In Chapter 1 (page 26), we learned that the mean x¯ is the average of the observations in a sample. Recall that a random variable X is a numerical outcome of a random process. The mean can be considered the average of many replications of X. This is true for discrete and continuous random variables.

If we think of a random process as corresponding to the population, then the mean of the random variable is a characteristic of this population. Here is an example.

If you play Tri-State Pick 3 several times, we would—as usual—call the mean of the actual amounts you win x¯. The mean in Example 4.33 is a different quantity: it is the long-run average winnings you expect if you play a very large number of times.

Just as probabilities are an idealized description of long-run proportions, the mean of a probability distribution describes the long-run average outcome. We can’t call this mean x¯, so we need a different symbol. The common symbol for the mean of a probability distribution is μ, the Greek letter mu. We used μ in Chapter 1 for the mean of a Normal distribution, so this is not a new notation. We will often be interested in several random variables, each having a different probability distribution with a different mean.

To remind ourselves that we are talking about the mean of X, we often write μX rather than simply μ. In Example 4.33, μX=$0.50. Notice that, as often happens, the mean is not a possible value of X. You will often find the mean of a random variable X called the expected value of X. This term can be misleading because we don’t necessarily expect one observation on X to be close to its expected value.

The mean of any discrete random variable is found just as in Example 4.33. It is an average of the possible outcomes—but it is a weighted average in which each outcome is weighted by its probability. Because the probabilities add to 1, we have total weight 1 to distribute among the outcomes. An outcome that occurs half the time has probability one-half and gets one-half the weight in calculating the mean. Here is the general definition.

Suppose that the random digits in Example 4.34 had a different probability distribution. In Example 4.15 (page 215), we described Benford’s law as a probability distribution that describes first digits of numbers in many real situations. Let’s calculate the mean for Benford’s law.

Figure 4.13 locates the means of X and V on the two probability histograms. Because the discrete uniform distribution of Figure 4.13(a) is symmetric, the mean lies at the center of symmetry. We can’t precisely locate the mean of the right-skewed distribution of Figure 4.13(b) by eye; calculation is needed.

What about continuous random variables? The probability distribution of a continuous random variable X is described by a density curve. Chapter 1 (page 49) showed how to find the mean of the distribution: it is the point at which the area under the density curve would balance if it were made out of solid material. The mean lies at the center of symmetric density curves such as the Normal curves. Exact calculation of the mean of a distribution with a skewed density curve requires advanced mathematics.¹² The idea that the mean is the balance point of the distribution applies to discrete random variables as well, but in the discrete case, we have a formula that gives us this point.

We would like to estimate the mean height μ of the population of all American women between the ages of 18 and 24 years. This μ is the mean of the random variable X obtained by choosing a young woman at random and measuring her height. To estimate μ, we choose an SRS of young women and use the sample mean x¯ to estimate the unknown population mean μ. In the language of Section 5.1 (page 271), μ is a parameter, and x¯ is a statistic.

Statistics obtained from probability samples are random variables because their values vary in repeated sampling. The sampling distributions of statistics are just the probability distributions of these random variables.

It seems reasonable to use x¯ to estimate μ. An SRS should fairly represent the population, so the mean x¯ of the sample should be somewhere near the mean μ of the population. Of course, we don’t expect x¯ to be exactly equal to μ, and we realize that if we choose another SRS, the luck of the draw will probably produce a different x¯.

If x¯ is rarely exactly right and varies from sample to sample, why is it nonetheless a reasonable estimate of the population mean μ? If we keep on adding observations to our random sample, the statistic x¯ is guaranteed to get as close as we wish to the parameter μ and then stay that close. We have the comfort of knowing that if we can afford to keep on measuring more women, eventually we will estimate the mean height of all young women very accurately. This remarkable fact is called the law of large numbers. It is remarkable because it holds for any population, not just for some special class such as Normal distributions.

The behavior of x¯ is similar to the idea of probability. In the long run, the proportion of outcomes taking any value gets close to the probability of that value, and the average outcome gets close to the distribution mean. Figure 4.1 (page 205) shows how proportions approach probability in one example. Here is an example of how sample means approach the distribution mean.

