Determining Normality by yurtgc548

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									                     Determining Normality
• Issues
     – Standard deviation is commonly used to express the
       magnitude of imprecision. (Eg. ±s or ±3s limits)
     – Standard deviation implies that the error distribution is
       normal
     – 68.3% of readings would be included in ±s if the data are
       normally distributed or 99.7% of readings for ±3s limits
     – This is not true if the distribution is not normal.
• Methods for determining normality
     – 1. Qualitative plotting. Plot on probability paper. Determine
       if data fall on a straight line
     – 2. Use 2 goodness of fit test.
     – 3. Use Kolmogorov-Smirnov one-sample test
          (see Steel and Torrie)

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                                     2 Goodness of Fit test
• Concept:
          Number of readings
                                                   Normal distribution




                                     -s    m        -s        Value


                                                                  How well does the actual
                Number of readings




                                                                  data fit the normal curve?

                                                                   Actual distribution



                                           m              Value




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                   2 Goodness of Fit test
• Concept -cont-
     – Sum the square of deviations of the actual data curve from
       the normal curve at n points and divide by the sum by n
     – This result has a 2 distribution
     – Test to see if this result is higher or lower than expected
• Method               n groups
                                  no  ne 2
                2     i 1         ne
                no  number _ observed _ per _ group
                ne  number _ expected _ per _ group




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                       2 Goodness of Fit test
• Method -cont-
     –    1. Sort data by value
     –    2. Divide the data into groups (see table 3.1 Doeblin)
     –    3. Count the number observed (no) in each group
     –    4. Use a table of cumulative standard normal distribution to
          find number expected (ne)
            • a) Find x, s for total sample population
            • b) compute w            x  m where x is the top of the
                                 w
              range                     
           • c) Determine for each range, the probability of data falling in
             the range, ( pr)
               – This is the cumulative probability at the start of the range
                 minus the end of the range

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                      2 Goodness of Fit test
        • d) calculate the number expected (ne) as the probability
          above times the total n. ( ntotalpr)
     – Sum the value of : n  n 2 for each group range
                                    o        e
                                        ne


     – Use a chi square table to determine if the value falls in the
       shaded area. If it does reject that the data are not from a
       normal distribution
          • Degrees of freedom = number of groups minus 3
              (mean, std dev, -1)




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                       2 Goodness of Fit test



          Chi Square




                          5     10        15
                       Degrees of Freedom


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