Capability¶

Normal-theory process capability answers one question: given the spec limits, how much of the tolerance does the process consume? mfgQC reports the full index family — \(C_p\), \(C_{pk}\) (with \(C_{pu}\)/\(C_{pl}\)), \(P_p\), \(P_{pk}\), and \(C_{pm}\) — names the sigma estimator it used, attaches confidence intervals because small-sample point estimates are overconfident, and checks the assumptions that make the numbers meaningful in the first place.

This page pins every formula and estimator to mfgqc/capability.py. The default method="normal" never transforms your data; for skewed processes see Non-normal capability.

qc  = mfgqc.load(df, measure="width", subgroup="lot", subgroup_size=5).spec(lower=1.0, upper=2.0, target=1.5)
cap = qc.capability()

1. The formulas mfgQC computes¶

Let \(\mu\) be the sample mean, \(\hat\sigma_{\text{within}}\) the short-term (within-subgroup) sigma, \(\hat\sigma_{\text{overall}}\) the long-term (ordinary sample) sigma, and \(T\) the target. With lower/upper spec limits \(\text{LSL}\)/\(\text{USL}\):

Within-subgroup indices (_indices, computed at \(\hat\sigma_{\text{within}}\)):

\[ C_p = \frac{\text{USL} - \text{LSL}}{6\,\hat\sigma_{\text{within}}} \]

\[ C_{pu} = \frac{\text{USL} - \mu}{3\,\hat\sigma_{\text{within}}}, \qquad C_{pl} = \frac{\mu - \text{LSL}}{3\,\hat\sigma_{\text{within}}}, \qquad C_{pk} = \min\!\big(C_{pu},\, C_{pl}\big) \]

Overall (performance) indices — the same formulas evaluated at \(\hat\sigma_{\text{overall}}\):

\[ P_p = \frac{\text{USL} - \text{LSL}}{6\,\hat\sigma_{\text{overall}}}, \qquad P_{pk} = \min\!\left(\frac{\text{USL} - \mu}{3\,\hat\sigma_{\text{overall}}},\; \frac{\mu - \text{LSL}}{3\,\hat\sigma_{\text{overall}}}\right) \]

Taguchi index \(C_{pm}\) — a two-sided index that penalizes being off-target, built on the overall sigma about the target:

\[ C_{pm} = \frac{\text{USL} - \text{LSL}}{6\,\tau}, \qquad \tau = \sqrt{\hat\sigma_{\text{overall}}^{\,2} + (\mu - T)^2} \]

Which indices appear depends on the spec you set

\(C_p\), \(P_p\), and \(C_{pm}\) require both limits. With only one limit set, mfgQC reports the relevant one-sided index (\(C_{pu}\) or \(C_{pl}\)) and folds it straight into \(C_{pk}\); the two-sided indices come back n/a. \(C_{pm}\) additionally requires target= in .spec(...). This mirrors _indices returning None for any index whose limit is absent.

2. The sigma estimator mfgQC selects¶

The within/overall distinction is the heart of capability and the reason mfgQC reports both families instead of one number:

Within-subgroup (\(C_p\)/\(C_{pk}\)) estimates the process's inherent short-term spread — the variation between consecutive parts, with shift and drift between subgroups filtered out. It answers "what is this process capable of?"
Overall (\(P_p\)/\(P_{pk}\)) uses the ordinary sample standard deviation across all data, so it captures everything — short-term noise plus every shift, drift, and tool change over the study. It answers "what did the customer actually receive?"

When \(C_{pk} \gg P_{pk}\), your process is stable in the short run but wandering over time — a hunt-for-special-causes signal, not a capable process.

