CHAPTER SEVEN:
Samples
Previously, it was stated that there are two types of
statistics: inferential and descriptive. The preferred would be
descriptive, because it is based upon a population. However,
this is seldom possible. Most statistical analyses are
inferential, based upon a sample. It is imperative, then, to
understand the assumptions and principles of samples. We shall
begin with some crucial definitions.
TERMS:
The more fundamental principles that govern the behavior of
samples are the Law of Large Numbers, the Central Limit Theorem
and the features of The Sampling Distribution of the Mean.
A Sampling Distribution- A theoretical probability
distribution which would be comprised of all the potential
calculations of a statistic, from an infinite variety of
samples.
If comprised of the infinite variety of all possible means
from all possible samples of a population, that would be the
sampling distribution of the mean (u_).
X
The Law of Large Numbers- As the sample size increases, the
error of estimate reduces; because the standard error (SE) of
the sampling distribution of the mean is (SE) = / n. Therefore,
as N ↑, SE ↓ for the sampling distribution of the mean (u_).
If u = 500 and = 100: X
when n = 4, SE = 100/ 4 = 100/2 = 50
when n = 25, SE = 100/ 25 = 100/5 = 20
when n = 100, SE = 100/ 100 = 100/10 = 10
The Central Limit Theorem (CLT) - As the sample size
increases,the error reduces and the sampling distribution of the
mean quickly approaches normal. This is true regardless of the
shape of the original population from which it was drawn. For
samples with N > 30, the mean of the sampling distribution of
the mean is the best unbiased estimate of the population true
mean (u); and the SE of the sampling distribution of the mean is
the best unbiased estimate of the SE and has the formula:
SE = Sigma (σ) / the square root of N
The value of these principles, is the assurance that means can
be compared. In fact, Z-scores for a mean of a sample can be
calculated, with a minor adjustment in the formula.
For a mean
Z = X - u / SE
Caution! It would not be appropriate to compare means based upon
the binomial approximation to the mean.
TYPES OF SAMPLING:
Random Samples-A subset of the referent population,in which all
members of the population have an equal and constant likelihood
of being selected; that is, sampling with replacement.
In this way, all potential confounders may be distributed across
groups in a non-systematic way, determined by chance factors
alone.
Block Randomization-A type of random sampling which collects
subjects in blocks or proportions according to some factor or
demographic characteristic.
Random Assignment-The distribution of subjects, who were
randomly selected, across experimental conditions to assure a
pattern of assignment determined by chance factors alone.