column d3 in Statistical Process Control
(sqrt of variance of the Range in a sample of size n 
taken from a normal distribution).

cfr http://www.ct-yankee.com/spc/
cfr http://en.wikipedia.org/wiki/User:NorwegianBlue/refdesk/statistics
(15.2 days of calculation on moderate workstation)
Accuracy is seriously degraded to 14 digits, even down to 11 digits).

\[Alpha][xn_]:= 1/2*(Erf[xn/Sqrt[2]] + 1)

Table[Sqrt[2 NIntegrate[1-\[Alpha][x1]^k-(1-\[Alpha][xn])^k+(\[Alpha][x1]-\[Alpha][xn])^k,{x1,-10,10},{xn,-10,x1},
AccuracyGoal->20,PrecisionGoal->20,WorkingPrecision->40]
-NIntegrate[1-(1-\[Alpha][x])^k-\[Alpha][x]^k,{x,-10,10},AccuracyGoal->20,PrecisionGoal->20,WorkingPrecision->40]^2],{k,2,25}]


{  0.85250246642742 , 
    0.88836800404520 , 
    0.8798082028250 , 
    0.8640819410995 , 
    0.8480396861175 , 
    0.8332053356223 , 
    0.819831489792 , 
    0.8078342745533 , 
    0.797050673519 , 
    0.787314620550 , 
    0.778478341203 , 
    0.770416202064 , 
    0.763023095625 , 
    0.756211429728 , 
    0.749908089410 , 
    0.74405178396 , 
    0.73859085338 , 
    0.73348149552 , 
    0.72868634571 , 
    0.72417334072 , 
    0.71991480843 , 
    0.71588673549 , 
    0.71206817515 ,  
    0.70844076589  }

column d2 in Statistical Process Control
(mean of the Range in a sample of size n 
taken from a normal distribution).
This is much faster (200 sec)

Table[NIntegrate[1-(1-\[Alpha][x])^k-\[Alpha][x]^k,{x,-10,10},
AccuracyGoal->20,PrecisionGoal->20,WorkingPrecision->40],{k,2,25}]

{1.12837916709551257389616, 
1.69256875064326886084423, 
2.058750746007928264114, 
2.325928947281039225545, 
2.53441272122294259530, 
2.7043567512138087985, 
2.8472006120905555062, 
2.9700263244184740126, 
3.07750546167034571206, 
3.1728727038160003379, 
3.2584552797438259806, 
3.3359803540982549961, 
3.4067631081999530429870, 
3.4718268898820748673343, 
3.5319827861095759345629, 
3.5878839617653817470903, 
3.6400637579374442093, 
3.688963023207649316212, 
3.734950119596640969474, 
3.778335829842620968883, 
3.819384643362832652246, 
3.858323423285006873245, 
3.895348148451356269514, 
3.930629219507113161515}

Wouter
2010/02/20