Exact Probability Limits for the R Chart

Compute exact equal-tail probability control limits for the R chart without constructing a plot. The limits are based on the distribution of the relative range $W = R / \sigma$ under normality, using the Studentized range distribution implemented by stats::qtukey().

Usage

r_exact_limits(sigma, n, alpha = ALPHA)

Arguments

sigma: Positive finite scalar. The in-control process standard deviation. When sigma is estimated from Phase I data, the returned limits are plug-in estimates and do not incorporate the additional uncertainty of Phase I estimation.
n: Integer subgroup size, at least 2.
alpha: Nominal false-alarm probability per subgroup. Defaults to 0.0027. One half is placed in each tail.

Value

A named list with components:

lcl: Lower control limit.
ucl: Upper control limit.
center: $d_2(n) \sigma$, the expected value of the range under normality.
sigma: The supplied sigma.
n: The supplied subgroup size.
alpha: The supplied nominal false-alarm probability.

Details

The exact limits are $$LCL = \sigma \, F_W^{-1}(\alpha/2; n)$$ and $$UCL = \sigma \, F_W^{-1}(1 - \alpha/2; n),$$ where $F_W^{-1}$ is the quantile function of $W = R / \sigma$.

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 and do not account for Phase I sampling variability.

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 alpha is not between 0 and 1.

References

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

# Known-sigma limits for subgroup size n = 5, sigma = 2, alpha = 0.0027
r_exact_limits(sigma = 2, n = 5)
#> $lcl
#> [1] 0.7930562
#> 
#> $ucl
#> [1] 10.7548
#> 
#> $center
#> [1] 4.651858
#> 
#> $sigma
#> [1] 2
#> 
#> $n
#> [1] 5
#> 
#> $alpha
#> [1] 0.0027
#> 

# Custom alpha
r_exact_limits(sigma = 1, n = 10, alpha = 0.01)
#> $lcl
#> [1] 1.334927
#> 
#> $ucl
#> [1] 5.417616
#> 
#> $center
#> [1] 3.077505
#> 
#> $sigma
#> [1] 1
#> 
#> $n
#> [1] 10
#> 
#> $alpha
#> [1] 0.01
#>