Conventional Three-Sigma Limits for the R Chart

Compute the conventional three-sigma (Shewhart) control limits for the R chart without constructing a plot. The limits are based on the constants $d_2(n)$ and $d_3(n)$, the mean and standard deviation of the relative range $W = R / \sigma$ under normality.

Usage

r_shewhart_limits(sigma, n, nsigmas = SIGMA_MULT)

Arguments

sigma: Positive finite scalar. The in-control process standard deviation. When estimated from Phase I data, the returned limits are plug-in estimates.
n: Integer subgroup size, at least 2.
nsigmas: Positive numeric scalar. The number of sigma units for the control limits. Defaults to 3.

Value

A named list with components:

lcl: Lower control limit (zero-truncated).
ucl: Upper control limit.
center: $d_2(n) \sigma$, the expected value of the range under normality.
nsigmas: The supplied multiplier.
sigma: The supplied sigma.
n: The supplied subgroup size.

Details

The conventional limits are $$LCL = \max\{0, d_2(n) - k \, d_3(n)\} \, \sigma$$ and $$UCL = \{d_2(n) + k \, d_3(n)\} \, \sigma,$$ where $k$ is the number of sigma units (default 3). Because the distribution of the relative range is not normal, the actual false-alarm probability of these limits can be substantially larger than the nominal 0.0027 associated with three-sigma limits for a normal variate. Use alpha.risk to evaluate the exact false-alarm probability.

Phase convention

These limits are computed for a known or separately estimated sigma. When sigma is estimated from a Phase I reference sample, the limits are plug-in limits.

Decision rule

A subgroup range $R$ signals out of control when $R < LCL$ or $R > UCL$. Equality to a limit is treated as in control.

Errors

An error is raised when sigma is NA, NaN, or non-positive; when n is smaller than 2, non-integer, or non-finite; or when nsigmas is non-positive.

References

Montgomery, D. C. (2009). Introduction to Statistical Quality Control, 6th ed. Wiley.

Barbosa, E. P., Gneri, M. A., and Meneguetti, A. (2013). Range control charts revisited: Simpler Tippett-like formulae, its practical implementation, and the study of false alarm. Communications in Statistics - Simulation and Computation, 42(2), 247–262. doi:10.1080/03610918.2011.639967 .

Examples

# Conventional three-sigma limits for n = 5, sigma = 2
r_shewhart_limits(sigma = 2, n = 5)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 9.83635
#> 
#> $center
#> [1] 4.651858
#> 
#> $nsigmas
#> [1] 3
#> 
#> $sigma
#> [1] 2
#> 
#> $n
#> [1] 5
#> 

# Custom multiplier
r_shewhart_limits(sigma = 1, n = 10, nsigmas = 2)
#> $lcl
#> [1] 1.483404
#> 
#> $ucl
#> [1] 4.671607
#> 
#> $center
#> [1] 3.077505
#> 
#> $nsigmas
#> [1] 2
#> 
#> $sigma
#> [1] 1
#> 
#> $n
#> [1] 10
#>