Constructing Charts for Individual Measurements and Moving 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)

the th individual measurement

mean of the individual measurements, computed as , where is the number of individual measurements

number of consecutive measurements used to calculate the moving ranges (by default, )

moving range computed for the th subgroup (corresponding to the th individual measurement). If , then is assigned a missing value. Otherwise,

     

This formula assumes that are nonmissing.

average of the nonmissing moving ranges, computed as

     

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 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 , 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 .

Central Lines

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 .

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 the limits

The following table provides the formulas for the limits:

Table 15.20 Limits for Individual Measurements and Moving Range Charts

Control Limits

Individual Measurements Chart

LCL lower control limit =

 

UCL upper control limit =

Moving Range Chart

LCL lower control limit =

 

UCL upper control limit =

Probability Limits

Individual Measurements Chart

LCL lower control limit =

 

UCL upper control limit =

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 15.20.

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 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.