Rolling a fair four-sided die and plotting the running average after each roll: it wanders at first, then settles onto the expectation 2.5 — the population mean. The shaded band is plus or minus one standard error of the sample mean, not a bound that every run must stay inside.
A fair four-sided die. Its long-run average was decided when the die was made — the rolls only reveal it.
How often each face has come up — they even out as n grows: