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 ith subgroup 

range of measurements in ith subgroup 

sample size of ith subgroup 
N 
number of subgroups 

weighted average of subgroup means 

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

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

th percentile of the standard normal distribution 

th percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation 
Each point on the chart indicates the value of a subgroup mean (). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the mean plotted for this subgroup is

Each point on the R chart indicates the value of a subgroup range (). For example, the range plotted for the tenth subgroup is .
On an chart, by default, the central line indicates an estimate of , which is computed as

If you specify a known value () for , the central line indicates the value of .
On an R chart, by default, the central line for the ith 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 . Note that the central line varies with .
You can compute the limits in the following ways:
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 17.72: Limits for and R Charts
Control Limits 


Chart 
LCL = lower limit = 
UCL = upper limit = 

R Chart 
LCL = lower limit = 
UCL = upper limit = 

Probability Limits 

Chart 
LCL = lower limit = 
UCL = upper limit = 

R Chart 
LCL = lower limit = 
UCL = upper limit = 
The formulas for R charts assume that the data are normally distributed. If standard values and are available for and , respectively, replace with and with in Table 17.72. Note that the limits vary with and that the probability limits for are asymmetric around the central line.
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 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 a LIMITS= data set.
Specify with the SIGMA0= option or with the variable _STDDEV_
in a LIMITS= data set.