Control charts¶

This page specifies the control-chart constants mfgQC uses and the limit formulas for every chart family it computes — variables (I-MR, X̄-R, X̄-S), attributes (p, np, c, u), the time-weighted charts (EWMA, CUSUM), and the standardized short-run chart. The run-rules engine that flags out-of-control points has its own page: Run rules.

Everything below is pinned to the implementation in mfgqc/constants.py (_CONTROL_TABLE, control_constant()) and mfgqc/control_charts.py. Where the code makes a choice a textbook might make differently, that choice is called out.

The control-chart constants¶

mfgQC keeps the standard SQC factor table as plain transcribed data, not values computed on the fly. That is deliberate: a practitioner can check every number against the published table they already trust (Montgomery, Introduction to Statistical Quality Control, Appendix VI). They are looked up by subgroup size \(n\) via control_constant(name, n) for \(n = 2 \ldots 25\); outside that range the function raises a ValueError.

Constant	What it is	Used by
\(d_2\)	mean of the relative range \(W = R/\sigma\) — converts \(\bar{R}\) to a \(\sigma\) estimate via \(\hat\sigma = \bar{R}/d_2\)	I-MR (with \(d_2\) at \(n=2\)); within-\(\sigma\) in capability
\(d_3\)	standard deviation of the relative range \(W\)	dispersion-limit width (tabulated; reported, not used directly in the R-chart formula below)
\(A_2\)	factor for X̄ limits from \(\bar{R}\)	X̄-R location limits
\(A_3\)	factor for X̄ limits from \(\bar{S}\)	X̄-S location limits
\(D_3, D_4\)	lower/upper factors for the range limits from \(\bar{R}\)	R-chart limits
\(B_3, B_4\)	lower/upper factors for the s limits from \(\bar{S}\)	S-chart limits
\(c_4\)	bias-correction for the sample standard deviation (\(E[s] = c_4\,\sigma\))	unbiased \(\sigma\) from \(\bar{S}\) in capability

Table excerpt (\(n = 2 \ldots 10\))¶

These are the exact values in _CONTROL_TABLE, read straight from the code:

\(n\)	\(d_2\)	\(d_3\)	\(A_2\)	\(A_3\)	\(D_3\)	\(D_4\)	\(B_3\)	\(B_4\)	\(c_4\)
2	1.128	0.853	1.880	2.659	0.000	3.267	0.000	3.267	0.7979
3	1.693	0.888	1.023	1.954	0.000	2.574	0.000	2.568	0.8862
4	2.059	0.880	0.729	1.628	0.000	2.282	0.000	2.266	0.9213
5	2.326	0.864	0.577	1.427	0.000	2.114	0.000	2.089	0.9400
6	2.534	0.848	0.483	1.287	0.000	2.004	0.030	1.970	0.9515
7	2.704	0.833	0.419	1.182	0.076	1.924	0.118	1.882	0.9594
8	2.847	0.820	0.373	1.099	0.136	1.864	0.185	1.815	0.9650
9	2.970	0.808	0.337	1.032	0.184	1.816	0.239	1.761	0.9693
10	3.078	0.797	0.308	0.975	0.223	1.777	0.284	1.716	0.9727

The full table runs to \(n = 25\). Note that \(D_3\) and \(B_3\) are \(0\) below \(n = 7\) and \(n = 6\) respectively — for small subgroups the lower dispersion limit is clamped to zero, which is what the table encodes.

These are tabulated factors, not live computations

The constants are standard published values transcribed to 3–4 decimal places. mfgQC does not recompute them from \(d_2 = E[W]\) integrals at run time. If you need a value to more decimals, or for an \(n\) outside \(2 \ldots 25\), that is a deliberate boundary, not a bug.

Variables charts¶

Variables charts track a continuous measurement. mfgQC computes the location panel (the process center) and a dispersion panel (the spread) together, and runs the run rules on the location series. The dispersion panel is additionally checked for the beyond-limits rule only (Rule 1), so a subgroup whose range or s blows up is flagged even when its mean stays inside the location limits.