_within_sigma picks the within estimator from the subgroup structure and records the choice in sigma_used:

Subgroup structure	Estimator	`sigma_used` label
Equal subgroups, size \(n \ge 2\)	\(\hat\sigma_{\text{within}} = \dfrac{\bar R}{d_2(n)}\)	`within (R-bar/d2)`
Individuals (\(n = 1\))	\(\hat\sigma_{\text{within}} = \dfrac{\overline{MR}}{d_2(2)}\)	`within (MR-bar/d2)`
Unequal subgroup sizes	pooled within-subgroup SD (below)	`within (pooled)`
No usable subgroups	falls back to overall	`overall`

where \(\bar R\) is the mean subgroup range, \(\overline{MR}\) the mean moving range of the individuals series, and \(d_2\) is the bias-correction constant from mfgqc/constants.py (e.g. \(d_2(2)=1.128\), \(d_2(5)=2.326\); Montgomery, Appendix VI). The pooled estimator is the usual degrees-of-freedom-weighted root mean square of the per-subgroup variances:

\[ \hat\sigma_{\text{pooled}} = \sqrt{\frac{\sum_{j}\,(n_j - 1)\,s_j^2}{\sum_{j}\,(n_j - 1)}} \]

over subgroups \(j\) with \(n_j \ge 2\).

\(P_p\)/\(P_{pk}\) always use the overall sample SD

Regardless of subgroup structure, the performance indices are computed at \(\hat\sigma_{\text{overall}} = s\) (values.std(ddof=1)). They never use a within estimator. The within and overall families are reported side by side and never conflated — this is correctness watch-list item #1 in the source.

The result object carries sigma_within, sigma_overall, and the string sigma_used so a report builder can show exactly which estimator drove \(C_{pk}\). Consume these from summary()/to_dict(), never by parsing the report text.

3. Confidence intervals¶

Small-\(n\) capability point estimates are biased toward looking better than the process is. mfgQC reports normal-theory CIs (default 95%, alpha=0.05) so you read a \(C_{pk}\) as a range, not a false-precision decimal. From _capability_cis:

\(C_p\) — exact \(\chi^2\) interval (Montgomery eq. 8.19):

\[ \widehat{C_p}\sqrt{\frac{\chi^2_{\alpha/2,\,n-1}}{n-1}} \;\le\; C_p \;\le\; \widehat{C_p}\sqrt{\frac{\chi^2_{1-\alpha/2,\,n-1}}{n-1}} \]

\(C_{pk}\) — large-sample normal approximation (Montgomery eq. 8.21): the half-width factor is

\[ m = z_{1-\alpha/2}\,\sqrt{\frac{1}{9 n\,\widehat{C_{pk}}^{\,2}} + \frac{1}{2(n-1)}}, \qquad \big(\widehat{C_{pk}}(1 - m),\;\; \widehat{C_{pk}}(1 + m)\big) \]

When CIs are omitted

CIs require \(n \ge 2\), and the \(C_{pk}\) interval additionally requires \(\widehat{C_{pk}} \neq 0\). Confidence intervals are computed for the normal method only. Non-normal methods (boxcox, clements, johnson) return None and the report prints CI: n/a (non-normal method) rather than fabricating a normal-theory interval — see Non-normal capability.

4. Assumptions — and which ones mfgQC checks¶

Assumption	Why it matters	Checked by mfgQC?
Normality	\(C_p\)/\(C_{pk}\) map to fraction-defective only if the data are normal	Yes — Anderson-Darling, reported
Process in statistical control	a capability index on an unstable process is meaningless	No — you must establish it first with a control chart
Adequate subgroup count	\(\bar R / d_2\) needs enough subgroups to be stable	Yes — flags fewer than 25

Normality (checked, reported). mfgQC runs an Anderson-Darling test (check_normality) and reports the verdict at \(\alpha = 0.05\). The binary passed/failed comes from the direct AD test; alongside it mfgQC reports an est. Cpk impact magnitude — the relative shift in \(C_{pk}\) between the normal method and an auto-fit non-normal method (_cpk_shift). That context tells you whether the non-normality actually moves the number you care about, but it never flips the verdict: grossly non-normal data fail even if the coincidental \(C_{pk}\) shift is small. If it fails, mfgQC recommends a non-normal method — it does not silently switch. See Reading the assumption report.

