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What is meant by the mean or average of a quantity? Well, suppose that we
wished to calculate the average age of undergraduates at the University of Texas at Austin.
We could go to the central administration building and find
out how many eighteen yearolds, nineteen yearolds, etc. were currently
enrolled. We would then write something like

(9) 
where is the number of enrolled eighteen yearolds, etc.
Suppose that we were to pick a student at random and then ask ``What is
the probability of this student being eighteen?'' From what we have
already discussed, this probability is defined

(10) 
where
is the total number of enrolled
students. (Actually, this definition is only accurate in the limit that
is very
large.)
We can now see that the average age takes
the form

(11) 
Well, there is nothing special about the age distribution of students
at UT Austin. So, for a general variable , which can take on any one of
possible values ,
, with corresponding probabilities
,
,
the mean or average value of , which
is denoted
, is defined as

(12) 
Suppose that is some function of . Then, for each of
the possible values of there is a corresponding value
of which occurs with the same probability. Thus, corresponds
to and occurs with the probability , and so on. It follows from
our previous definition that the mean value of is
given by

(13) 
Suppose that and are two general functions of . It follows that

(14) 
so

(15) 
Finally, if is a general constant then

(16) 
We now know how to define the mean value of the general variable .
But, how can we characterize the scatter around the mean value?
We could investigate the deviation of from its mean value
,
which is denoted

(17) 
In fact, this is not a particularly interesting quantity, since its average
is zero:

(18) 
This is another way of saying that the average deviation from the
mean vanishes. A more interesting quantity is the square of the
deviation. The average value of this quantity,

(19) 
is usually called the
variance.
The variance is a positive number,
unless there is no scatter at all in the
distribution, so that all possible values of correspond to the
mean value
, in which case it is zero.
The following general relation
is often useful

(20) 
giving

(21) 
The variance of
is proportional to the square of the scatter
of around its mean value. A more useful measure of the scatter
is given by the square root of the variance,

(22) 
which is usually called the standard deviation of . The
standard deviation is essentially the width of the range over which
is distributed around its mean value
.
Next: Continuous Probability Distributions
Up: Probability Theory
Previous: Combining Probabilities
Richard Fitzpatrick
20100720