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I am trying to figure out what it means if a population mean value is fixed.

gung - Reinstate Monica
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user112808
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    Assume there is a population (distribution), which has a (population) mean value. You may not know what the mean value or the distribution are, but they are what they are, and (depending on context, and true in the context you appear to be using it in) do not change (over time, say), therefore the population mean (value) is fixed, even though you may not know its value. By contrast, a sample mean from this same population could change whenever you draw a new sample from the population, and therefore is random and not fixed.. – Mark L. Stone Apr 20 '16 at 00:07
  • It means that a frequentist statistician is talking – Henry Oct 21 '16 at 17:47

2 Answers2

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Consider a finite population of values -- let's say you have a billion of them

The mean of that population is a single value -- you can compute it. If you recalculate it again on the same population, you use all the same values in your mean, so it's the same number every time.

By contrast if you take new samples the sample mean is different from one sample to the next, because the samples contain different individuals.

Glen_b
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  • The thing missing from this answer is, how could a population mean not be a fixed number? Is the statement just stating the obvious, or is there some subtlety to it? If it's obvious, why say it at all? – user541686 Apr 20 '16 at 04:30
  • @Mehrdad Because it's not obvious for everyone. Some people might think it's obvious that the population mean is a random variable distributed around the sample mean, for example. – user253751 Apr 20 '16 at 09:36
  • @Mehrdad There may be more to this than either the question or its answers have revealed so far. What is the mean weight of the Chinese people? After you have answered that one, answer it again. During the intervening time some of the people have died, some have been born, and everybody has gained or lost weight through eating, metabolism, breathing, etc. A natural response to that is to say "but wait--that's a changing population!" Fair enough. But in that case what would it mean to assert "the population" has a fixed mean weight? Would such a statement have any meaning at all? – whuber Apr 20 '16 at 13:09
  • @whuber: I don't get it, if the population is changing then wouldn't having a fixed mean weight mean that at any infinitesimally small instant in time if you took two samples as large as the population then you would be guaranteed to that they would have the same mean? Which to me is again pretty obvious; I don't understand why it would be stated at all. – user541686 Apr 20 '16 at 17:37
  • @Mehrdad Such a complete census is physically impossible. The issues raised by this example go to understanding exactly what a "population" is, how it is defined, and the extent to which our mathematical models accurately reflect a real phenomenon, process, or population. The model you invoke, although an excellent one, is already sufficiently complex to justify asking exactly what a "fixed mean value" really is. – whuber Apr 20 '16 at 18:43
  • @whuber: I guess I don't really understand your reasoning. You define things in math that you cannot physically observe all the time. I don't see where the problem is. – user541686 Apr 20 '16 at 18:57
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Here's how it finally made sense to me.

When we say something like, "human male height is normally distributed", we really mean it. We actually believe that, whoever is in charge of this thing, has a book, and on a page of that book is written something like:

Human males. Distributed like $N(70, 5)$, in units of inches.

When whoever is in charge needs a new male, they fire up their random number generator

Ok, I need a new male. (Generates sample point), ok, 72 inches.

We don't get to see the book, we just get to see a bunch of males. So the best we can do is use this bunch of males to attempt to infer what is in the book.

What is in the book is the population parameter. What we get by using statistics and data is a parameter estimate.

Matthew Drury
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