Example 4.36 Heights of young women.

The distribution of the heights of all young women is close to the Normal distribution with mean 64.5 inches and standard deviation 2.5 inches. Suppose that μ=64.5 were exactly true.

Figure 4.14 shows the behavior of the mean height x¯ of n women chosen at random from a population whose heights follow the N(64.5,2.5) distribution. The graph plots the values of x¯ as we add women to our sample. The first woman drawn had height 64.21 inches, so the line starts there. The second had height 64.35 inches, so for n=2 the mean is

x¯=64.21+64.352=64.28

This is the second point on the line in the graph.

At first, the graph shows that the mean of the sample changes as we take more observations. Eventually, however, the mean of the observations gets close to the population mean μ=64.5 and settles down at that value. The law of large numbers says that this always happens.

The mean μ of a random variable is the average value of the variable in two senses. By its definition, μ is the average of the possible values, weighted by their probability of occurring. The law of large numbers says that μ is also the long-run average of many independent observations on the variable. The law of large numbers can be proved mathematically starting from the basic laws of probability.

The law of large numbers says broadly that the average results of many independent observations are stable and predictable. The gamblers in a casino may win or lose, but the casino will win in the long run because the law of large numbers says what the average outcome of many thousands of bets will be. An insurance company deciding how much to charge for life insurance and a fast-food restaurant deciding how many beef patties to prepare also rely on the fact that averaging over many individuals produces a stable result. It is worth the effort to think a bit more closely about such an important fact.

We occasionally need to apply a linear transformation, a+bX, to a random variable. Similarly, there are situations where our interest is in the sum or difference between two random variables, X+Y or X−Y. The means of the transformed variables can be found easily from the means of the original random variables. Here are some examples.

In more formal language, the number of dimples on a refrigerator is a random variable X that varies as we inspect one refrigerator after another. We know only that the mean number of dimples is μX=0.7. The number of paint sags is a second random variable Y having mean μY=1.4. (As usual, the subscripts keep straight which variable we are talking about.) The total number of both dimples and sags is another random variable, the sum X+Y. Its mean μX+Y is the average number of dimples and sags together. It is just the sum of the individual means μX and μY.

That’s an important rule for how means of random variables behave. Here’s another example describing a different rule.

The point of these examples is that means behave like averages. Here are the rules written in more formal language.

Example 4.42 How many courses?

In Exercise 4.35 (page 233), you described the probability distribution of the number of courses taken in the fall by students at a small liberal arts college. Here is the distribution:

Courses in the fall	1	2	3	4	5	6
Probability	0.06	0.06	0.12	0.20	0.41	0.15

For the spring semester, the distribution is a little different:

Courses in the spring	1	2	3	4	5	6
Probability	0.06	0.08	0.15	0.25	0.34	0.12

For a randomly selected student, let X be the number of courses taken in the fall semester and let Y be the number of courses taken in the spring semester. The means of these random variables are

μX=(1)(0.06)+(2)(0.06)+(3)(0.12)+(4)(0.20)+(5)(0.41)+(6)(0.15)=4.29μY=(1)(0.06)+(2)(0.08)+(3)(0.15)+(4)(0.25)+(5)(0.34)+(6)(0.12)=4.09

The mean course load for the fall is 4.29 courses, and the mean course load for the spring is 4.09 courses. We assume that these distributions apply to students who earned credit for courses taken in the fall and the spring semesters. The mean of the total number of courses taken for the academic year is X+Y. Using Rule 2, we calculate the mean of the total number of courses:

μZ=μX+μY=4.29+4.09=8.38

Note that it is not possible for a student to take 8.38 courses in an academic year. This number is the mean of the probability distribution.

The mean is a measure of the center of a distribution. A basic numerical description requires, in addition, a measure of the spread or variability of the distribution. The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center. Just as for the mean, we need a distinct symbol to distinguish the variance of a random variable from the variance s2 of a data set. We write the variance of a random variable X as σX2. Once again, the subscript reminds us which variable we have in mind.

