17.2 Dispersion & Qualitative vs Quantitative Analysis

17.2 Dispersion & Qualitative vs Quantitative Analysis

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Dispersion: why spread beats “average alone”

Two processes can share the same mean yet deliver opposite quality outcomes. Process A is tightly clustered around the target; Process B swings widely. Customers feel Process B as defects, scrap, and complaints—even when the long-run mean looks fine. Dispersion (variability) is therefore as important as central tendency in audit evidence.

Common dispersion measures:

Range

R = x_max − x_min

Worked example: hardness readings 42, 45, 44, 48, 43 → R = 48 − 42 = 6. Easy for a quick check of a small sample; weak for large n because one extreme sets R regardless of the bulk of the data.

Variance and standard deviation (sample)

Sample variance:

$s^2 = \dfrac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}$

Sample standard deviation:

$s = \sqrt{s^2}$

Using n − 1 (Bessel’s correction) produces an unbiased estimate of population variance for random samples. Population formulas use N and $\mu$; auditors almost always see sample s.

Worked mini-calculation — standard deviation

Data: 10, 12, 14, 16, 18

1. Mean $\bar{x}$ = 70 / 5 = 14
2. Deviations: −4, −2, 0, +2, +4
3. Squared: 16, 4, 0, 4, 16 → sum = 40
4. $s^2$ = 40 / (5 − 1) = 10
5. $s$ = √10 ≈ 3.16

Interpretation in audit language: “Typical observations sit about 3 units from the mean,” not “the process always varies by exactly 3.” For roughly normal data, about 68% of points lie within $\bar{x} \pm s$, about 95% within $\bar{x} \pm 2s$ (empirical rule)—a useful sanity check, not a law for every distribution.

Coefficient of variation (optional but useful)

CV = (s / $\bar{x}$) × 100% when mean ≠ 0. CV compares relative variation across different scales (e.g., two product weights with different targets). Auditors use it carefully—CV can mislead near zero means.

Frequency distributions

A frequency distribution counts how many observations fall into defined categories or bins.

Steps auditors should expect in good analysis:

1. Choose classes (bins) of equal width for continuous data, or natural categories for discrete/attribute data
2. Tally frequencies and relative frequencies (percentages)
3. Display as a table and/or histogram
4. Read shape: center, spread, skew, gaps, outliers, multimodality

Mini frequency table — audit finding severity (example)

This is quantitative counts of qualitative labels. It immediately shows whether risk is concentrated in minors or whether criticals demand escalation.

Histogram audit questions

• Is the process centered near target or specs?
• Is spread wide relative to tolerance?
• Are there two peaks (mixed streams)?
• Is there a cliff at a specification (inspection bias / sorting)?
• Are tails heavy (rare but severe events)?

A histogram that piles up just inside the USL may indicate inspection sorting rather than a capable process—prime audit follow-up.

Qualitative analysis (coding and grouping)

Qualitative data are non-numeric: interview notes, procedure language, observation descriptions, open-ended customer comments, root-cause narratives. Auditors still analyze them systematically:

1. Collect raw notes with source, time, and context
2. Code segments with labels (theme, clause, risk type, process step)
3. Group related codes into categories or affinity groups
4. Count or map frequencies of codes (qualitative → semi-quantitative)
5. Trace high-frequency or high-risk themes to objective evidence

Examples of disciplined qualitative work in audits:

• Affinity diagram of recurring “workarounds” mentioned by operators
• Coding CAPA narratives for “training only” vs. “system change”
• Grouping supplier complaints by failure mode language before any mean is computed

Strengths: captures context, intent, and process knowledge numbers miss.
Risks: bias, selective listening, vague codes, treating opinions as facts. Mitigate with dual review, clear code definitions, and triangulation with records and observation.

Quantitative analysis (patterns in numbers)

Quantitative analysis uses measured variables—continuous (time, dimension, temperature) or discrete/count (defects, late lots)—and applies descriptive statistics, charts, and comparisons.

Auditor applications:

• Trend charts of scrap %, OTD, first-pass yield
• Comparison of means/medians and SDs across shifts or suppliers
• Control charts and capability metrics (later sections)
• Pareto of defect counts (quantitative frequencies of qualitative categories)

Strengths: reproducibility, clear thresholds, easy comparison.
Risks: wrong measure, bad sampling, garbage-in data, over-precision, ignoring measurement system error.

Qualitative + quantitative together

Strong audit conclusions usually blend both: numbers show magnitude; qualitative work explains mechanism and system context.

Systemic vs. isolated problems

Dispersion and pattern analysis help classify problems—critical for CAPA vs. firefighting.

Exam framing: “Systemic” does not mean “everyone is guilty.” It means the process design, methods, or environment repeatedly produce the condition. “Isolated” means a specific assignable circumstance, not the steady-state process signature.

Mini case — systemic vs. isolated

• Site scrap mean is 1.0% for six months (s small). In week 27, scrap jumps to 8% for three days after a wrong material was issued, then returns to 1.0%. Pattern: isolated special event on top of a stable baseline—still serious, but CAPA should address material control, not only “operator training on quality awareness.”
• Another site scrap stays at 4–6% every week for a year with many small causes. Pattern: systemic high variation or off-center process—needs system-level improvement, not 50 disconnected one-off corrections.

Auditor practice checklist (V.C.2–3)

1. Request raw or binned data, not only a single KPI tile
2. Compute or verify center and spread (and n, period)
3. Use a frequency view for shape and concentration
4. Code qualitative notes with defined categories; group before concluding
5. Stratify by the factors that could mix streams
6. Judge systemic vs. isolated from pattern stability and scope, then align recommendations

Common traps

• Equating low mean defect rate with low risk when SD is large
• Using range on large mixed datasets as if it were robust
• Calling interview consensus “quantitative proof” without counts or records
• Averaging away a bad shift by plant-wide SD/mean
• Treating every defect as special cause (or every spike as common cause) without pattern evidence

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