I-MR (individuals + moving range)¶

For one measurement per time point (\(n = 1\)). Spread is estimated from the moving range of consecutive points, \(MR_i = |x_i - x_{i-1}|\), and converted to \(\sigma\) using \(d_2\) at \(n = 2\) (because each moving range spans two observations):

\[ \bar{MR} = \frac{1}{k-1}\sum_{i=2}^{k} MR_i, \qquad \hat\sigma = \frac{\bar{MR}}{d_2}, \quad d_2 = 1.128 \]

Panel	CL	UCL	LCL
Individuals (I)	\(\bar{x}\)	\(\bar{x} + 3\hat\sigma\)	\(\bar{x} - 3\hat\sigma\)
Moving range (MR)	\(\bar{MR}\)	\(D_4\,\bar{MR}\), \(\ D_4 = 3.267\)	\(0\)

The MR series is aligned to points \(2 \ldots k\) (the first point has no moving range and plots as NaN). kind="i" is an accepted alias for "i_mr", and an explicit I-MR chart on a plain measure column defaults each row to its own size-1 subgroup rather than erroring.

X̄-R (subgroup mean and range)¶

For constant subgroup size \(2 \le n \le 10\). Spread per subgroup is the range \(R = \max - \min\); \(\bar{R}\) is the mean range, \(\bar{\bar{x}}\) the grand mean.

Panel	CL	UCL	LCL
Mean (X̄)	\(\bar{\bar{x}}\)	\(\bar{\bar{x}} + A_2\bar{R}\)	\(\bar{\bar{x}} - A_2\bar{R}\)
Range (R)	\(\bar{R}\)	\(D_4\,\bar{R}\)	\(D_3\,\bar{R}\)

X̄-S (subgroup mean and standard deviation)¶

Same structure as X̄-R but the per-subgroup spread is the sample standard deviation \(s\) (computed with ddof=1), and \(\bar{S}\) is the mean of those. This is the preferred variables chart for larger subgroups.

Panel	CL	UCL	LCL
Mean (X̄)	\(\bar{\bar{x}}\)	\(\bar{\bar{x}} + A_3\bar{S}\)	\(\bar{\bar{x}} - A_3\bar{S}\)
Std dev (S)	\(\bar{S}\)	\(B_4\,\bar{S}\)	\(B_3\,\bar{S}\)

For the run-rules engine, the location-panel sigma is recovered from the limit width itself — \(\sigma_{\text{loc}} = (\text{UCL} - \text{CL})/3\) — so the zone tests use the same \(\pm 3\sigma\) band the chart draws.

Attributes charts¶

Attributes charts track counts or proportions of defectives/defects. All four use 3σ limits with the lower limit clamped at zero (np.clip(cl - 3*sd, 0, None)). Each must be requested explicitly with kind= — they are never inferred.

The sample size for p/np/u comes from n=: a column name (per-point sizes, which produce stepped limits when they vary), an int (constant size), or None (falls back to a size role, else 1).

Chart	Tracks	Point	CL	\(3\sigma\) half-width
p	proportion defective, variable \(n\)	\(p_i = c_i / n_i\)	\(\bar{p} = \dfrac{\sum c_i}{\sum n_i}\)	\(3\sqrt{\dfrac{\bar{p}(1-\bar{p})}{n_i}}\)
np	count defective, constant \(n\)	\(c_i\)	\(n\bar{p}\)	\(3\sqrt{n\bar{p}(1-\bar{p})}\)
c	defects per unit, constant area	\(c_i\)	\(\bar{c}\)	\(3\sqrt{\bar{c}}\)
u	defects per unit, variable area	\(u_i = c_i / n_i\)	\(\bar{u} = \dfrac{\sum c_i}{\sum n_i}\)	\(3\sqrt{\dfrac{\bar{u}}{n_i}}\)

For p and u the half-width depends on \(n_i\), so the limits step up and down with each point's sample size; a point is flagged only if it falls outside its own limits. The np chart enforces constant \(n\) (it raises if sizes vary) because \(n\bar{p}\) has no meaning otherwise. Attributes charts have a location panel only — there is no dispersion panel.

