Process Capability Analysis and the Normal Distribution

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Process Capability Analysis and the Normal
Distribution

• BUAD820
• Data Analysis and Quality Management

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Background

• A soda bottle manufacturer wishes to determine
  process capability of the machine that
  manufactures soda bottles. To do this, data on
  an important characteristic of soda bottle, its
  bursting strength, is collected

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Raw (sorted) data
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Normal distribution

• Process capability analysis assumes that process
  data follow the normal distribution
• Thus, we should determine whether or not bursting
  strengths follow the normal distribution
• There are several ways to check this assumption

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Normal distribution check 1 Stem and leaf
display
Comment Majority of the data points are in the
middle, with few in the tails distribution
appears symmetrical like the normal distribution
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Normal distribution check 2 Frequency
distribution
Comment Data appears to follow the normal
distribution
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Normal distribution check 3 Histogram
Comment Data resemble the normal distribution
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Normal distribution check 4 Box and whiskers
plot
Comment Data resemble the normal distribution
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Normal distribution check 5 The empirical rule

• Bursting strength data have a mean of 264 and
  standard deviation of 32
• 71 observations are within 1 std. dev. Of the
  mean (i.e. between 232 and 296)
• 94 observations are within 2 std. dev. Of the
  mean (i.e. between 200 and 328)
• All observations are within 3 std. dev. Of the
  mean (i.e. between 168 and 360)

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Expected versus observed
Range Expected (empirical rule) Observed from data
Within 1 std. dev. 68 71
Within 2 std. dev. 95 94
Within 3 std. dev. gt99 All
Comments The bursting strength data
reasonably satisfy the empirical rule
suggesting that they come from a normal
distribution
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Normal distribution check 6 Normal probability
(QQ) plot

• This is a more formal test where data are
  arranged low to high
• For each observation, a z-score is calculated
• A chart is generated showing data values along
  Y-axis and Z-scores along X-axis
• For data from a normal distribution, the chart
  should resemble a straight line

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Normal probability plot (QQ plot)
Comments A reasonably straight line may be
drawn through these points, suggesting that the
bursting strength data follow normal distribution
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Normal distribution requirements

• In practice, we should be able to make at least a
  reasonable case that bursting strengths follow
  the normal distribution a perfect fit is not
  necessary
• Worst case scenario is that the data distribution
  does not resemble the normal distribution
• Process capability analysis should not be used if
  the data do not reasonably resemble the normal
  distribution