SCHART Statement: SHEWHART Procedure

Constructing Charts for Standard Deviations

The following notation is used in this section:

$\sigma $

Process standard deviation (standard deviation of the population of measurements)

$s_{i}$

Standard deviation of measurements in ith subgroup

\[ s_{i} = \sqrt {(1/(n_ i-1))( (x_{i1} - \bar{X_{i}})^2 + \cdots + (x_{in_{i}} - \bar{X_{i}})^2)} \]

$n_{i}$

Sample size of ith subgroup

$c_{4}(n)$

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 s chart indicates the value of a subgroup standard deviation ($s_{i}$). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 13, the value plotted for this subgroup is

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

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 error of $s_{i}$ 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 $s_{i}$ exceeds the limits

The following table provides the formulas for the limits:

Table 18.46: Limits for s Charts

Control Limits

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

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 assume that the data are normally distributed. If a standard value $\sigma _{0}$ is available for $\sigma $, replace $\hat{\sigma }$ with $\sigma _{0}$ in Table 18.46. Note that the upper and lower limits vary with $n_{i}$ 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 $\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 $\sigma _{0}$ with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.