AP Statistics Section 7.2A Mean & Standard Deviation of a Probability Distribution

The mean,,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. The mean of the random variable X, denoted by ____, is found using the formula: In words, Value of x….. Probability….. 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 random variable ( __ ), use the formula: Recall that the standard deviation of 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.

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.

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

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,, 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 to estimate ? The answer is IT DEPENDS! We don’t expect, but what could we do to increase the reasonableness of using to estimate ?

The Law of Large Numbers says, broadly anyway, that as the size of an SRS increases, the mean of the sample,, 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.