Let's say we have a very small population of {2, 4, 9}. We could imagine those are ages of three children. So first, the population stats:

μ = (2+4+9)/3 = 5

σ^2 =

((2-5)^2 + (4-5)^2 + (9-5)^2) / 3

= ((-3)^2 + (-1)^2 + 4^2) / 3

= (9 + 1 + 4) / 3

= 8.67

This part is clear. But now, let's say I take a sample of n=2 from that population. There are only 3 possible samples:

- {2, 4}
- {2, 9}
- {4, 9}

- {2, 4}; mean = 3
- {2, 9}; mean = 5.5
- {4, 9}; mean = 6.5

This is where things stop making sense. If I apply my "normal" approach to variance among these sample means, I get this:

Var(X¯) =

((3-5)^2 + (5.5-5)^2 + (6.5-5)^2) / 3

= ((-2)^2 + (0.5)^2 + (1.5)^2) / 3

= ( 4 + 0.25 + 2.25) / 3

= 2.17

BUT if I use the formula σ^2/n, the result is = 8.67/2 = 4.33. That's twice the result I got when applying my "normal" approach to variance. What is wrong in my math?

Any guidance would be VERY appreciated!