The definition of the variance σX2 of a random variable is similar to the definition of the variance s2 given in Chapter 1 (page 35). That is, the variance is an average value of the squared deviation (X−μX)2 of the variable X from its mean μX.

For discrete random variables, the average we use is a weighted average in which each outcome is weighted by its probability. Calculating this weighted average is straightforward. For continuous random variables, however, advanced mathematics is needed. Here is the formula for the discrete case.

What are the facts for variances that parallel Rules 1, 2, and 3 for means? The mean of a sum of random variables is always the sum of their means, but this addition rule is true for variances only in special situations. The following example explains why.

If random variables are independent, this kind of association between their values is ruled out, and their variances do add. Two random variables X and Y are independent if knowing that any event involving X alone did or did not occur tells us nothing about the occurrence of any event involving Y alone.

Probability models often assume independence when the random variables describe outcomes that appear unrelated to each other. You should ask in each instance whether the assumption of independence seems reasonable.

When random variables are not independent, the variance of their sum depends on the correlation between them as well as on their individual variances. In Chapter 2, we met the correlation r between two observed variables measured on the same individuals. We defined the correlation r as an average of the products of the standardized x and y observations. The correlation between two random variables is defined in the same way. We won’t give the details; it is enough to know that the correlation between two random variables has the same basic properties as the correlation r calculated from data. We use ρ, the Greek letter rho, for the correlation between two random variables. The correlation ρ is a number between −1 and 1 that measures the direction and strength of the linear relationship between two variables. The correlation between two independent random variables is zero.

With the correlation at hand, we can state the rules for manipulating variances.

Because a variance is the average of squared deviations from the mean, multiplying X by a constant b multiplies σX2 by the square of the constant. Adding a constant a to a random variable changes its mean but does not change its variability. The variance of X+a is, therefore, the same as the variance of X. Because the square of −1 is 1, the addition rule says that the variance of a difference between independent random variables is the sum of the variances. For independent random variables, the difference X−Y is more variable than either X or Y alone because variations in both X and Y contribute to variation in their difference.

As with data, we prefer the standard deviation to the variance as a measure of the variability of a random variable. Rule 2 for variances implies that standard deviations of independent random variables do not add. To combine standard deviations, use the rules for variances. For example, the standard deviations of 2X and −2X are both equal to 2σX because this is the square root of the variance 4σX2.

If you buy a Pick 3 ticket, your winnings are W=X−1 because the dollar you paid for the ticket must be subtracted from the payoff. Let’s find the mean and variance for this random variable.

Suppose now that you buy a $1 ticket on each of two different days. The payoffs X and Y on the two tickets are independent because separate drawings are held each day. Your total payoff is X+Y. Let’s find the mean and standard deviation for this payoff.

Suppose that we buy 20 lottery tickets. The same rules for means and variances of independent random variables works for any number of random variables. Therefore, the mean for 20 tickets is 20×$0.50=$10.00. The variance is 20×249.75=4995, and the standard deviation is 4995=70.68.

When we add random variables that are correlated, we need to use the correlation for the calculation of the variance but not for the calculation of the mean. Here is an example.

The negative correlation in Example 4.51 is due to the fact that, in this household, natural gas is used for heating, and electricity is used for air-conditioning. So, when it is warm, the electricity charges are high, and the natural-gas charges are low. When it is cool, the reverse is true. This causes the standard deviation of the sum to be less than it would be if the two bills were uncorrelated (see Exercise 4.68, page 252).

There are situations where we need to combine several of our rules to find means and standard deviations. Here is an example.

The correlation in this example illustrates an unfortunate fact about compliance and having an adequate diet. Some of the subjects in this study have diets that provide an adequate amount of calcium, while others do not. The positive correlation between compliance and dietary intake tells us that those who have relatively high dietary intakes are more likely to take the assigned supplements. On the other hand, subjects with relatively low dietary intakes, the ones who need the supplement the most, are less likely to take the assigned supplements.