Standard Deviation

• Statistical simulation of the standard deviation with R-Project
  

   x1 <- c(rep(15,2),rep(30,4),rep(40,10),rep(50,10),rep(60,3),rep(75,1));
   x2 <- (x1-40)/5;
  ni<- table(x1);
  
  n<-sum(ni)
  
  s1 <-sd(x1)*sqrt((n-1)/n)
  s1
  [1] 12.52664
  
  s2 <- sd(x2)*sqrt((n-1)/n)
  s2
  [1] 2.505328
  
  s2*5
  [1] 12.52664
  

  
  
  
  

• Statistical simulation of the standard deviation with Mathematica
  

  x1 := Join[ConstantArray[15, 2], ConstantArray[30, 4], 
     ConstantArray[40, 10], ConstantArray[50, 10], ConstantArray[60, 3], 
     ConstantArray[75, 1]];
  x2 := (x1-40)/5;
     
  ni := Tally [x1];
  
  n := Sum[ni[[i, 2]], {i, Length[ni]}];
  
  s1:=StandardDeviation[x1]  Sqrt[(n - 1)/n];
  s2:=StandardDeviation[x2]  Sqrt[(n - 1)/n];
  
  N[s1]
  N[s2]
  N[s2 5]
  
  
  12.5266
  2.50533
  12.5266