Determining Normality

1
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

Normal distribution
Number of readings
m
-s
-s
Value
How well does the actual data fit the normal
curve?
Number of readings
Actual distribution
m
Value
3
?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

4
?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 where x
  is the top of the 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

5
?2 Goodness of Fit test

• d) calculate the number expected (ne) as the
  probability above times the total n. ( ntotalpr)
• Sum the value of for each
  group range
• 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