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Control Chart Constants¶

This page defines the unbiasing constants behind the sigma estimators and control limits: the range constants \(d_2\), \(d_3\), \(d_4\), the standard deviation constants \(c_4\), \(c_5\), the MSSD constant \(c_4'\), and the classical derived factors.

Notation¶

Term	Description
\(n\)	subgroup size (or moving range length, when applied to moving ranges)
\(R\)	range of \(n\) independent standard normal observations
\(\Gamma\)	gamma function
\(k\)	sigma multiplier for control limits (usually 3)

Range constants (\(d_2\), \(d_3\), \(d_4\))¶

The range constants describe the distribution of the range \(R\) of \(n\) independent standard normal observations:

\[ d_2(n) = \mathrm{E}[R], \qquad d_3(n) = \sqrt{\mathrm{Var}(R)}, \qquad d_4(n) = \operatorname{median}(R) \]

So \(R/d_2(n)\) is an unbiased estimate of \(\sigma\) from a subgroup range, \(d_3(n)\) scales the spread of a range, and \(\widetilde{R}/d_4(n)\) is the corresponding estimate from a median range. Quantum XL evaluates these from precomputed full-precision tables for \(n = 2\) to \(100\).

Standard deviation constant (\(c_4\))¶

\[ c_4(n) = \sqrt{\frac{2}{n - 1}}\; \frac{\Gamma\!\left(\frac{n}{2}\right)}{\Gamma\!\left(\frac{n - 1}{2}\right)} \]

\(c_4(n)\) is the expected value of the sample standard deviation of \(n\) independent standard normal observations, so \(s/c_4(n)\) is an unbiased estimate of \(\sigma\). \(\Gamma\) is the gamma function. Quantum XL evaluates the formula directly (a log-gamma form is used for large \(n\); the value approaches \(1\) as \(n\) grows and is taken as \(1\) for extremely large \(n\)).

Standard deviation of the sample standard deviation (\(c_5\))¶

\[ c_5(n) = \sqrt{1 - c_4(n)^2} \]

\(c_5(n)\) is the standard deviation of the sample standard deviation of \(n\) standard normal observations, computed from \(c_4\).

MSSD constant (\(c_4'\))¶

\(c_4'(N)\) is the unbiasing constant for the square root of MSSD estimator: dividing the raw MSSD estimate by \(c_4'(N)\) makes it unbiased for \(\sigma\). Quantum XL looks the value up from a table for \(N = 2\) to \(500\) (at \(N = 500\) the value is \(0.999124\)); for \(N > 500\) the constant is taken as \(1\).

Derived control chart factors¶

The classical control chart factors are assembled from \(d_2\), \(d_3\), \(c_4\), and \(c_5\) with the sigma multiplier \(k\) (usually \(3\)):

\[ A = \frac{k}{\sqrt{n}}, \qquad A_2 = \frac{k}{d_2\sqrt{n}}, \qquad A_3 = \frac{k}{c_4\sqrt{n}} \]

\[ D_1 = \max(0,\; d_2 - k\,d_3), \qquad D_2 = d_2 + k\,d_3, \qquad D_3 = \max\!\left(0,\; 1 - k\,\frac{d_3}{d_2}\right), \qquad D_4 = 1 + k\,\frac{d_3}{d_2} \]

\[ B_3 = \max\!\left(0,\; 1 - \frac{k\,c_5}{c_4}\right), \qquad B_4 = 1 + \frac{k\,c_5}{c_4}, \qquad B_5 = \max(0,\; c_4 - k\,c_5), \qquad B_6 = c_4 + k\,c_5, \qquad E_2 = \frac{k}{d_2} \]

Precision and compatibility modes¶

Quantum XL's default is full precision: \(d_2\), \(d_3\), \(d_4\) from full-precision tables and \(c_4\) from the gamma formula. Two compatibility modes reproduce software that uses rounded published tables: one rounds the full-precision values to the published number of digits (\(d_2\) and \(d_4\) to 3, \(d_3\) to 4, \(c_4\) to 6), and one uses the older published tables directly (\(d_2\) for \(n \le 50\); \(d_3\) and \(d_4\) tabulated to \(n = 25\) with a cubic polynomial approximation for \(n = 26\) to \(50\); \(c_4\) tabulated to \(n = 100\), then the gamma formula rounded to 6 digits).

Used by¶

Process Sigma Estimation: \(d_2\), \(d_3\), \(d_4\), \(c_4\), and \(c_4'\) appear in every within-sigma estimator and its weights.
Cpk/Histogram Math Details: through the sigma estimates behind each capability block.
Laney Overdispersion: \(d_2(2) = 1.1284\) divides the moving range of the z-scores.
The variable control charts use the derived factors for their limits; their math pages are planned.

References¶

Wheeler, D. J., and Chambers, D. S. (1992). Understanding Statistical Process Control (2nd ed.). SPC Press.
Montgomery, D. C. (2013). Introduction to Statistical Quality Control (7th ed.). Wiley.
Tippett, L. H. C. (1925). On the extreme individuals and the range of samples taken from a normal population. Biometrika, 17(3/4), 364-387.
Harter, H. L. (1960). Tables of range and studentized range. Annals of Mathematical Statistics, 31(4), 1122-1147.
Duncan, A. J. (1986). Quality Control and Industrial Statistics (5th ed.). Irwin.