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 --