This post is to demonstrate how individuals in a population act around mean, with an emphasis on how the mean behaves in groups. Keep in mind, the individuals in this demonstration DO NOT display random variation which is a key component in many of the assumptions about statistics. HOWEVER, some key things can be learned from this.
Start by loading the Desmos page here. (It will open in a new tab.)
Notice how the individual value varies. It goes back and forth across the center, the value of the true mean of the individuals µ. The green line indicates the mean of the sample values. Since there is only one value at the start, the green line will always be through the individual point.
This is not statistically precise here, but for this demonstration, it is assumed that the distance from the center µ to the rightmost line is σ, the amount of variation from the mean that can be expected from individuals.
Now, go to the left side of the page and start adding values. (Change the c values from 0 to 1.) More dots will appear, each varying at different speeds, but all centered about the same mean µ.
The lines on the left and right indicate the range of values the mean will take most of the time. Notice, the lines move closer together as we increase the sample size. Also notice that the green line tends to stay closer and closer to µ each time you increase the sample size. These lines are placed precisely by dividing σ by the square root of n.
Remember, 
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σx̄ is how much the mean x̄ of a group varies about the true mean of the samples µx̄. σx̄ is found by dividing the amount individuals vary about the mean, σ, by the square root of n.
As n increases, σx̄ decreases. This is illustrated in Desmos: when you add more individuals into the sample, the right and left bars narrow, showing that the sample mean (the green line) tends to vary less.
Notice that even as we add more and more ‘randomly’ varying individuals, the green line x̄ oscillates about the center line µ. Consider why this demonstrates that µx̄ = µ.
