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 XSCHART Statement

## Constructing Charts for Means and Standard Deviations

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 si standard deviation of the measurements xi1, ... ,xini in the i th subgroup ni sample size of i th subgroup N number of subgroups weighted average of subgroup means zp 100p th percentile of the standard normal distribution c4(n) expected value of the standard deviation of n independent normally distributed variables with unit standard deviation c5(n) standard error of the standard deviation of n independent observations from a normal population with unit standard deviation 100p th percentile (0

### Plotted Points

Each point on an chart indicates the value of a subgroup mean (). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 13, the mean plotted for this subgroup is
Each point on an s chart indicates the value of a subgroup standard deviation (si). For example, the standard deviation plotted for the tenth subgroup is

### 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 the s chart, by default, the central line for the i th subgroup indicates an estimate for the expected value of si, 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 si 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 si exceeds the limits

The following table provides the formulas for the limits:

Table 44.22: Limits for and s Charts
 Control Limits Chart LCL = lower limit UCL = upper limit s Chart LCL = lower limit UCL = upper limit =

 Probability Limits Chart LCL = lower limit UCL = upper limit s Chart LCL = lower limit UCL = upper limit

The formulas for s charts assume that the data are normally distributed. If standard values and are available for and , respectively, replace with and with in Table 44.22. Note that the limits vary with ni and that the probability limits for si are asymmetric about 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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