Constructing Charts for Standard Deviations

SCHART Statement

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 i^th subgroup
$s_{i} = \sqrt{(1/(n_i-1))( (x_{i1} - \bar{X_{i}})^2 + ... + (x_{in_{i}} - \bar{X_{i}})^2)}$

n_i sample size of i^th subgroup

c₄(n) expected value of the standard deviation of n independent normally distributed variables with unit standard deviation

c₅(n) standard error of the standard deviation of n independent observations from a normal population with unit standard deviation

$\chi^2_{p}(n)$ 100p^th 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 i^th 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 40.21: Limits for s Charts

Control Limits

LCL = lower limit $= {max}(c_{4}(n_{i})\hat{\sigma} - kc_{5}(n_{i})\hat{\sigma},0)$

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 40.21. 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.

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$\sigma$	process standard deviation (standard deviation of the population of measurements)
s_i	standard deviation of measurements in i^th subgroup $s_{i} = \sqrt{(1/(n_i-1))( (x_{i1} - \bar{X_{i}})^2 + ... + (x_{in_{i}} - \bar{X_{i}})^2)}$
n_i	sample size of i^th subgroup
c₄(n)	expected value of the standard deviation of n independent normally distributed variables with unit standard deviation
c₅(n)	standard error of the standard deviation of n independent observations from a normal population with unit standard deviation
$\chi^2_{p}(n)$	100p^th percentile (0<p<1) of the $\chi^2$ distribution with n degrees of freedom