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The SHEWHART Procedure

Constructing Charts for Medians and Ranges

The following notation is used in this section:

process mean (expected value of the population of measurements)

process standard deviation (standard deviation of the population of measurements)

mean of measurements in th subgroup

range of measurements in th subgroup

sample size of th subgroup

the number of subgroups

th measurement in the th subgroup,

th largest measurement in the th subgroup. Then

     

weighted average of subgroup means

median of the measurements in the th subgroup:

     

average of the subgroup medians:

     

median of the subgroup medians. Denote the th largest median by so that .

     

standard error of the median of independent, normally distributed variables with unit standard deviation (the value of can be calculated with the STDMED function in a DATA step)

100th percentile of the distribution of the median of independent observations from a normal population with unit standard deviation

expected value of the range of independent normally distributed variables with unit standard deviation

standard error of the range of independent observations from a normal population with unit standard deviation

100th percentile of the standard normal distribution

100th percentile of the distribution of the range of independent observations from a normal population with unit standard deviation

Plotted Points

Each point on a median chart indicates the value of a subgroup median (). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is . Each point on a range chart indicates the value of a subgroup range (). For example, the value plotted for the tenth subgroup is .

Central Lines

On a median chart, the value of the central line indicates an estimate for , which is computed as

  • by default

  • when you specify MEDCENTRAL=AVGMEAN

  • when you specify MEDCENTRAL=MEDMED

  • when you specify with the MU0= option

On the range chart, by default, the central line for the th subgroup indicates an estimate for the expected value of , which is computed as , where is an estimate of . If you specify a known value () for , the central line indicates the value of . The central line on the range chart varies with .

Control Limits

You can compute the limits

  • as a specified multiple () of the standard errors of and above and below the central line. The default limits are computed with (these are referred to as limits).

  • as probability limits defined in terms of , a specified probability that or exceeds its limits

The following table provides the formulas for the limits:

Table 13.32 Limits for Median and Range Charts

Control Limits

Median Chart

LCL lower limit =

 

UCL upper limit =

Range Chart

LCL lower control limit =

 

UCL upper control limit =

Probability Limits

Median Chart

LCL lower limit =

 

UCL upper limit =

Range Chart

LCL lower limit =

 

UCL upper limit =

In Table 13.32, replace with if you specify MEDCENTRAL=AVGMEAN, and replace with if you specify MEDCENTRAL=MEDMED. Replace with if you specify with the MU0= option, and replace with if you specify with the SIGMA0= option.

The formulas assume that the data are normally distributed. Note that the limits for both charts vary with and that the probability limits for are asymmetric around the central line.

You can specify parameters for the limits as follows:

  • Specify 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 a constant nominal sample size for the control limits 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.

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