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¶

A subgroup is a small set of parts measured together under the same conditions, for example five consecutive parts off one machine. You declare how rows group into subgroups when you load the data; see How subgrouping works. 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.

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 cannot be trusted. Run a control chart first, confirm there are no out-of-control signals, and only then compute capability.

Subgroup sufficiency (checked, reported). The within estimator pools the spread inside each subgroup, so it needs enough subgroups to be stable. This is the number of subgroups, not the size of each one and not the total sample size: twenty subgroups of five is twenty subgroups, even though that is a hundred measurements. 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 source for SPC and capability. See the Bibliography.