Constructing Charts for Means and Ranges :: SAS/QC(R) 12.3 User's Guide

Constructing Charts for Means and Ranges

The following notation is used in this section:

$\mu$	process mean (expected value of the population of measurements)
$\sigma$	process standard deviation (standard deviation of the population of measurements)
$\bar{X}_{i}$	mean of measurements in ith subgroup
$R_{i}$	range of measurements in ith subgroup
$n_{i}$	sample size of ith subgroup
N	number of subgroups
$\overline{\overline{X}}$	weighted average of subgroup means
$d_{2}(n)$	expected value of the range of n independent normally distributed variables with unit standard deviation
$d_{3}(n)$	standard error of the range of n independent observations from a normal population with unit standard deviation
$z_{p}$	$100 \times p$ th percentile of the standard normal distribution
$D_{p}(n)$	$100 \times p$ 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 $\bar{X}$ chart indicates the value of a subgroup mean ( $\bar{X}_{i}$ ). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the mean plotted for this subgroup is

$\bar{X}_{10}=\frac{12 + 15 + 19 + 16 + 14}{5} = 15.2$

Each point on the R chart indicates the value of a subgroup range ( $R_{i}$ ). For example, the range plotted for the tenth subgroup is $R_{10}=19-12=7$ .

Central Lines

On an $\bar{X}$ chart, by default, the central line indicates an estimate of $\mu$ , which is computed as

$\hat{\mu } = \overline{\overline{X}} = \frac{n_{1}\bar{X}_{1} + \cdots + n_{N}\bar{X}_{N}}{n_{1} + \cdots + n_{N}}$

If you specify a known value ( $\mu _{0}$ ) for $\mu$ , the central line indicates the value of $\mu _{0}$ .

On an R chart, by default, the central line for the ith subgroup indicates an estimate for the expected value of $R_{i}$ , which is computed as $d_{2}(n_{i})\hat{\sigma }$ , where $\hat{\sigma }$ is an estimate of $\sigma$ . If you specify a known value ( $\sigma _{0}$ ) for $\sigma$ , the central line indicates the value of $d_{2}(n_{i})\sigma _{0}$ . Note that the central line varies with $n_{i}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard errors of $\bar{X_{i}}$ and $R_{i}$ above and below the central line. The default limits are computed with k = 3 (these are referred to as $3\sigma$ limits).
as probability limits defined in terms of $\alpha$ , a specified probability that $\bar{X_{i}}$ or $R_{i}$ exceeds the limits

The following table provides the formulas for the limits.

Table 17.72: Limits for $\bar{X}$ and R Charts

Control Limits
$\bar{X}$ Chart	LCL = lower limit = $\overline{\overline{X}} - k\hat{\sigma }/ \sqrt {n_{i}}$
	UCL = upper limit = $\overline{\overline{X}} + k\hat{\sigma }/ \sqrt {n_{i}}$
R Chart	LCL = lower limit = $\mbox{max}(d_{2}(n_{i})\hat{\sigma } - kd_{3}(n_{i})\hat{\sigma },0)$
	UCL = upper limit = $d_{2}(n_{i})\hat{\sigma } + kd_{3}(n_{i})\hat{\sigma }$
Probability Limits
$\bar{X}$ Chart	LCL = lower limit = $\overline{\overline{X}} - z_{\alpha /2}(\hat{\sigma }/ \sqrt {n_{i}})$
	UCL = upper limit = $\overline{\overline{X}} + z_{\alpha /2}(\hat{\sigma }/ \sqrt {n_{i}})$
R Chart	LCL = lower limit = $D_{\alpha /2}\hat{\sigma }$
	UCL = upper limit = $D_{1-\alpha /2}\hat{\sigma }$

The formulas for R charts assume that the data are normally distributed. If standard values $\mu _{0}$ and $\sigma _{0}$ are available for $\mu$ and $\sigma$ , respectively, replace $\overline{\overline{X}}$ with $\mu _{0}$ and $\hat{\sigma }$ with $\sigma _{0}$ in Table 17.72. Note that the limits vary with $n_{i}$ and that the probability limits for $R_{i}$ 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 $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $n_{i} \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $\mu _{0}$ with the MU0= option or with the variable _MEAN_ in a LIMITS= data set.
Specify $\sigma _{0}$ with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.

XRCHART Statement: SHEWHART Procedure

Constructing Charts for Means and Ranges

Plotted Points

Central Lines

Control Limits