Student Investigation 6.1

Predicting Height from the Length of Limb Bones

Background

The formation of bone by the activity of osteoblasts and the addition of minerals and salts is known as the process of ossification. This process begins in the fetus before birth and continues to some extent until about the age of 25. Although every person's bone development schedule is unique to that individual (just as the schedule at which they begin to walk or when they begin to talk is very individual and unique), a general timetable, Table 1, has been developed that indicates the various ages at which certain bones for the "average" person will complete ossification.

As you can see, during your high school years, the bones of the upper limbs and shoulder area are completing ossification. Also, by the ages of about 18 to 23, the bones of the lower limbs have been fully formed. In the average person, once ossification of the upper and lower limbs is complete, the size of the two major arm bones (the humerus and the radius) as well as the size of the two major leg bones (the femur and the tibia) have grown to a length that is proportional to the person's height. In fact, so precise is the relationship between these various bones and height that anthropologists and forensic scientists, with one dried bone as a clue, can closely estimate its owner's former living height.

Table I. Ossification timetable in humans.
Ossification Timetable
Third month of embryonic development Ossification in long bones beginning
Fourth month Most primary ossification centers have appeared in thediaphyses of bone.
Birth to 5 years Secondary ossification centers appear in the epiphyses
5 years to 12 years in females, or 5 to 14 years in males Ossification is spreading rapidly from the ossifcatio centers and various bones are becoming ossified
17 to 20 years Bone of upper limbs and scapulae becoming completely ossified
18 to 23 years Bone of the lower limbs and os coxae become completely ossified
23 to 25 years Bone of the sternum, clavicles, and bertebrae become completely ossified
By 25 years Nearly all bones are completely ossified

For your information, an anthropologist is a person involved in studying humans in relation to their physical character, distribution, origin, racial background, social structure, and culture. Anthropologists are often involved with digging for and finding the remains of people who lived years and years ago. Certain kinds of remains, such as bones, can serve to provide information about the physical characteristics of these ancient people. A forensic scientist is a person who was trained to take evidence from crime scenes and determine from that evidence who may have been involved in the crime. Both of these groups know that certain bones serve as excellent sources of information to determine a person's height. Such a determination is accomplished using a set of standard equations that contain the scientific information gathered from hundreds of studies. For instance, if a 17.9 inch femur is found, this value is inserted into the following equation:

Height in inches = (1.880 x femur length) + 32.010
and the result is that the femur belonged to a person who had a height of five and a half feet, assuming this person was a male. If the person were a female, a different equation must be used:
Height in inches = (1.945 x femur length ) + 28.679
and the result is that the femur belonged to a woman whose height was about five feet, four inches. Now, let's look at the two equations above.

Both follow a very important form that is equivalent to the equation of a line:

y = mx + b where: y = the height of the person,
m = the slope of a graph of the line,
x = the length of the person's limb bone, and
b = the y-intercept on a graph of the line.
Any equation of the form y = mx + b has a graph that is a straight line. The y-intercept, b, is the y value when x = 0. Thus, the point (O,b) lies on the graph of the straight line. The number m is called the slope, and it is an indication of how the line slants. The greater the slope, the steeper the line slants upward, from left to right. If the slope is negative, the line slants downward from left to right (Figure 11).
Figure 11. To determine the slope of a line on a graph, divide the number of vertical units by the number of horizontal units, or remember the relationship rive/run (for positive slopes) and fall/run (for negative slopes). The y-intercept is indicated on the graphs.

The fact that the length of the limb bones and its associated height is described by a linear equation such as this means that if you were tograph, for a group of different height individuals, the length of a particular limb bone along the x-axis and their height along the y-axis, the points on the graph should fall generally along a straight line. However, the slope of each line will always be positive.

You must be wondering why we have gone into detail about determining the equation of a line. Well, for this exercise, you will be doing just that. But that's not all. A small competition between groups is involved here! Let's begin.

Materials

Tape measures
Graph paper
Rulers

Procedure

Everyone should read all of the steps before beginning.

Step 1

Break into groups based on your teacher's instruction. For each person in the group, two measurements will be taken:
  • the length of the upper arm limb (the humerus), and
  • the height of each person.
For the measurement of the humerus, bend the subject's arm at the elbow and feel the tuberosity, or "knot," on the side of the elbow. This is essentially one of the ends of the humerus. Also, feel at the shoulder a similar "knot;" this should be the top of the humerus. Carefully measure the entire length of this bone. Then measure the subject's height very carefully. Record this data for each person before you begin the next step.

Step 2

The data for each person should include the two measurements. Graph the data with the length of the humerus along the x-axis and the subject's height along the y-axis. Once all of the data is graphed, draw a straight line along the data points and extend it so that your line intersects with the y-axis. This point will be the y-intercept. Next determine the slope of the line based on the model in Figure 11. Finally, determine the equation of your line.

Step 3

After each group has completed their data collection and has determined the equation for their lines, it will be time to determine which group came up with the most accurate equation. Select two or three "new" people from another group and switch them with two or three members of your own group. Select people of different heights. Each group will then measure the length of the humerus bone for each of the "new" persons and determine their height using two methods:
  • first, using your graphs, your group will estimate height, and
  • second, using your line equations, each group will calculate each new person's height.
Record this data. Then measure the person's height to see how closely your graph and your line equation were able to predict the height of these students. Which group came up with the most accurate graph? Which group came up with the most accurate line equation? They (there may be two groups) are the "winners!" The other groups will have to perform an adaptation of the very famous song "Dry Bones" to the rest of the class! The words to this song are shown at the end of this exercise.

Step 4

Based on your experiences, each group must answer the following questions.

Questions

  1. What are the sources of error that might have been responsible for:
    • data points not lining up,
    • the graph not being accurate enough to estimate the "new" person's height, and
    • the equation not being accurate enough to calculate the "new" person's height.
  2. What kind of sample population would yield the best set of data points and, therefore, the best line equation?
  3. How will the age of the sample population affect your results? How old must your subjects be to determine height from the radius, the tibia, and the femur?

Dry Bones

Adapted from the original African American Spiritual

The foot bone connected to the leg bone,
The leg bone connected to the knee bone,
The knee bone connected to the hip bone,
The hip bone connected to the back bone,
The back bone connected to the shoulder bone,
The shoulder bone connected to the neck bone,
The neck bone connected to the jaw bone,
The jaw bone connected to the head bone,
Now hear us sing our song.
Them bones, them bones gonna walk around,
Them bones, them bones gonna walk around,
Them bones, them bones gonna walk around,
Now hear us sing our song.

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