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Handout 4
Defining the Population Mean and Population Variance
In the last handout the subject of statistics was defined to be the art and science of
making inferences about a population on the basis of a sample.
But what sort of inferences
about a population does one seek to make ?
One often seeks to estimate the
population
mean
and the
population variance
.
So the point of taking a sample and computing a
sample mean and a sample variance is to
estimate
the population mean and population
variance, respectively.
The foregoing discussion is getting ahead of itself, in the sense that it presumes that
the terms
population
mean and
population
variance have already been defined – which they
haven’t.
There are two cases to consider : the case in which the population is finite, and the
case in which the population is infinite.
Finite versus
Infinite Populations
In the case of a finite population, say of size N, the definition of a population mean
is the obvious analogy to the definition of a sample mean given in Handout 1 : the population
mean is
.
For an infinite population this clearly makes no sense at all!
Infinite
populations are not just the province of pure mathematicians spinning abstract theories for
the sake of abstraction itself ; the case of an infinite population is quite common in scientific
settings.
Suppose one is investigating the wing span ( in millimeters ) of the progeny resulting
from the cross breeding of two strains of fruit fly.
What is the population under investigation?
The fruit flies bred by the investigator ?
The fruit flies bred by all researchers in the US that
year ?
The fruit flies bred by anyone, anyplace to date ?
If it’s a truly general scientific
investigation, the population of interest is the set of all wing spans of all
potential
fruit flies that
have been or
could be
bred, anytime or anyplace ; — which is, by the nature of the concept, an
infinite set.
And any scientific investigation of sufficient
generality
to be of significant enduring
interest, is likely to be about an infinite population.
Developing a Definition for the Case of Discrete Random Variables
We will consider the problem of defining the population mean in the special case of a
discrete
random variable : that is, a random variable that takes on a finite number of
different values.
( For example, consider
the random variable X that takes on the value 1 if a
coin toss comes up heads, and the value 0 if the coin toss comes up tails.
This is a discrete
random variable since there are only two different values the random variable
takes on ; but one
can, at least conceptually, keep tossing
the coin from here to eternity, and hence the population
of interest
is infinite. )
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View Full DocumentThe goal is to
develop
a definition which makes sense whether the population is
finite or infinite. We’re not going to just plop down an unmotivated definition !! The motivation
will be provided by a consideration of how to efficiently
compute a sample mean when one
has socalled ‘grouped’ data.
Keep in mind that the computational aspect is not what is of
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 Fall '08
 Dinwoodie
 Statistics, Variance

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