Constructing Charts for Medians and Ranges :: SAS/QC(R) 12.1 User's Guide

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Constructing Charts for Medians 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

the number of subgroups

$x_{ij}$

jth measurement in the ith subgroup, $j=1,2,3, \ldots , n_{i}$

$x_{i(j)}$

jth largest measurement in the ith subgroup. Then

$x_{i(1)} \leq x_{i(2)} \leq \ldots \leq x_{i(n_{i})}$

$\overline{\overline{X}}$

weighted average of subgroup means

$M_{i}$

median of the measurements in the ith subgroup:

$M_ i = \left\{ \begin{array}{ll} x_{i((n_ i + 1)/2)} & \mbox{if $n_{i}$ is odd} \\ (x_{i(n_ i/2)} + x_{i((n_ i/2)+1)})/2 & \mbox{if $n_{i}$ is even} \end{array} \right.$

$\bar{M}$

average of the subgroup medians:

$\bar{M} = (n_1M_1 + \ldots + n_ NM_ N)/(n_1+\ldots +n_ N)$

$\tilde{M}$

median of the subgroup medians. Denote the jth largest median by $M_{(j)}$ so that $M_{(1)} \leq M_{(2)}\leq \ldots \leq M_{(N)}$ .

$\tilde{M} = \left\{ \begin{array}{ll} M_{((N+1)/2)} & \mbox{if \Mathtext{N} is odd} \\ (M_{(N/2)} + M_{(N/2)+1})/2 & \mbox{if \Mathtext{N} is even} \end{array} \right.$

standard error of the median of n independent, normally distributed variables with unit standard deviation (the value of can be calculated with the STDMED function in a DATA step)

$Q_{p}(n)$

100pth percentile (0 < p < 1) of the distribution of the median of n independent observations from a normal population with unit standard deviation

$d_{2}(n)$

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

$z_{p}$

100pth percentile of the standard normal distribution

$D_{p}(n)$

100pth percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation

Plotted Points

Each point on a median chart indicates the value of a subgroup median ( $M_{i}$ ). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is $M_{10} = 15$ . Each point on a range chart indicates the value of a subgroup range ( $R_{i}$ ). For example, the value plotted for the tenth subgroup is $R_{10}=19-12=7$ .

Central Lines

On a median chart, the value of the central line indicates an estimate for $\mu$ , which is computed as

$\bar{M}$ by default
$\overline{\overline{X}}$ when you specify MEDCENTRAL=AVGMEAN
$\tilde{M}$ when you specify MEDCENTRAL=MEDMED
$\mu _{0}$ when you specify $\mu _0$ with the MU0= option

On the range 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 $d_{2}(n_{i})\sigma _{0}$ . The central line on the range chart varies with $n_{i}$ .

Control Limits

You can compute the limits

as a specified multiple (k) of the standard errors of $M_{i}$ 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 $M_{i}$ or $R_{i}$ exceeds its limits

The following table provides the formulas for the limits:

Table 17.32: Limits for Median and Range Charts

Control Limits
Median Chart	LCL = lower limit = $\bar{M} - k\hat{\sigma }e_{M}(n_ i)$
	UCL = upper limit = $\bar{M} + k\hat{\sigma }e_{M}(n_ i)$
Range Chart	LCL = lower control limit = $\mbox{max}(d_{2}(n_{i})\hat{\sigma } - kd_{3}(n_{i})\hat{\sigma },0)$
	UCL = upper control limit = $d_{2}(n_{i})\hat{\sigma } + kd_{3}(n_{i})\hat{\sigma }$
Probability Limits
Median Chart	LCL = lower limit = $\bar{M} - Q_{\alpha /2}(n_ i)\hat{\sigma }$
	UCL = upper limit = $\bar{M} + Q_{1-\alpha /2}(n_ i)\hat{\sigma }$
Range Chart	LCL = lower limit = $D_{\alpha /2}\hat{\sigma }$
	UCL = upper limit = $D_{1-\alpha /2}\hat{\sigma }$

In Table 17.32, replace $\bar{M}$ with $\overline{\overline{X}}$ if you specify MEDCENTRAL=AVGMEAN, and replace $\bar{M}$ with $\tilde{M}$ if you specify MEDCENTRAL=MEDMED. Replace $\bar{M}$ with $\mu _0$ if you specify $\mu _0$ with the MU0= option, and replace $\hat{\sigma }$ with $\sigma _0$ if you specify $\sigma _0$ with the SIGMA0= option.

The formulas assume that the data are normally distributed. Note that the limits for both charts 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 the LIMITS= data set.
Specify $\sigma _{0}$ with the SIGMA0= option or with the variable _STDDEV_ in the LIMITS= data set.

MRCHART Statement: SHEWHART Procedure

Constructing Charts for Medians and Ranges

Plotted Points

Central Lines

Control Limits