Mathematics 505D

Data Analysis and Probability

Summer 2011

Projects

Body Measurements

A researcher (Dr. A. Garth Fisher) took measurements of 252 people to test the claim that who you can estimate a person’s body fat from tables using their age and various skin-fold measurements obtained by using a caliper. Also there are predictive equations for body fat using body circumference measurements (e.g. abdominal circumference) and/or skin-fold measurements. Dr. Fisher has given permission to freely distribute the data and for its use for non-commercial purposes. The data is given in a separate spreadsheet. Every measurement other than age, weight and height refers to the circumference of that body part in centimeters.

The Measure of Body fat was estimated using accepted methods from other body measurements that are not included in the table. Here are questions to consider:

1. Does this data show that any single measurement (age, weight, height, neck, chest, abdomen, hip, thigh, knee, ankle, biceps, forearm, or wrist) is a good indicator of the percentage of body fat?

2. Does this change if you restrict you attention to the people under age 30, or to the people over age 60?

3. You might expect that weight and these circumferences are connected. For which of these body parts is there evidence in the data of such a relationship?

4. You might also expect that height and these circumferences are connected. For which of these body parts is there evidence in the data of such a relationship?

You do not need to understand all the details of the methods used to gather the data, generating some of the estimate of body fat to answer these questions. If you are interested in these details, here is Dr. Fisher’s summary description of the data:

Short Summary:

Lists estimates of the percentage of body fat determined by underwater weighing and various body circumference measurements for 252 men.

A variety of popular health books suggest that the readers assess their health, at least in part, by estimating their percentage of body fat. In Bailey (1994), for instance, the reader can estimate body fat from tables using their age and various skin-fold measurements obtained by using a caliper. Other texts give predictive equations for body fat using body circumference measurements (e.g. abdominal circumference) and/or skin-fold measurements. See, for instance, Behnke and Wilmore (1974), pp. 66-67; Wilmore (1976), p. 247; or Katch and McArdle (1977), pp. 120-132).

Percentage of body fat for an individual can be estimated once body density has been determined. Folks (e.g. Siri (1956)) assume that the body consists of two components - lean body tissue and fat tissue. Letting

D = Body Density (gm/cm³)

A = proportion of lean body tissue

B = proportion of fat tissue (A+B=1)

a = density of lean body tissue (gm/cm³)

b = density of fat tissue (gm/cm³)

We have D = 1/[(A/a) + (B/b)] solving for B we find B = (1/D)*[ab/(a-b)] - [b/(a-b)].

Using the estimates a=1.10 gm/cm³ and b=0.90 gm/cm³ (see Katch and McArdle (1977), p. 111 or Wilmore (1976), p. 123) we come up with "Siri's equation":

Percentage of Body Fat (i.e. 100*B) = 495/D - 450.

Volume, and hence body density, can be accurately measured a variety of ways. The technique of underwater weighing "computes body volume as the difference between body weight measured in air and weight measured during water submersion. In other words, body volume is equal to the loss of weight in water with the appropriate temperature correction for the water's density" (Katch and McArdle (1977), p. 113). Using this technique,

Body Density = WA/[(WA-WW)/c.f. - LV]

where WA = Weight in air (kg) and WW = Weight in water (kg)

c.f. = Water correction factor (=1 at 39.2 deg F as one-gram of water occupies exactly one cm³ at this temperature, =.997 at 76-78 deg F) LV = Residual Lung Volume (liters) (Katch and McArdle (1977), p. 115). Other methods of determining body volume are given in Behnke and Wilmore (1974), p. 22 ff.

The variables listed below, from left to right, are:

Density determined from underwater weighing

Percent body fat from Siri's (1956) equation

Age (years)

Weight (lbs)

Height (inches)

Neck circumference (cm)

Chest circumference (cm)

Abdomen 2 circumference (cm)

Hip circumference (cm)

Thigh circumference (cm)

Knee circumference (cm)

Ankle circumference (cm)

Biceps (extended) circumference (cm)

Forearm circumference (cm)

Wrist circumference (cm)

These data are used to produce the predictive equations for lean body weight given in the abstract "Generalized body composition prediction equation for men using simple measurement techniques", K.W. Penrose, A.G. Nelson, A.G. Fisher, FACSM, Human Performance Research Center, Brigham Young University, Provo, Utah 84602 as listed in _Medicine and Science in Sports and Exercise_, vol. 17, no. 2, April 1985, p. 189. (The predictive equations were obtained from the first 143 of the 252 cases that are listed below).

The data were generously supplied by Dr. A. Garth Fisher who gave permission to freely distribute the data and use for non-commercial purposes.

References:

Bailey, Covert (1994). _Smart Exercise: Burning Fat, Getting Fit_, Houghton-Mifflin Co., Boston, pp. 179-186.

Behnke, A.R. and Wilmore, J.H. (1974). _Evaluation and Regulation of Body Build and Composition_, Prentice-Hall, Englewood Cliffs, N.J.

Siri, W.E. (1956), "Gross composition of the body", in _Advances in Biological and Medical Physics_, vol. IV, edited by J.H. Lawrence and C.A. Tobias, Academic Press, Inc., New York.

Katch, Frank and McArdle, William (1977). _Nutrition, Weight Control, and Exercise_, Houghton Mifflin Co., Boston.

Wilmore, Jack (1976). _Athletic Training and Physical Fitness: Physiological Principles of the Conditioning Process_, Allyn and Bacon, Inc., Boston.

Roger W. Johnson

Department of Mathematics & Computer Science

South Dakota School of Mines & Technology

501 East St. Joseph Street

Rapid City, SD 57701

email address: rwjohnso@silver.sdsmt.edu

web address: http://silver.sdsmt.edu/~rwjohnso