# 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:
Normal distribution

-s    m        -s        Value

How well does the actual

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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