Constructing Range Charts

RCHART Statement

Constructing Range Charts

The following notation is used in this section:

$\sigma$ process standard deviation (standard deviation of the population of measurements)

R_i range of measurements in i^th subgroup

n_i sample size of i^th subgroup

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

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 an R chart indicates the value of a subgroup range (R_i). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is R₁₀=19-12=7.

Central Line

By default, the central line for the i^th subgroup indicates an estimate of 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 error of 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 R_i exceeds the limits

The following table provides the formulas for the limits:

Table 39.21: Limits for R Charts

Control Limits

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

LCL = lower limit $= D_{\alpha/2}\hat{\sigma}$

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

The formulas assume that the data are normally distributed. Note that the control limits vary with n_i and that the probability limits for R_i are asymmetric around the central line. If a standard value $\sigma_{0}$ is available for $\sigma$ , replace $\hat{\sigma}$ with $\sigma_{0}$ in Table 39.21.

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)
R_i	range of measurements in i^th subgroup
n_i	sample size of i^th subgroup
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
D_p(n)	100p^th percentile of the distribution of the range of n independent observations from a normal population with unit standard deviation