Establish statistical control before capability

mfgQC does not test for statistical control inside capability(). A capability index computed on an out-of-control process is not interpretable — the within-sigma estimate is contaminated and the fraction-defective projection is fiction. Run a control chart first, confirm there are no out-of-control signals, and only then compute capability.

Subgroup sufficiency (checked, reported). When a within estimator is in use, mfgQC counts the subgroups and fails the check below 25, with the recommendation "Only N subgroups; >=25 recommended for a stable within-sigma estimate." It still computes the index — it warns, it does not refuse.

5. Worked example¶

Twenty subgroups of five, spec \([1.0, 2.0]\) with target \(1.5\):

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))
cap = qc.capability()
print(cap.report())

Process Capability (method=normal)
==================================
n = 100   mean = 1.4992
sigma (within)  = 0.1105
sigma (overall) = 0.10556
Cp/Cpk sigma    = within (R-bar/d2)

Cp  = 1.508  95% CI (1.3, 1.72)
Cpk = 1.506  95% CI (1.29, 1.73)   (Cpu=1.51, Cpl=1.506)
Pp  = 1.579    Ppk = 1.576   (Ppu=1.581, Ppl=1.576)
Cpm = 1.579

Assumption checks:
  [PASS] normality (Anderson-Darling): AD=0.301, p=0.572; est. Cpk impact 5.0%; n=100
  [FAIL] subgroup_sufficiency (subgroup count >= 25): subgroup count 20; n=20

Recommendations:
  - Only 20 subgroups; >=25 recommended for a stable within-sigma estimate.

Reading it:

The estimator is named. \(C_p\)/\(C_{pk}\) used within (R-bar/d2) — equal subgroups of 5, so \(\hat\sigma_{\text{within}} = \bar R / d_2(5)\) with \(d_2(5)=2.326\).
Both families show. \(C_{pk}=1.506\) (short-term) vs \(P_{pk}=1.576\) (long-term). They are close here because the process is stable.
CIs are wide. \(C_{pk}=1.506\) with a 95% CI of \((1.29,\,1.73)\) — even with 100 measurements the index is only pinned to one decimal.
The guardrails fired. Normality passes; subgroup count (20) is flagged below 25. mfgQC tells you and recommends a fix; it does not alter the calculation.

The flat dashboard dict is available without parsing text:

cap.summary()
# {'method': 'normal', 'n': 100, 'mean': 1.4992..., 'sigma_within': 0.11051...,
#  'sigma_overall': 0.10556..., 'Cp': 1.50813..., 'Cp_CI_low': 1.29824...,
#  'Cp_CI_high': 1.71768..., 'Cpk': 1.50581..., 'Cpk_CI_low': 1.28613...,
#  'Cpk_CI_high': 1.72549..., 'Pp': 1.57883..., 'Ppk': 1.57640...,
#  'Cpm': 1.57879..., 'confidence': 95, 'normality_passed': True}

Other subgroup structures¶

The same call adapts the estimator to the data — verified output:

INDIVIDUALS (subgroup_size=1)  sigma_used: within (MR-bar/d2)
  sigma_within=0.55286  sigma_overall=0.57708   Cpk=1.1884  Ppk=1.1385

UNEQUAL subgroups               sigma_used: within (pooled)
  sigma_within=0.32545  sigma_overall=0.30076

Note that in the individuals case the within and overall sigmas differ even though both summarize the same series: within (MR-bar/d2) uses the moving range (\(\overline{MR}/d_2(2)\), \(d_2(2)=1.128\)), filtering point-to-point variation only.

6. Source standard¶

mfgQC's capability indices, the \(\bar R/d_2\) and \(\overline{MR}/d_2\) within-sigma estimators, the \(d_2\) constants, and both confidence-interval formulas (eq. 8.19 for \(C_p\), eq. 8.21 for \(C_{pk}\)) are pinned to Montgomery, Introduction to Statistical Quality Control — mfgQC's primary build oracle for SPC and capability. See the Bibliography.