's hat size, of course, 
models that use major and minor axis length are not too practical - it's 
hard to measure through someone's head! An interesting rule of thumb 
mentioned by a salesperson at the store where the hat data was collected is 
that one can estimate one's hat size by measuring the hand from the base of 
the palm to the tip of the middle finger, in inches.

It seems that the formula hat size equals circumference (in inches) divided 
by pi is true regardless of where the hat was made (Italy or the U.S.) or 
the manufacturer (Beaver, Borsalino, Dobbs, Stetson).

REFERENCES:

Beyer, W., editor (1981), CRC Standard Mathematical Tables, 25th edition, 
  CRC Press, Boca Raton, Florida, p. 144 and p. 435.

Lord, John (1995), Sizes: The Illustrated Encyclopedia, Harper Collins, 
  New York, New York, p. 117.

SUBMITTED BY:

Roger Johnson
Department of Mathematics & Computer Science
South Dakota School of Mines & Technology
501 East St. Joseph Street
Rapid City, SD  57701
Roger.Johnson@sdsmt.edu

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