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 MRCHART Statement

## Constructing Charts for Medians and Ranges

The following notation is used in this section:
 process mean (expected value of the population of measurements) process standard deviation (standard deviation of the population of measurements) mean of measurements in i th subgroup Ri range of measurements in i th subgroup ni sample size of i th subgroup N the number of subgroups xij j th measurement in the i th subgroup, j = 1,2,3, ... , ni xi(j) j th largest measurement in the i th subgroup. Then weighted average of subgroup means Mi median of the measurements in the i th subgroup: average of the subgroup medians: median of the subgroup medians. Denote the j th largest median by M(j) so that . eM(n) standard error of the median of n independent, normally distributed variables with unit standard deviation (the value of eM(n) can be calculated with the STDMED function in a DATA step) Qp(n) 100p th percentile (0

### Plotted Points

Each point on a median chart indicates the value of a subgroup median (Mi). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is M10 = 15. Each point on a range chart indicates the value of a subgroup range (Ri). For example, the value plotted for the tenth subgroup is R10=19-12=7.

### Central Lines

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

• by default
• when you specify MEDCENTRAL=AVGMEAN
• when you specify MEDCENTRAL=MEDMED
• when you specify with the MU0= option

On the range chart, by default, the central line for the i th subgroup indicates an estimate for the expected value of Ri, which is computed as ,where is an estimate of . If you specify a known value () for , the central line indicates the value of .The central line on the range chart varies with ni.

### Control Limits

You can compute the limits

• as a specified multiple (k) of the standard errors of Mi and Ri above and below the central line. The default limits are computed with k=3 (these are referred to as limits).
• as probability limits defined in terms of , a specified probability that Mi or Ri exceeds its limits

The following table provides the formulas for the limits:

Table 36.22: Limits for Median and Range Charts
 Control Limits Median Chart LCL = lower limit UCL = upper limit Range Chart LCL = lower control limit UCL = upper control limit = Probability Limits Median Chart LCL = lower limit UCL = upper limit Range Chart LCL = lower limit UCL = upper limit

In Table 36.22, replace with if you specify MEDCENTRAL=AVGMEAN, and replace with if you specify MEDCENTRAL=MEDMED. Replace with if you specify with the MU0= option, and replace with if you specify with the SIGMA0= option.

The formulas assume that the data are normally distributed. Note that the limits for both charts vary with ni and that the probability limits for Ri 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 with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
• Specify a constant nominal sample size for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
• Specify with the MU0= option or with the variable _MEAN_ in the LIMITS= data set.
• Specify with the SIGMA0= option or with the variable _STDDEV_ in the LIMITS= data set.

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