Is there a built-in function for studentized range or do I have to build it myself?
Is there a test of normality for the studentized range?
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I do not think there's one built-in, but it's fairly easy to write a function for it:
studentizedRange[data_List] := (Max[data] - Min[data])/StandardDeviation[data]
But I would also check out the ANOVA package for related functionality.
Studentizing has been dealt with adequately by Eli. As to your second question: Normality tests can be performed with DistributionFitTest.