Each attributes chart attaches an over/under-dispersion check (chi-square against the binomial for p/np, against the Poisson for c/u). It reports the dispersion ratio and warns; it does not switch models.

Time-weighted and short-run charts¶

These exist for situations a Shewhart chart handles poorly. One line each — see the source for full parameter detail.

EWMA (qc.ewma_chart(...), mfgqc/timeseries_charts.py) — exponentially weighted moving average for detecting small sustained shifts. Recursion \(z_i = \lambda x_i + (1-\lambda)z_{i-1}\) with time-varying limits \(\mu_0 \pm L\sigma\sqrt{\frac{\lambda}{2-\lambda}\big(1-(1-\lambda)^{2i}\big)}\). Key parameters: smoothing \(\lambda\) (default \(0.1\)) and limit width \(L\) in sigma (default \(2.7\)).
CUSUM (qc.cusum_chart(...)) — tabular two-sided cumulative-sum chart for the same small-shift regime. Accumulates \(C^+\) and \(C^-\) deviations and signals when either exceeds the decision interval \(H = h\sigma\). Key parameters: reference value \(k\) (slack, default \(0.5\sigma\)) and \(h\) (default \(5\sigma\)).
Short-run / standardized (qc.short_run_chart(by=...)) — pools several part numbers onto one chart by plotting the standardized deviation \(z = (x - \text{target}_{\text{part}})/\sigma_{\text{part}}\) against fixed \(\pm 3\) limits. target is a scalar, a {part: target} map, or None (each part's own mean). It checks homogeneity of within-part variance across the pooled parts.

EWMA and CUSUM assume a known in-control baseline

Both require an in-control \(\mu_0\) and \(\sigma\). mfgQC defaults \(\mu_0\) to the spec target if set (else the sample mean) and \(\sigma\) to the I-chart estimate \(\bar{MR}/d_2\), and records which source it used. For a trustworthy chart, supply \(\mu_0\) and \(\sigma\) from a stable baseline study rather than letting them be estimated from the data you are charting.

Chart-kind inference¶

When you call qc.control_chart() with no kind=, mfgQC infers a variables chart from the subgroup sizes. The rule is exactly (_infer_variable_kind):

Subgroup size \(n\)	Inferred chart
all \(n = 1\)	`i_mr`
\(2 \le n \le 10\)	`xbar_r`
\(n > 10\)	`xbar_s`

The threshold between X̄-R and X̄-S is \(n \le 10\) (X̄-R) versus \(n > 10\) (X̄-S). The inferred choice is recorded in the provenance history (inferred: true in the step params) and named in the report. Attributes charts are never inferred — you must pass kind="p", "np", "c", or "u" explicitly.

Guardrail: no silent switching

Inference only fills in a missing kind. If you request a kind explicitly, mfgQC computes that kind or raises — it never quietly substitutes a different chart. For how to override the inferred choice, see Choosing a control chart.

Assumptions and what mfgQC checks¶

Assumption	Why it matters	Checked?
Rational subgrouping	Subgroups must be formed so within-subgroup variation captures only common cause; this underlies the \(\bar{R}/d_2\) and \(\bar{S}\) sigma estimates.	Not checked — it is a sampling-design decision the engineer owns.
Independence	Autocorrelated points make Shewhart limits too tight and produce false alarms.	Checked. A lag-1 autocorrelation test (`check_independence`) runs on the location series; it reports \(r_1\), a p-value, and — if it fails — recommends a time-series chart (EWMA). It self-flags `low_power` for \(n < 30\).
Approximate normality of subgroup means	The \(\pm 3\sigma\) limits assume the plotted statistic is roughly normal; for X̄ charts the CLT makes this mild even for moderately skewed data.	Not tested directly on the variables charts; relied on via the CLT. (Per-point normality of individuals on the I-MR chart is the stronger assumption — judge it with a normality check.)
Binomial / Poisson counts	p/np assume binomial counts; c/u assume Poisson defects.	Checked. A chi-square over/under-dispersion test runs on every attributes chart and reports the dispersion ratio.

