Conventional Shewhart Limits for the S Chart

Compute conventional Shewhart limits for subgroup standard deviations without constructing a plot. Under normality, $E(S)=c_4(n)\sigma$ and $SD(S)=\sigma\sqrt{1-c_4(n)^2}$. With the default theoretical center, the limits are therefore $$LCL=\max\{0,c_4(n)\sigma-k\sigma\sqrt{1-c_4(n)^2}\}$$ and $$UCL=c_4(n)\sigma+k\sigma\sqrt{1-c_4(n)^2},$$ where $k$ is nsigmas.

Usage

s_shewhart_limits(
  sigma,
  n,
  nsigmas = SIGMA_MULT,
  side = c("two.sided", "upper"),
  center = NULL
)

Arguments

sigma: Positive finite scalar. The in-control process standard deviation, known or estimated separately.
n: Integer subgroup size(s), each at least 2.
nsigmas: Positive finite scalar giving the number of standard-error units used for the limits. Defaults to 3.
side: Either "two.sided" or "upper". For an upper chart the lower limit is zero.
center: Optional finite numeric center. It may be a scalar or have the same length as n. If omitted, $c_4(n)\sigma$ is used.

Value

A named list with components lcl, ucl, center, sigma, n, nsigmas, side, and method.

Details

The optional center argument exists to reproduce the historical qcc S-chart convention, which centers the limits on the observed weighted mean of subgroup standard deviations while estimating sigma separately. When center = NULL, the theoretical center $c_4(n)\sigma$ is used.

Legacy qcc convention

The current qcc::limits.S() implementation computes std.dev * sqrt(1 - c4(sizes)^2) as the standard error of the plotted S statistic and centers conventional limits on the chart center supplied by qcc. Consequently, cchart.S(type = "n") passes its Phase I weighted S center explicitly to this function. This preserves the historical IQCC/qcc numerical behavior rather than silently replacing the sample center by $c_4(n)\hat\sigma$.

References

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

Scrucca, L. qcc: Quality Control Charts. R package.

Examples

s_shewhart_limits(sigma = 2, n = 5)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 3.927256
#> 
#> $center
#> [1] 1.879971
#> 
#> $sigma
#> [1] 2
#> 
#> $n
#> [1] 5
#> 
#> $nsigmas
#> [1] 3
#> 
#> $side
#> [1] "two.sided"
#> 
#> $method
#> [1] "shewhart"
#> 
s_shewhart_limits(sigma = 2, n = 5, center = 1.9)
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 3.947285
#> 
#> $center
#> [1] 1.9
#> 
#> $sigma
#> [1] 2
#> 
#> $n
#> [1] 5
#> 
#> $nsigmas
#> [1] 3
#> 
#> $side
#> [1] "two.sided"
#> 
#> $method
#> [1] "shewhart"
#>