Constructing Charts for Means and Standard Deviations

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
$s_{i}$	standard deviation of the measurements $x_{i1},\ldots ,x_{in{i}}$ in the ith subgroup $s_{i} = \sqrt {( (x_{i1} - \bar{X_{i}})^2 + \cdots + (x_{in_{i}} - \bar{X_{i}})^2) / (n_ i-1)}$
$n_{i}$	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
$c_{4}(n)$	expected value of the standard deviation of n independent normally distributed variables with unit standard deviation
$c_{5}(n)$	standard error of the standard deviation of n independent observations from a normal population with unit standard deviation
$\chi ^{2}_{p}(n)$	100pth percentile $(0<p<1)$ of the $\chi ^{2}$ distribution with n degrees of freedom

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 13, the mean plotted for this subgroup is

$\bar{X}_{10}=\frac{12 + 15 + 19 + 16 + 13}{5} = 15$

Each point on an s chart indicates the value of a subgroup standard deviation ( $s_{i}$ ). For example, the standard deviation plotted for the tenth subgroup is

$s_{10}= \sqrt {((12-15)^2 + (15-15)^2 + (19-15)^2 + (16-15)^2 + (13-15)^2)/4 } = 2.739$

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 the s chart, by default, the central line for the ith subgroup indicates an estimate for the expected value of $s_ i$ , which is computed as $c_{4}(n_{i})\hat{\sigma }$ , 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 $c_{4}(n_{i})\sigma _{0}$ . 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 $s_{i}$ 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 $s_{i}$ exceeds the limits

The following table provides the formulas for the limits:

Table 17.78: Limits for $\bar{X}$ and s Charts

Control Limits
$\bar{X}$ Chart	LCL = lower limit = $\overline{\overline{X}} - k\hat{\sigma }/ \sqrt {n_{i}}$
	UCL = upper limit = $\overline{\overline{X}} + k\hat{\sigma }/ \sqrt {n_{i}}$
s Chart	LCL = lower limit = $\mbox{max}(c_{4}(n_{i})\hat{\sigma } - kc_{5}(n_{i})\hat{\sigma },0)$
	UCL = upper limit = $c_{4}(n_{i})\hat{\sigma } + kc_{5}(n_{i})\hat{\sigma }$

Probability Limits
$\bar{X}$ Chart	LCL = lower limit = $\overline{\overline{X}} - z_{\alpha /2}(\hat{\sigma }/ \sqrt {n_{i}})$
	UCL = upper limit = $\overline{\overline{X}} + z_{\alpha /2}(\hat{\sigma }/ \sqrt {n_{i}})$
s Chart	LCL = lower limit = $\hat{\sigma }\sqrt {\chi ^{2}_{\alpha /2}(n_ i - 1)/(n_ i-1)}$
	UCL = upper limit = $\hat{\sigma }\sqrt {\chi ^{2}_{1-\alpha /2}(n_ i - 1)/(n_ i-1)}$

The formulas for s 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 17.78. Note that the limits vary with $n_{i}$ and that the probability limits for $s_{i}$ 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 $\alpha$ 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.

XSCHART Statement: SHEWHART Procedure

Constructing Charts for Means and Standard Deviations

Plotted Points

Central Lines

Control Limits