True to mfgQC's guardrail design, these checks report and recommend — they never silently switch methods. An autocorrelation failure does not auto-convert your X̄-R chart to an EWMA; it tells you to consider one.

Worked example¶

A 20-subgroup, size-5 dataset. With no kind=, \(n = 5\) falls in \(2 \le n \le 10\), so mfgQC infers X̄-R:

import numpy as np, pandas as pd, mfgqc

rng = np.random.default_rng(7)
df = pd.DataFrame({
    "width": np.round(rng.normal(1.52, 0.12, size=100), 3),
    "lot":   np.repeat(np.arange(1, 21), 5),     # 20 subgroups of 5
})
qc = (mfgqc.load(df, measure="width", subgroup="lot", subgroup_size=5)
           .spec(lower=1.0, upper=2.0, target=1.5))

print(qc.control_chart())

Control Chart: xbar_r (inferred); rules=nelson
==============================================
Xbar: CL=1.4992  UCL=1.6475  LCL=1.3509
R: CL=0.25705  UCL=0.5434  LCL=0

Out-of-control signals: none (process in control)

Assumption checks:
  [PASS] independence (lag-1 autocorrelation): r=0.193, p=0.387; n=20 [low power]

You can read the formulas straight off the numbers. With \(\bar{R} = 0.25705\) and \(n = 5\) (so \(A_2 = 0.577\), \(D_4 = 2.114\), \(D_3 = 0\)):

\(\text{UCL}_{\bar{x}} = 1.4992 + 0.577 \times 0.25705 = 1.6475\)
\(\text{UCL}_R = 2.114 \times 0.25705 = 0.5434\), \(\ \text{LCL}_R = 0\)

An I-MR chart on single observations, showing the dispersion-panel beyond-limits flag and the \(\bar{MR}/d_2\) sigma:

rng = np.random.default_rng(3)
df = pd.DataFrame({"visc": np.round(rng.normal(50, 2, size=24), 2)})
qc = mfgqc.load(df, measure="visc")
print(qc.control_chart(kind="i_mr"))

Control Chart: i_mr (specified); rules=nelson
=============================================
Individual: CL=49.913  UCL=56.915  LCL=42.912
MR: CL=2.6326  UCL=8.6007  LCL=0

Out-of-control signals: 1
  point 2 (dispersion): nelson_1 - one point beyond control limits

Assumption checks:
  [PASS] independence (lag-1 autocorrelation): r=-0.182, p=0.371; n=24 [low power]

Here \(\hat\sigma = \bar{MR}/d_2 = 2.6326 / 1.128 = 2.334\), so the individuals limits are \(49.913 \pm 3 \times 2.334\), and the MR upper limit is \(D_4 \bar{MR} = 3.267 \times 2.6326 = 8.6007\). Point 2's moving range exceeds it.

A p-chart with a constant sample-size column:

df = pd.DataFrame({"defs": [3,5,2,4,6,1,3,4,2,5], "n": [100]*10})
qc = mfgqc.load(df, measure="defs")
print(qc.control_chart(kind="p", n="n"))

Control Chart: p (specified); rules=nelson
==========================================
Proportion (p): CL=0.035  UCL=0.090134  LCL=0

Out-of-control signals: none (process in control)

Assumption checks:
  [PASS] dispersion (chi-square dispersion): dispersion ratio 0.74, p=0.672; n=10 [low power]

\(\bar{p} = 35/1000 = 0.035\); the half-width is \(3\sqrt{0.035 \times 0.965 / 100} = 0.0551\), giving \(\text{UCL} = 0.0901\) and a clamped \(\text{LCL} = 0\).

Source and standard¶

Constants table and variables/attributes limit formulas: Montgomery, D. C., Introduction to Statistical Quality Control (Wiley) — mfgQC's primary build oracle for SPC. The constants follow the conventional SQC tables (Appendix VI). See the Bibliography. These are the same standard tabulated factors published in ASTM control-chart references; mfgQC cites Montgomery as the transcription source.
The run-rules engine (Nelson / Western Electric) is documented separately on Run rules and cites Nelson (1984) and the Western Electric Statistical Quality Control Handbook (1956).