Constructing Charts for Means and Ranges

XRCHART Statement

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 i^th subgroup

R_i range of measurements in i^th subgroup

n_i sample size of i^th subgroup

N number of subgroups

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

d₂(n) expected value of the range of n independent normally distributed variables with unit standard deviation

d₃(n) standard error of the range of n independent observations from a normal population with unit standard deviation

z_p 100p^th percentile of the standard normal distribution

D_p(n) 100p^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 R₁₀=19-12=7.

Central Lines

On an $\bar{X}$ chart, by default, the central line indicates an estimate of $\mu$ , which is computed as

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

If you specify a known value ( $\mu_{0}$ ) for $\mu$ ,the central line indicates the value of $\mu_{0}$ .

On an R chart, by default, the central line for the i^th 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

The following table provides the formulas for the limits:

Table 43.22: 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 $= {max}(d_{2}(n_{i})\hat{\sigma} - kd_{3}(n_{i})\hat{\sigma},0)$
	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 43.22. 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.

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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
R_i	range of measurements in i^th subgroup
n_i	sample size of i^th subgroup
N	number of subgroups
$\overline{\overline{X}}$	weighted average of subgroup means
d₂(n)	expected value of the range of n independent normally distributed variables with unit standard deviation
d₃(n)	standard error of the range of n independent observations from a normal population with unit standard deviation
z_p	100p^th percentile of the standard normal distribution
D_p(n)	100p^th percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation

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 $= D_{1-\alpha/2}\hat{\sigma}$