MORE ABOUT MEANS OF MEANS

So far, we have seen that for a sufficiently large sample size , the distribution of means is normal. That tells us a lot about how likely different means are, but only if we know what the mean and standard deviation of the distribution are. The mean of the "real" distribution of means is the population mean. The mean of the means is the mean? What does that mean? It means (one step at a time, now): Suppose you could take an infinite number of samples and calculate the average for each one. Suppose you could then calculate the average of your averages. What you would get is the same number as if you just went ahead and took the average of the whole population. That really is not surprising at all. For example, if 50% of all the people in a population agree with a statement, then:

• The true population mean is 50%. We just said that: 50% of all the people in a population agree.

• The mean of the distribution of sample means from that population is 50%, too.

Similarly, if the average response of a given study in the population is 100, then the mean of the distribution of means from the population is also 100. It does not matter how large the samples are, whether you have ten-case samples or 10,000-case samples. Nor does it matter whether the response is normally distributed. The mean of the distribution of means is the population mean.