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Where:
= the mean of the X-bars T = specification target Sigma is an estimated sigma or sigma of the individual values, depending on how the histogram is configured or whether you are calculating Cpk (estimated) or Ppk (actual).
Actual % out-of-spec = % above upper spec + % below lower spec
To find actual % above spec:
To find actual % below spec:
Related topic: The formula for theoretical percent out of spec
To find theoretical % above spec:
Locate the value for Zupper in a standard normal table.
To find theoretical % below spec:
Locate the value for Zlower in a standard normal table.
Given:
And assuming:
In theory:
If the Z value from the standard normal table is multiplied by 100, then this process is expected to produce X% of its output outside specification.
Show me: The standard normal table The formulas for Target Z
Related topic: The formula for actual percent out of spec
Alpha value = 5%
Sigma of the individuals (actual sigma)
Estimated sigma
X-bar control limits are based on either range or sigma, depending on which chart it is paired with. When the X-bar chart is paired with a range chart, the most common (and recommended) method of computing control limits based on 3 standard deviations is:
Upper control limit
Lower control limit:
k is the number of subgroups
Lower control limit