# function for the average or expected range?; CORECTION

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## function for the average or expected range?; CORECTION

 Hi, All: ** My previous email on this subject seemed to contain an error;  check the correction:       Is there a function in R to compute the expected range of a sample of size n from some distribution?  I ask, because I was recently asked about the control chart constant 'd2', which is the expected range for a sample of size n from a standard normal.       There is a fairly simple formula for the expected value of the range, given, e.g., in Kendall and Stuart (1969) The Advanced Theory of Statistics, vol. 1, Distribution Theory, 3rd ed. (Hafner, expression (14.82), sec. 14.25:  The exact distribution of the range, p. 339). Unfortunately, either I don't understand this formula, or it's wrong. Using expression (14.1) in the same reference, I get the following:      E(R) = n*integral{-Inf to Inf of [(F(x))**(n-1) - (1-F(x))**(n-1)]dF(x).       Thanks,       Spencer Graves ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: function for the average or expected range?; CORECTION

 The "ptukey" and "qtukey" functions may be what you want (or at least in the right direction). You could also easily estimate this by simulation. Hope this helps, -- Gregory (Greg) L. Snow Ph.D. Statistical Data Center Intermountain Healthcare [hidden email] (801) 408-8111     > -----Original Message----- > From: [hidden email] > [mailto:[hidden email]] On Behalf Of Spencer Graves > Sent: Friday, March 21, 2008 11:13 AM > To: [hidden email] > Subject: [R] function for the average or expected range?; CORECTION > > Hi, All: > > ** My previous email on this subject seemed to contain an > error;  check the correction: > >       Is there a function in R to compute the expected range > of a sample of size n from some distribution?  I ask, because > I was recently asked about the control chart constant 'd2', > which is the expected range for a sample of size n from a > standard normal. > >       There is a fairly simple formula for the expected value > of the range, given, e.g., in Kendall and Stuart (1969) The > Advanced Theory of Statistics, vol. 1, Distribution Theory, > 3rd ed. (Hafner, expression (14.82), sec. 14.25:  The exact > distribution of the range, p. 339). > Unfortunately, either I don't understand this formula, or it's wrong. > Using expression (14.1) in the same reference, I get the following: > >      E(R) = n*integral{-Inf to Inf of [(F(x))**(n-1) - > (1-F(x))**(n-1)]dF(x). > >       Thanks, >       Spencer Graves > > ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-help> PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html> and provide commented, minimal, self-contained, reproducible code. > ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: function for the average or expected range?; CORECTION

 Hi, Greg:       Thanks very much for the reply.       1.  The 'ptukey' and 'qtukey' function are the distribution of the studentized range, not the range.  I tried "sum(ptukey(x, 2, df=Inf, lower=FALSE))*.1" and got 1.179 vs. 1.128 in the standard table of d2 for n = 2 observations per subgroup.       2.  I tried simulation and found that I needed 1e7 or 1e8 random normal deviates to get the accuracy of the published table.       3.  Then I programmed in Excel the integral over seq(-5, 5, .1) using a correction to the formula I got from Kendall and Stuart and got the exact numbers in the published table except in one case where it was off by 1 in the last significant digit.       Thanks again,       Spencer Greg Snow wrote: > The "ptukey" and "qtukey" functions may be what you want (or at least in > the right direction). > > You could also easily estimate this by simulation. > > Hope this helps, > > ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: function for the average or expected range?; CORECTION

 Why do the integration in Excel instead of using the integrate function in R?  The R function will allow integration from -Inf to Inf.   What was the correction to the formula?  The last one you showed  looked like the difference between the average min and average max, but did not take into account the correlation between the max and min (going from memory, don't have my theory books handy). For large n the correlation is probably small enough that it makes a good approximation. ________________________________ From: Spencer Graves [mailto:[hidden email]] Sent: Fri 3/21/2008 3:39 PM To: Greg Snow Cc: [hidden email] Subject: Re: [R] function for the average or expected range?; CORECTION Hi, Greg:       Thanks very much for the reply.       1.  The 'ptukey' and 'qtukey' function are the distribution of the studentized range, not the range.  I tried "sum(ptukey(x, 2, df=Inf, lower=FALSE))*.1" and got 1.179 vs. 1.128 in the standard table of d2 for n = 2 observations per subgroup.       2.  I tried simulation and found that I needed 1e7 or 1e8 random normal deviates to get the accuracy of the published table.       3.  Then I programmed in Excel the integral over seq(-5, 5, .1) using a correction to the formula I got from Kendall and Stuart and got the exact numbers in the published table except in one case where it was off by 1 in the last significant digit.       Thanks again,       Spencer Greg Snow wrote: > The "ptukey" and "qtukey" functions may be what you want (or at least in > the right direction). > > You could also easily estimate this by simulation. > > Hope this helps, > >           [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.