much editorial commentary to this effect. It turned out,

however, that Landon’s opponent, Franklin Roosevelt,

won the election by a landslide. Members of the Digest

did not realize that they had worked with a biased

sample—only affluent Americans could afford tele-

phones at the time of the Great Depression and thus be

listed in telephone books. This was a class of voter that

was more likely to vote Republican. Consequently, the

Digest’s prediction was erroneous. The publication

folded in 1937 due to both the sampling fiasco and the

difficult times of the depression.

Today, a number of sampling methods are com-

monly used to help ensure that no bias occurs. These

methods include:

Random Sampling

Each subject of the population is assigned a number,

and numbers are generated randomly with the aid of a

computer to select members.

Systematic Sampling

Each subject of the population is assigned a number,

and, starting at a random number, every kth member

from then on is selected. For example, one might select

every 23rd person, starting with the 533rd member.

Stratified Sampling

When a population is naturally divided into groups

(such as male/female, or age by decade), selecting a ran-

dom sample from within each group produces what is

called a “stratified sample.” Samples produced this way

are used to ensure that representatives of each subgroup

are present in the study. For example, in a study involv-

ing college freshmen and sophomores, one might select

25 students at random from each group—freshman

males, freshman females, sophomore males, and sopho-

more females—to make a sample of 100 students.

Cluster Sampling

If an intact subgroup of a population is used as a rep-

resentative sample of the entire population, then the

sample is called a cluster sample. For example, the set

of all freshman females might be used to represent the

population of all college students for the purposes of

one study, or the 12 eggs in one carton of eggs as rep-

resentative of all the eggs handled by a particular

supermarket.

See also

BIAS

;

STATISTICS

:

DESCRIPTIVE

.

population models In biology the term population

means the number of individuals or organisms living in

a certain area. For example, the population of Aus-

tralia is the number of individuals currently living on

that continent, and the population of a laboratory

yeast culture is the number of organisms present in a

particular petri dish. A population model is a mathe-

matical theory used to describe, or predict, how a pop-

ulation size changes over time.

Interest in how populations grow was stimulated

in the late 18th century when Thomas Malthus

(1766–1834) published An Essay on the Principle of

Population as it Affects the Future Improvement of Soci-

ety. Malthus developed a simple model that yielded the

troubling conclusion that eventually the human popula-

tion would reach a size that cannot be sustained with the

food resources available on this planet. Although his

model oversimplifies matters and has proved to be incor-

rect for making long-term predictions, the Malthusian

model is still useful for understanding short-term

growth. His model is developed as follows:

Let P(t) be the population size at time tand

assume that over one unit of time (a minute, or

a day, or a year) that a certain percentage, say b

percent, of the population gives birth to off-

spring, and another percentage, say dpercent

of the population, dies. (The number bis called

the birth rate and dis called the death rate.)

Thus after one unit of time, the population

increases by the amount bP(t) – dP(t).We have:

P(t+ 1) – P(t) = (b– d)P(t)

This says that the rate of change of population

size is given by a constant (b– d) times the

population size. Assuming that the population

function P(t) can, for the sake of convenience,

be regarded as continuously changing with

time, this final statement can be interpreted, in

CALCULUS

, as a formula:

= kP

where k= b– d. (The constant kis called the

growth rate.) Thus P(t) is a function whose

derivative is a constant times itself. Only expo-

nential functions have this property and so:

P(t) = P(0)ekt

dP

––

dt

population models 407