XRCHART Statement: SHEWHART Procedure

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

Table 18.60 provides the formulas for the limits.

Table 18.60: 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 18.60. 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.