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 XRCHART Statement

## Constructing Charts for Means 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 i th subgroup Ri range of measurements in i th subgroup ni sample size of i th subgroup N number of subgroups weighted average of subgroup means d2(n) expected value of the range of n independent normally distributed variables with unit standard deviation d3(n) standard error of the range of n independent observations from a normal population with unit standard deviation zp 100p th percentile of the standard normal distribution Dp(n) 100p th percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation

### Plotted Points

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 (Ri). For example, the range plotted for the tenth subgroup is R10=19-12=7.

### Central Lines

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 i th subgroup indicates an estimate for the expected value of Ri, 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 ni.

### Control Limits

You can compute the limits in the following ways:

• as a specified multiple (k) of the standard errors of and Ri 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 Ri exceeds the limits

The following table provides the formulas for the limits:

Table 43.22: 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 43.22. Note that the limits vary with ni and that the probability limits for Ri 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.

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