Constructing Charts for Means and Standard Deviations

XSCHART Statement

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 i^th subgroup

s_i standard deviation of the measurements x_i1, ... ,x_ini in the i^th subgroup
$s_{i} = \sqrt{( (x_{i1} - \bar{X_{i}})^2 + ... + (x_{in_{i}} - \bar{X_{i}})^2) / (n_i-1)}$

n_i sample size of i^th subgroup

N number of subgroups

$\overline{\overline{X}}$ weighted average of subgroup means

z_p 100p^th percentile of the standard normal distribution

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 $\bar{X}$ 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

$\hat{\mu} = \overline{\overline{X}} = \frac{n_{1}\bar{X}_{1} + ... + n_{N}\bar{X}_{N}} {n_{1} + ... + n_{N}}$

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 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 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 44.22: 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 $= {max}(c_{4}(n_{i})\hat{\sigma} - kc_{5}(n_{i})\hat{\sigma},0)$
	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 44.22. 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.

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$\mu$	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 i^th subgroup
s_i	standard deviation of the measurements x_i1, ... ,x_ini in the i^th subgroup $s_{i} = \sqrt{( (x_{i1} - \bar{X_{i}})^2 + ... + (x_{in_{i}} - \bar{X_{i}})^2) / (n_i-1)}$
n_i	sample size of i^th subgroup
N	number of subgroups
$\overline{\overline{X}}$	weighted average of subgroup means
z_p	100p^th percentile of the standard normal distribution
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

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 $= \hat{\sigma}\sqrt{\chi^2_{1-\alpha/2}(n_i - 1)/(n_i-1)}$