The following notation is used in this section:
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process mean (expected value of the population of measurements) |
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process standard deviation (standard deviation of the population of measurements) |
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the ith individual measurement |
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mean of the individual measurements, computed as , where N is the number of individual measurements |
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n |
number of consecutive measurements used to calculate the moving ranges (by default, n = 2) |
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moving range computed for the ith subgroup (corresponding to the ith individual measurement). If , then is assigned a missing value. Otherwise,
This formula assumes that are nonmissing. |
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average of the nonmissing moving ranges, computed as
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expected value of the range of n independent normally distributed variables with unit standard deviation |
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standard error of the range of n independent observations from a normal population with unit standard deviation |
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100pth percentile (0 < p < 1) of the standard normal distribution |
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100pth percentile (0 < p < 1) of the distribution of the range of n independent observations from a normal population with unit standard deviation |
Each point on an individual measurements chart, indicates the value of a measurement ().
Each point on a moving range chart indicates the value of a moving range (). With n = 2, for example, if the first three measurements are 3.4, 3.7, and 3.6, the first moving range is missing, the second moving range is , and the third moving range is .
By default, the central line on an individual measurements chart indicates an estimate for , which is computed as . If you specify a known value () for , the central line indicates the value of .
The central line on a moving range chart indicates an estimate for the expected moving range, computed as where . If you specify a known value () for , the central line indicates the value of .
You can compute the limits
as a specified multiple (k) of the standard errors of and above and below the central line. The default limits are computed with k = 3 (these are referred to as limits).
as probability limits defined in terms of , a specified probability that or exceeds the limits
The following table provides the formulas for the limits:
Table 17.20: Limits for Individual Measurements and Moving Range Charts
Control Limits |
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Individual Measurements Chart |
LCL = lower control limit = |
UCL = upper control limit = |
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Moving Range Chart |
LCL = lower control limit = |
UCL = upper control limit = |
Probability Limits |
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Individual Measurements Chart |
LCL = lower control limit = |
UCL = upper control limit = |
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Moving Range Chart |
LCL = lower control limit = |
UCL = upper control limit = |
The formulas assume that the measurements are normally distributed. Note that the probability limits for the moving range are asymmetric about the central line. If standard values and are available for and , replace with and with in Table 17.20.
You can specify parameters for the limits as follows:
Specify k with the SIGMAS= option or with the variable _SIGMAS_
in a LIMITS= data set.
Specify with the ALPHA= option or with the variable _ALPHA_
in a LIMITS= data set.
Specify n with the LIMITN= option or with the variable _LIMITN_
in a LIMITS= data set.
Specify with the MU0= option or with the variable _MEAN_
in the LIMITS= data set.
Specify with the SIGMA0= option or with the variable _STDDEV_
in the LIMITS= data set.