QUALITATIVE AND QUANTITATIVE SAMPLINGalready selected for the sample. For almost all prac-
tical purposes in social science, random sampling
is without replacement.
We can see the logic of simple random sampling
with an elementary example: sampling marbles
from a jar. Let us say I have a large jar full of 5,000
marbles, some red and some white. The marble is
my sampling element, the 5,000 marbles are my
population (both target and ideal), and my sample
size is 100. I do not need a sampling frame because
I am dealing with small physical objects. The pop-
ulation parameter I want to estimate is the percent-
age of red marbles in the jar.
I need a random process to select 100 marbles.
For small objects, this is easy; I close my eyes, shake
the jar, pick one marble, and repeat the procedure
100 times. I now have a random sample of marbles.
I count the number of red marbles in my sample to
estimate the percentage of red versus white marbles
in the population. This is a lot easier than counting
all 5,000 marbles. My sample has 52 white and
48 red marbles.
Does this mean that the population parameter
is exactly 48 percent red marbles? Maybe or maybe
not; because of random chance, my specific sample
might be off. I can check my results by dumping the
100 marbles back in the jar, mixing the marbles, and
drawing a second random sample of 100 marbles.
On the second try, my sample has 49 white marbles
and 51 red ones. Now I have a problem. Which is
correct? You might ask how good this random
sampling business is if different samples from the
same population can yield different results. I repeat
the procedure over and over until I have drawn
130 different samples of 100 marbles each (see
Chart 1 for results). Most people might find it eas-
ier to empty the jar and count all 5,000 marbles, but
I want to understand the process of sampling. The
results of my 130 different samples reveal a clear
pattern. The most common mix of red and white
marbles is 50/50. Samples that are close to that split
are more frequent than those with more uneven
splits. The population parameter appears to be 50
percent white and 50 percent red marbles.
Mathematical proofs and empirical tests
demonstrate that the pattern found in Chart 1
always appears. The set of many different samples
is my sampling distribution. It is a distribution of
different samples. It reveals the frequency of dif-
ferent sample outcomes from many separate ran-
dom samples. This pattern appears if the sample
size is 1,000 instead of 100, if there are 10 colors
of marbles instead of 2, if the population has 100
marbles or 10 million marbles instead of 5,000,
and if the sample elements are people, automobiles,
or colleges instead of marbles. In fact, the “bell-
shaped” sampling distribution pattern becomes
clearer as I draw more and more independent ran-
dom samples from a population.
The sampling distribution pattern tells us that
over many separate samples, the true population
parameter (i.e., the 50/50 split in the preceding
example) is more common than any other outcome.
Some samples may deviate from the population
parameter, but they are less common. When we plot
many random samples as in the graph in Chart 1,
the sampling distribution always looks like a nor-
mal or bell-shaped curve. Such a curve is theoreti-
cally important and is used throughout statistics.
The area under a bell-shaped curve is well known
or, in this example, we can quickly figure out the
odds that we will get a specific number of marbles.
If the true population parameter is 50/50, standard
statistical charts tell what the odds of getting 50/50
or a 40/50 or any other split in a random sample are.
The central limit theoremfrom mathematics
tells us that as the number of different random
samples in a sampling distribution increases toward
infinity, the pattern of samples and of the popula-
tion parameter becomes increasingly predictable.
For a huge number of random samples, the sampling
distribution always forms a normal curve, and
the midpoint of the curve will be the population
parameter.Sampling distribution A distribution created by
drawing many random samples from the same
population.
Central limit theorem A mathematical relationship
that states when many random samples are drawn
from a population, a normal distribution is formed, and
the center of the distribution for a variable equals the
population parameter.