Conceptual Populations and Population Means

The clicker question today about the relationship between a sample mean (\(\)) and a population mean (\(\)) was a challenging one. Here's a bit more context for that question.

Suppose you weigh a rock on a scale several times, each time getting a slightly different reading. Assuming that the physical characteristics of the rock aren't changing, then these readings can be considered a random sample, taken from a conceptual population consisting of all the readings that the scale could in principle produce.

Finding the sample mean (\(\)) for this sample is straightforward: you add up the readings and divide by the number of readings. However, the population mean (\(\)) isn't so straightforward. It's not possible to add up all the readings that the scale could in principle produce, much less divide by the number of such readings.  That's why I objected to the statement in the clicker question that said that sample means and population means are calculated in the same way. You can't always "calculate" a population mean.

So what do we mean by "population mean" for conceptual populations? One way to think of it is to define the population mean as the mean of a sample that (somehow, miraculously) follows the population distribution perfectly. However, that requires defining a "population distribution," which we haven't done yet. For now, perhaps it's better to think of the population mean as the expected value of the population.

That rock has, in a sense, a "true" weight. That's the expected value of the population. We can't calculate that or even really know what it is. But we can calculate the mean of our sample of readings, and that sample mean is likely to be close to the "true" weight of the rock. Thus the population mean for this conceptual population is this "true" weight, since it's the expected value for any sample we might take.

Hope that helps, at least a little.

Image: "Dravite," Craig Elliott, Flickr (CC)

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