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AP Statistics Section 7.2A
Mean & Standard Deviation of a
Probability Distribution
The mean, x ,of a set of observations is
simply their ordinary average. The mean
of a random variable X is also an average of
the possible values of X, BUT we must take
into account that the various values of X
are not all equally likely.
The probability distribution shows k
possible values of the random variable X
along with the corresponding probabilities.
Value of x
x1
Probability
p1
x2
p2
x3
…..
p3
…..
xk
pk
X
The mean of the random variable X, denoted by ____,
is found using the formula:
 X  xi  pi  x1  p1  x2  p2      xk  pk
In words, multiply each value of X times its probability
and add all these products together.
The mean of a random variable X is
also called the expected value,
because it is the
“average” we would expect to get
with infinitely many trials.
To find the variance of a discrete
2
random variable ( __x ), use the
formula:
 x  ( xi   x )  pi
2
2
Recall that the standard deviation of X
 x is the
(__)
square root of the variance.
Example: Consider the probabilities at the right for the
number of games, X, it will take to complete the World
Series in any given year.
a) Find the mean of X and interpret this value in the
context of the problem.
 x  4(.1819)  5(.2121)  6(.2323)  7(.3737)  5.7978
The average number of games that would be played in
a very large number of World Series is 5.7978
Example: Consider the probabilities at the right for the
number of games, X, it will take to complete the World
Series in any given year.
b) Find the standard deviation of X.
 x 2  (4  5.7978) 2  .1819  (5  5.7978) 2  .2121      1.2725
 x  1.2725  1.128
TI83/84:
Put X’s in L1 and P(X)s in L2
STAT / CALC 1:1-Var Stats ENTER
1-Var Stats L1, L2
x  5.7978
x  1.128
Suppose we would like to estimate
the mean height,  ,of the
population of all American women
between the ages of 18 and 24
years.
To estimate  , we choose an SRS of
young women and use the sample
mean, x, as our best estimate of  .
Statistics, such as the mean,
obtained from samples are random
variables because their values
would vary in repeated sampling.
The sampling distribution of a statistic
is really just the probability
distribution of the random variable.
We will discuss sampling distributions
in detail in Chapter 9.
The important question is this: Is it reasonable
to use x to estimate  ?
The answer is IT DEPENDS!
We don’t expect x   , but what could
we do to increase the reasonableness of
using x to estimate  ?
The Law of Large Numbers says, broadly
anyway, that as the size of an SRS
increases, the mean of the sample, x ,
eventually approaches the mean of the
population,  , and then stays close.
Casinos, fast-food restaurants and
insurance companies rely on this
law to ensure steady profits.
Many people incorrectly believe in
the “law of small numbers” (i.e.
they expect short term behavior to
show the same randomness as
long term behavior).
Remember that it is only in the
long run that the regularity
described by probability and the
law of large numbers takes over.