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mean of X (the outcome of a single die) is = 3.5, as seen in Figure 6-1a. The mean of , denoted , equals 3.5 as well. The average of a single roll is the same as the average of all possible sample means from 10 rolls.
In general, the mean of this population of all possible sample means is the same as the mean of the entire population. Notationally speaking, you write . This makes sense; the average of the averages from all samples is the average of the population that the samples came from.
Using subscripts on we can distinguish which mean we're talking about. The mean of X (the individuals in the population) or the mean of (all possible sample means from the population) is denoted .
Standard error of a sampling distribution
The values in any population deviate from their mean (people have different heights, and so on). Variability in a population
of individuals ( X ) is measured in standard deviations (see Chapter 2). Sample means vary because you're not sampling the whole population, only a subset. Variability in the sample mean ( ) is measured in terms of standard errors .
Error here doesn't mean there's been a mistake — it means there is a gap between the population and sample results.
The standard error of the sample means is denoted by . Its
formula is , where is population standard deviation and
n is sample size. In the next sections you see the effect each of has on the standard error.
Sample size and standard error
Because n is in the denominator of its formula, the standard error decreases as n increases. It makes sense that having more data gives less variation (and more precision) in your results.
A visual can help you see what's happening here with respect to gaining precision in as n increases. Suppose X is the time it takes for a worker to type and send 10 letters of recommendation. Suppose X has a normal distribution with mean 5 minutes and standard deviation 2 minutes. Figure 6-2a shows the picture of the distribution of X .
Now take a random sample of 10 workers, measure their times, and find the average, each time. Repeat this process over and over, and graph all of the possible results for all possible samples. Figure 6-2b shows the picture of the distribution of . Notice that it's still centered at 10 (which we expected) and that its variability is smaller; the standard
error in this case is . The average times are
closer to 10 than the individual times shown in Figure 6-2a. That's because average times for 10 individuals don't change as much as individual times do.
Now take random samples of 50 workers and find their means. This sampling distribution is shown in Figure 6-2c. The variation is even smaller here than it was for n = 10; the standard
error of in this case is . The average times here
are even closer to 10 than the ones from Figure 6-2b. Larger sample sizes mean more precision and less change from sample to sample.

    Figure 6-2: Distributions of a) individual times; b) average times for 10 individuals; c) average times for 50 individuals.
     
Population standard deviation and standard error
In the standard error formula for , you see that the
popula-tion standard deviation, , is in the numerator. That means as the population standard deviation increases, the standard error of the sample means increases. Mathematically this makes sense; how about statistically?
Suppose you have two ponds of fish (call them Pond #1 and Pond #2), and you want to find the average length of all the fish in each pond. Suppose you know that the fish lengths in Pond #1 have a mean of 20 inches and a standard deviation of 2 inches (see Figure 6-3a). Suppose the fish in Pond #2 also average 20 inches, but have a standard deviation of 5 inches (see Figure 6-3b). Comparing Figures 6-3a and 6-3b you see they have the same shape and mean, but the fish in Pond #2 are more variable than in Pond #1.
Now suppose you take a sample of 100 fish from Pond #1, find the mean length of the fish, and repeat this process over and over.