Shewhart, W. A. (1926), "Quality Control Charts", Bell System Technical Journal 5(4), 593-603, doi:10.1002/j.1538-7305.1926.tb04213.x. The paper was opened directly (Internet Archive scan of BSTJ 5:4) and read as rendered page images: the four-step procedure of specification, estimation, distribution and test is on p. 597; Figure 3, p. 600, is the Inspection Engineering analysis sheet, whose final column computes 3*sigma for each monitored statistic – 3 sigma_xbar = .0198010, 3 sigma_sigma = .0180106, 3 sigma_k = .0599001, 3 sigma_beta2 = .119800 – and p. 601 describes the resulting dotted lines in Figure 4 as "the limits within which the different statistics should lie, if the product had been controlled". The multiplier three is therefore taken from the primary source, not from later practice.
Value
List with estimate (number of signals), alerts,
z, lcl, ucl, n_alerts, alarm_rate,
false_alarm_prob, arl0, n, k, method.
Details
CITATION LIMIT. The book-length treatment, Economic Control of Quality of Manufactured Product (1931), where the choice of three is argued at length rather than merely applied, is lending-restricted on the Internet Archive and could not be opened; nothing is attributed to it here.
The chart signals when |x_t - mu| > k sigma. With k = 3 and a Gaussian
in-control distribution the false-alarm probability per point is
2(1 - Phi(3)) = 0.0026998, so the in-control average run length is
about 370 points: monitor a stable process daily for a year and expect
roughly one alarm that means nothing. That number, not the 3, is what makes
the choice a trade-off, and it is returned alongside the decisions so the
alarm rate observed can be read against the alarm rate expected. A chart is a
classifier, and its false-positive rate is a property of the limit, not of
the data.
References
Shewhart, W. A. (1926), Bell System Technical Journal 5(4):593-603, doi:10.1002/j.1538-7305.1926.tb04213.x, Figure 3, p. 600.
Examples
Shewh(c(0, 1, 2, 4), 0, 1)$n_alerts # 1
#> [1] 1