Constructing Charts for Individual Measurements and Moving Ranges

IRCHART Statement

Constructing Charts for Individual Measurements and Moving 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)

X_i the i^th individual measurement

$\bar{X}$ mean of the individual measurements, computed as (X₁+ ... +X_N)/N, where N is the number of individual measurements

n number of consecutive measurements used to calculate the moving ranges (by default, n=2)

R_i moving range computed for the i^th subgroup (corresponding to the i^th individual measurement). If i<n, then R_i is assigned a missing value. Otherwise,
R_i = max(X_i,X_i-1,...,X_i-n+1) - min(X_i,X_i-1,...,X_i-n+1)
This formula assumes that X_i, X_i-1,...,X_i-n+1 are nonmissing.

$\bar{R}$ average of the nonmissing moving ranges, computed as
[(R_n + R_n+1 ... + R_N)/(N+1-n)]

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 (0<p<1) of the standard normal distribution

D_p(n) 100p^th percentile (0<p<1) of the distribution of the range of n independent observations from a normal population with unit standard deviation

Plotted Points

Each point on an individual measurements chart, indicates the value of a measurement (X_i).

Each point on a moving range chart indicates the value of a moving range (R_i). With n=2, for example, if the first three measurements are 3.4, 3.7, and 3.6, the first moving range is missing, the second moving range is |3.7-3.4|=0.3, and the third moving range is |3.6-3.7|=0.1.

Central Lines

By default, the central line on an individual measurements chart indicates an estimate for $\mu$ , which is computed as $\bar{X}$ .If you specify a known value ( $\mu_{0}$ ) for $\mu$ , the central line indicates the value of $\mu_{0}$ .

The central line on a moving range chart indicates an estimate for the expected moving range, computed as $d_{2}(n)\hat{\sigma}$ where $\hat{\sigma} = \bar{R}/d_2(n)$ .If you specify a known value ( $\hat{\sigma}_0$ ) for $\sigma$ , the central line indicates the value of $d_{2}(n)\sigma_{0}$ .

Control Limits

You can compute the limits

as a specified multiple (k) of the standard errors of 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 X_i or R_i exceeds the limits

The following table provides the formulas for the limits:

Table 34.22: Limits for Individual Measurements and Moving Range Charts

Control Limits

Individual Measurements Chart	LCL = lower control limit $= \bar{X} - k\hat{\sigma}$
	UCL = upper control limit $= \bar{X} + k\hat{\sigma}$
Moving Range Chart	LCL = lower control limit $= \max(d_{2}(n)\hat{\sigma} - kd_{3}(n)\hat{\sigma},0)$
	UCL = upper control limit = $d_{2}(n)\hat{\sigma} + kd_{3}(n)\hat{\sigma}$

Probability Limits

Individual Measurements Chart LCL = lower control limit $= \bar{X} - z_{\alpha/2}\hat{\sigma}$

UCL = upper control limit $= \bar{X} + z_{\alpha/2}\hat{\sigma}$

Moving Range Chart LCL = lower control limit $= D_{\alpha/2}(n)\hat{\sigma}$

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

The formulas assume that the measurements are normally distributed. Note that the probability limits for the moving range are asymmetric about the central line. If standard values $\mu_{0}$ and $\sigma_{0}$ are available for $\mu$ and $\sigma$ ,replace $\bar{X}$ with $\mu_{0}$ and $\hat{\sigma}$ with $\sigma_{0}$ in Table 34.22.

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 n 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.

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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)
X_i	the i^th individual measurement
$\bar{X}$	mean of the individual measurements, computed as (X₁+ ... +X_N)/N, where N is the number of individual measurements
n	number of consecutive measurements used to calculate the moving ranges (by default, n=2)
R_i	moving range computed for the i^th subgroup (corresponding to the i^th individual measurement). If i<n, then R_i is assigned a missing value. Otherwise, R_i = max(X_i,X_i-1,...,X_i-n+1) - min(X_i,X_i-1,...,X_i-n+1) This formula assumes that X_i, X_i-1,...,X_i-n+1 are nonmissing.
$\bar{R}$	average of the nonmissing moving ranges, computed as [(R_n + R_n+1 ... + R_N)/(N+1-n)]
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 (0<p<1) of the standard normal distribution
D_p(n)	100p^th percentile (0<p<1) of the distribution of the range of n independent observations from a normal population with unit standard deviation

Probability Limits

Individual Measurements Chart	LCL = lower control limit $= \bar{X} - z_{\alpha/2}\hat{\sigma}$
	UCL = upper control limit $= \bar{X} + z_{\alpha/2}\hat{\sigma}$
Moving Range Chart	LCL = lower control limit $= D_{\alpha/2}(n)\hat{\sigma}$
	UCL = upper control limit $= D_{1-\alpha/2}(n)\hat{\sigma}$