Constructing Charts for Means :: SAS/QC(R) 13.1 User's Guide

Constructing Charts for Means

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
$z_{p}$	100pth percentile of the standard normal distribution

Plotted Points

Each point on an $\bar{X}$ 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 value plotted for this subgroup is

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

Central Line

By default, the central line on an $\bar{X}$ chart indicates an estimate for $\mu$ , 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 $\mu _{0}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard error of $\bar{X_{i}}$ 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 $\bar{X_{i}}$ exceeds the limits

The following table provides the formulas for the limits:

Table 17.66: Limits for $\bar{X}$ Charts

Control Limits
LCL = lower limit = $\overline{\overline{X}} - k\hat{\sigma }/ \sqrt {n_{i}}$
UCL = upper limit = $\overline{\overline{X}} + k\hat{\sigma }/ \sqrt {n_{i}}$

Probability Limits
LCL = lower limit = $\overline{\overline{X}} - z_{\alpha /2}(\hat{\sigma }/ \sqrt {n_{i}})$
UCL = upper limit = $\overline{\overline{X}} + z_{\alpha /2}(\hat{\sigma }/ \sqrt {n_{i}})$

Note that the limits vary with $n_ i$ . 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 17.66.

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.

XCHART Statement: SHEWHART Procedure

Constructing Charts for Means

Plotted Points

Central Line

Control Limits