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