bar is generally used to denote means, but why because someone did it that way, presumably because it seemed like a good idea at the time, and then other people followed suit, as with almost any notational convention. m was often used for means of both distributions and of samples across a wide range of time; it's "re-invented" regularly. I always assumed the bar came from physics. The use of a bar over small x is discussed here: https://mathshistory.st-andrews.ac.uk/Miller/mathsym/stat/ (or see the older version of the page here http://www.math.hawaii.edu/~tom/history/stat.html ) ... scroll about 3/4 of the way down, to the section headed SYMBOLS IN STATISTICS and look at paragraph 2. It looks like it did indeed come from physics. Why are there competing conventions, anyways? Because people keep ignoring existing conventions in favor of ones they like for one reason or another (sometimes out of ignorance, sometimes with a pedagogical motive, sometimes to avoid a clash with some other convention, etc). Standards always multiply. Just recently (i.e. in the last few decades) it happened when ML people started adopting a lot of statistical methods and redefined all the terms and symbols (sometimes to match their own pre-existing terms, sometimes out of ignorance that there was already a good term/notation, sometimes for other reasons). Sadly, some of those conventions cause serious issues (like calling a regression coefficient a weight, leading to a serious clash when you need to talk about weighted regression).

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.

μ Population MeanThe term population mean, which is the average score of the population on a given variable, is represented by:μ = ( Σ Xi ) / NThe symbol ‘μ’ represents the population mean. The symbol ‘Σ Xi’ represents the sum of all scores present in the population (say, in this case) X1 X2 X3 and so on. The symbol ‘N’ represents the total number of individuals or cases in the population.Population Standard DeviationThe population standard deviation is a measure of the spread (variability) of the scores on a given variable and is represented by:σ = sqrt[ Σ ( Xi – μ )2 / N ]The symbol ‘σ’ represents the population standard deviation. The term ‘sqrt’ used in this statistical formula denotes square root. The term ‘Σ ( Xi – μ )2’ used in the statistical formula represents the sum of the squared deviations of the scores from their population mean.Population VarianceThe population variance is the square of the population standard deviation and is represented by:σ2 = Σ ( Xi – μ )2 / NThe symbol ‘σ2’ represents the population variance.Sample MeanThe sample mean is the average score of a sample on a given variable and is represented by:x_bar = ( Σ xi ) / nThe term “x_bar” represents the sample mean. The symbol ‘Σ xi’ used in this formula represents the represents the sum of all scores present in the sample (say, in this case) x1 x2 x3 and so on. The symbol ‘n,’ represents the total number of individuals or observations in the sample.Sample Standard DeviationThe statistic called sample standard deviation, is a measure of the spread (variability) of the scores in the sample on a given variable and is represented by:s = sqrt [ Σ ( xi – x_bar )2 / ( n – 1 ) ]The term ‘Σ ( xi – x_bar )2’ represents the sum of the squared deviations of the scores from the sample mean.Sample VarianceThe sample variance is the square of the sample standard deviation and is represented by:s2 = Σ ( xi – x_bar )2 / ( n – 1 )The symbol ‘s2’ represents the sample variance.Pooled Sample Standard DeviationThe pooled sample standard deviation is a weighted estimate of spread (variability) across multiple samples. It is represented by:sp = sqrt [ (n1 – 1) * s12 + (n2 – 1) * s22 ] / (n1 + n2 – 2) ]The term ‘sp’ represents the pooled sample standard deviation. The term ‘n1’ represents the size of the first sample, and the term ‘n2’ represents the size of the second sample that is being pooled with the first sample. The term ‘s12’ represents the variance of the first sample, and ‘s22’ represents the variance of the second sample.