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 telephones 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 commonly 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 random 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 involving college freshmen and sophomores, one might select
25 students at random from each group—freshman
males, freshman females, sophomore males, and sophomore females—to make a sample of 100 students.
Cluster Sampling
If an intact subgroup of a population is used as a representative 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 representative 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 Australia 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 mathematical theory used to describe, or predict, how a population 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 Society. Malthus developed a simple model that yielded the
troubling conclusion that eventually the human population 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 incorrect 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 t and
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 offspring, and another percentage, say d percent
of the population, dies. (The number b is called
the birth rate and d is 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 k is called the
growth rate.) Thus P(t) is a function whose
derivative is a constant times itself. Only exponential functions have this property and so:
P(t) = P(0)e
kt
dP
––
dt
population models 407
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 telephones 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 commonly 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 random 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 involving college freshmen and sophomores, one might select
25 students at random from each group—freshman
males, freshman females, sophomore males, and sophomore females—to make a sample of 100 students.
Cluster Sampling
If an intact subgroup of a population is used as a representative 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 representative 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 Australia 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 mathematical theory used to describe, or predict, how a population 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 Society. Malthus developed a simple model that yielded the
troubling conclusion that eventually the human population 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 incorrect 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 t and
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 offspring, and another percentage, say d percent
of the population, dies. (The number b is called
the birth rate and d is 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 k is called the
growth rate.) Thus P(t) is a function whose
derivative is a constant times itself. Only exponential functions have this property and so:
P(t) = P(0)e
kt
dP
––
dt
population models 407