(sum from i = 1 to 5) (X(i) - Xbar)^2 ~ 10 chi^2(4)
[chi^2(4) is the chi-squared distribution with 4 degrees of freedom]
From statistical tables,
P(chi^2(4) <= 0.2971) = 0.01
P(chi^2(4) >= 13.28) = 0.99
So the control limits are 10*0.2971 = 2.971 and 10*13.28 = 132.8.
--
If the actual stable variance is v then the probability of a false alarm (ie, the probability of an alarm when the process is stable) is
P(v*chi^2(4) <= 2.971 or v*chi^2(4) >= 132.8)
= F(2.971/v) + 1 - F(132.8/v)
where F is the CDF of chi^2(4). The attached graph shows that
- if v is more than 10 then the probability of a false alarm is more than 0.02,
- if v is less than 10, but more than about 7, then the probability of a false alarm is less than 0.02.