Measures of Dispersion

• Or how spread out is the data.
• Range = high value - low value
• Example

  Find the range of this data set
  
     5 9 3 12 20 1 3 5 3
  
     High = 20
     Low = 1
     Range = 20-1 = 19
     

• The standard deviation describes how spread out ALL of the data is.
• s² = Σ(x-x)²/(n-1)
• To do this:
  • Calculate x
  • Set up a table with the following columns
    • Data item (x)
    • x-x
    • (x-x)²
  • Write the data items in the x column
  • Calculate the values in the other two columns
  • Find the sum of the third column
  • Divide this by n-1
  • Take the square root.
• Example: Number 18, page 664

  x = (16+17+18+19+20)/5
       = 90/5
       = 18
  
     x     x-x  (x-x)2
    16     -2     4
    17     -1     1
    18      0     0
    19      1     1
    20      2     4
               ------
  	   sum  10   n = 5, n-1 = 4
  
  	   s2 = 10/4   
  	      = 2.5
             s = 1.58 
     

• Example: Number 24, page 664

  x = (6+6+6+6+7+7+7+4+8+3)/10
     =  60/10 
     = 6
  
     x     x-x  (x-x)2
     6      0    0
     6      0    0
     6      0    0
     6      0    0
     7      1    1
     7      1    1
     7      1    1
     4     -2    4
     3     -3    9
     8      2    4
               ----
         sum =   20
         sum/(n-1) = 20/9 = 2.22 
         s = 1.49
         

• Example: (I made this one up)

  Find s for the following data set
     100 101 102 103 104
  
  x = (100+101+102+103+104)/5
     = 510/5
     = 102
     x     x-x  (x-x)2
    100     -2     4
    101     -1     1
    102      0     0
    103      1     1
    104      2     4
             ---------
  	   sum  10   n = 5, n-1 = 4
  
  	   s2 = 10/4   
  	      = 2.5
             s = 1.58 
         

Homework