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 SCHART Statement

## Constructing Charts for Standard Deviations

The following notation is used in this section:
 process standard deviation (standard deviation of the population of measurements) si standard deviation of measurements in i th subgroup ni sample size of i th subgroup 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 s chart indicates the value of a subgroup standard deviation (si). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 13, the value plotted for this subgroup is

### Central Line

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 error of 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 si exceeds the limits

The following table provides the formulas for the limits:

Table 40.21: Limits for s Charts
 Control Limits LCL = lower limit UCL = upper limit =

 Probability Limits LCL = lower limit UCL = upper limit

The formulas assume that the data are normally distributed. If a standard value is available for ,replace with in Table 40.21. Note that the upper and lower limits vary with ni and that the probability limits 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 SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.

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