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I'm studying statistics using an introductory book and it mentioned these:

"The Greek symbol μ is the population mean."

"When evaluating the long-term results of statistical experiments, we often want to know the “average” outcome. This “long-term average” is known as the mean or expected value of the experiment and is denoted by the Greek letter μ."

Why do they use the same symbol and what's the difference between the two?

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    $\begingroup$ Because the mean of a random variable is the same as its expectation $E(X) = \mu_x.$ $\endgroup$
    – BruceET
    May 22, 2019 at 4:43
  • $\begingroup$ Personally I would not choose for $\mu$ as notation of population mean. The notation gives the impression that we are dealing with some real number which is not the case. The population mean is a random variable. $\endgroup$
    – drhab
    May 22, 2019 at 7:25
  • $\begingroup$ @drhab No, the population mean $\mu$ is not a random variable, but a fixed number (a parameter). The sample mean $\bar{X}$ is a random variable (a statistic) when the sample values are considered random variables. But the sample mean $\bar{x}$ of a given sample is a fixed number, and thus not a random variable. $\endgroup$
    – matthiash
    Aug 18, 2022 at 8:51

2 Answers 2

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First of all to answer a question you didn't ask, $\mu$ is the Greek equivalent of the latin $m$, which stands for mean.

Now for the question you did ask. If you have a random variable $X$, and let's assume $X$ is positive for simplicity, then you always have a mean $\mathbb EX$ (which could be infinite). The mean is computed mathematically, by integrating against the probability density function. Thus, both the variable $X$ and the mean $\mu=\mathbb EX$ are theoretical quantities. They describe the statitician's model of the quantity of interest.

On the other hand, the way experiments commonly work is that we collect a sequence of samples to try to nail down a more accurate model. Now the experiment as a whole can be thought of as a single random object, described mathematically by a probability distribution (or better yet, measure) on an infinite sequence space. The actual measurements taken can be written as an infinite sequence $(X_i)_{i\in\mathbb N}$. Now our model will usually posit that the measurements we take all have the same distribution ($X_i$ and $X_j$ have the same law, for all $i$ and $j$) and that the measurements are independent. In this case, the central limit theorem guarantees that if you compute the sample mean $$ \lim_{n\to\infty}\frac{X_1+\cdots+X_n}{n}, $$ this a priori random quantity will in fact converge (with probability $1$) to the theoretical mean $\mathbb EX_1$.

Thus, in the limit of a very large number of samples, there ceases to be a distinction between the theoretical mean of a single variable, and the sample mean of the whole population.

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  • $\begingroup$ +1 Nice explanation. $\endgroup$
    – drhab
    May 22, 2019 at 7:22
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Consider we have the data {x1, x2, x3, x4} with probabilities {p1, p2, p3, p4}

Expected value: $$E(x) = x1*p(x1) + x2*p(x2) + x3*p(x3) + x4*p(x4)$$

if probabilities are the same then: $E(x) = \frac{\sum xi}{4}$ that is the same as Mean (average of xis

if probabilities are not the same, then: the average of xis would be their weighted sum and that is again like E(x)

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