Probability Limits for p Charts

Compute normal or Cornish-Fisher probability limits for a p chart without constructing a plot. The Cornish-Fisher formulas retain the original proportion scale and correct the normal approximation using binomial cumulants.

Usage

pchart_limits(
  p,
  n,
  alpha = ALPHA,
  type = c("normal", "cf1", "cf2"),
  sides = c("two.sided", "upper"),
  truncate = TRUE
)

Arguments

p

In-control nonconforming proportion. A scalar strictly between zero and one.

n

Positive integer sample size or vector of sample sizes.

alpha

Nominal false alarm probability. Defaults to 0.0027.

type

Limit method: "normal", "cf1", or "cf2".

sides

Either "two.sided" or "upper".

truncate

Logical. If TRUE, limits are restricted to [0, 1].

Value

A list containing center, lcl, ucl, n, alpha, type, sides, npq, and applicable.

Details

type = "cf1" uses the first skewness correction proposed by Winterbottom (1993). type = "cf2" uses the operational two-adjustment limits in equation (8) of Joekes and Barbosa (2013). In particular, the second adjustment is evaluated with the positive normal quantile and is applied with the same sign to both limits. This convention reproduces the published limits and false-alarm risks in Tables 2 and 3.

The article recommends the normal approximation for approximately \(n p (1-p) >= 5\), CF1 for approximately \(n p (1-p) >= 0.25\), and CF2 for approximately \(n p (1-p) >= 0.08\). The returned applicable field reports whether each sample size meets the corresponding recommendation.

References

Winterbottom, A. (1993). Simple adjustment to improve control limits on attribute charts. Quality and Reliability Engineering International.

Joekes, S. and Barbosa, E. P. (2013). An improved attribute control chart for monitoring non-conforming proportion in high quality processes. Control Engineering Practice, 21, 407–412. doi:10.1016/j.conengprac.2012.12.005 .

Examples

pchart_limits(0.015, n = 20, type = "normal")
#> $center
#> [1] 0.015
#> 
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 0.09653924
#> 
#> $n
#> [1] 20
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $type
#> [1] "normal"
#> 
#> $sides
#> [1] "two.sided"
#> 
#> $npq
#> [1] 0.2955
#> 
#> $applicable
#> [1] FALSE
#> 
pchart_limits(0.015, n = 20, type = "cf1")
#> $center
#> [1] 0.015
#> 
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 0.1612048
#> 
#> $n
#> [1] 20
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $type
#> [1] "cf1"
#> 
#> $sides
#> [1] "two.sided"
#> 
#> $npq
#> [1] 0.2955
#> 
#> $applicable
#> [1] TRUE
#> 
pchart_limits(0.015, n = 20, type = "cf2")
#> $center
#> [1] 0.015
#> 
#> $lcl
#> [1] 0
#> 
#> $ucl
#> [1] 0.1303192
#> 
#> $n
#> [1] 20
#> 
#> $alpha
#> [1] 0.0027
#> 
#> $type
#> [1] "cf2"
#> 
#> $sides
#> [1] "two.sided"
#> 
#> $npq
#> [1] 0.2955
#> 
#> $applicable
#> [1] TRUE
#>