Pareto and Root Cause: Method

Pareto Principle, Stratification, Rate Tests, Control Charts of Counts and Fishbone Diagrams

The Pareto principle

Juran observed in the 1940s that in almost every collection of defects, complaints, costs or delays a small number of categories accounts for most of the total: the vital few against the useful many (he first wrote "trivial many" and later withdrew the word). He named the pattern after Pareto's income curve; the 80/20 ratio is a figure of speech, not a law, and the data decide where the break falls. The Pareto chart orders the categories by their contribution, draws each as a bar and runs the cumulative percentage across the top. Its purpose is to choose: the categories to the left of the cut are the ones whose removal changes the total, and effort spent on the right-hand tail changes little. The tool takes the categories either already tallied (name and count, with an optional cost) or as raw records with a header line, and tallies them itself.

Weighting

A count Pareto answers "what happens most often"; a cost Pareto answers "what hurts most", and the two orders can differ sharply when a rare defect scraps an expensive part. The tool draws the bars by count, by the summed cost of each category, or by count × severity, where the severity table gives each category a factor (a 1 to 10 rating, hours of rework, a weight of any kind). The vital few are the categories whose cumulative share first reaches the cut (80 % by convention), or the top n when a number is entered. Ties keep the order of the data. The table also shows the count and cost shares side by side whatever the weighting, so the two views can be compared.

Stratification

Stratification splits the data by a factor the process has in it, shift, machine, operator, supplier, line, product, and draws the Pareto of every stratum. If the category mix is the same in every stratum the factor is not part of the explanation; if one stratum carries a different mix the problem has been localised. The tool tests the difference on the r × c table of category × level with Pearson's X² = Σ (O − E)²/E and the likelihood-ratio statistic G² = 2 Σ O ln(O/E) on (r − 1)(c − 1) degrees of freedom, E being row total × column total / n under independence (Agresti 2007, Section 2.4). Cramér's V = √(X²/(n (min(r, c) − 1))) puts the strength on a 0 to 1 scale. The adjusted standardized residual of a cell, (O − E)/√(E (1 − p_row)(1 − p_col)) (Haberman 1973), is a standard normal score under independence: cells beyond ±2 are where the levels differ, and they point at the stratum to look into. Expected counts below 5 make the X² approximation rough; merge sparse categories or levels when many cells are small. The side-by-side bar chart shows the same thing by eye.

Before and after

Once a countermeasure is in place the question is whether the count really fell. Two periods are compared, either from a period column of the records, from a tallied period column, or from a second paste. For every category and for the total the tool treats the counts as Poisson with exposures t₁ and t₂ (units produced, hours, days; the units column summed over the period when there is one, otherwise the values entered, otherwise equal). Conditional on the total c₁ + c₂, the count c₁ is binomial with p = t₁/(t₁ + t₂) under equal rates, which gives the exact conditional test of Przyborowski and Wilenski (1940); its two-sided p-value sums the probabilities of every outcome at most as likely as the observed one. The normal approximation uses z = (c₁/t₁ − c₂/t₂)/√(λ̂ (1/t₁ + 1/t₂)) with the pooled rate λ̂; it agrees with the exact test for counts above about 20 and overstates the evidence below. (Krishnamoorthy and Thomson's E-test is a little more powerful than the conditional test; the tool reports the conditional one because its p-value is exact and simple to state.) A second question is whether a category's share of all occurrences changed, which it can do while its own rate holds still because the others moved: the two-proportion z test with the pooled proportion (Agresti 2007, Section 2.2) compares c₁/N₁ with c₂/N₂. With many categories tested at once a few 5 % results arise by chance; quote the exact p-value for small counts and read the category table as a whole.

Counts over time

When the records carry dates the occurrences are grouped by day, ISO week or month and put on a Shewhart chart of counts. The c chart plots the count per period about c̄ = Σc/m with the Poisson 3σ limits c̄ ± 3√c̄, the lower limit floored at zero; when the number of units inspected in each period is known the u chart plots occurrences per unit, u_i = c_i/n_i, about ū = Σc/Σn with limits ū ± 3√(ū/n_i) that narrow as n_i grows (Montgomery Chapter 7; NIST/SEMATECH 6.3.3.1). A period beyond a limit had an assignable cause and deserves its own search; a period below the lower limit is as useful, since it shows what the process can do. A chart with every point inside the limits says the rate is stable: it will not fall by exhortation, only by changing the system, which is what the Pareto and the cause analysis are for. The c chart's normal approximation is adequate when c̄ is at least 5; below about 9 there is no lower limit. Days without a record do not appear on the chart; give such production days a line with a count of 0.

Root cause: 5-Whys and the fishbone

The Pareto says where to look; it does not say why. Two of Ishikawa's seven tools take over. The 5-Whys (Ohno 1988) asks why of the symptom, then why of the answer, until a cause is reached that a decision can remove, usually a missing standard, a rule or a design, rather than a lapse that vigilance might catch next time; five is the typical depth, not a rule. The tool takes chains written as category -> cause -> why -> ... -> root cause, shows them as an indented table with the root cause marked, and uses the first cause of each chain in the summary. The cause-and-effect diagram (Ishikawa 1976) organises the candidate causes before any is confirmed: the effect in the head, the major categories as bones, the causes as ribs. Manufacturing uses the 6M, Man, Machine, Method, Material, Measurement and Environment (Mother Nature); service processes use the 4P or 4S (policies, procedures, people, plant; surroundings, suppliers, systems, skills); any names can be entered. A chain's bracketed prefix puts its first cause on that bone; a line Bone: cause; cause adds causes directly. The diagram is a map of hypotheses: the stratified Pareto, the record's cause column and the chains say which ribs are supported by data.

Reading the summary

The summary names the vital few with their share, the leading cause of each (from its 5-Whys chain, else the most frequent entry of the records' cause column), the stratification verdict, the before-and-after verdict and the periods beyond the control limits. That is the content of a one-page problem statement: what the problem is, how big each part is, where it is concentrated, whether it is changing, and what the first countermeasure should address.

Worked examples on the page

The default data are Montgomery's Pareto chart of the tank defect data (7th ed., Section 5.4, Figure 5.17): 19 defect categories found at the final inspection of tanks, 166 defects in all, with incorrect dimensions (36), parts damaged (34) and machining problems (29) at the head; the book notes that these three account for about 60 % of the total (59.6 %), and the 80 % cut falls after six categories (81.3 %). The 5-Whys chains under them are illustrative. The selector also loads an illustrative incident log of 200 records with date, cause, shift, machine, period and units inspected (stratification by shift, the Q1 to Q2 comparison with the units as exposure, the weekly u chart), and Montgomery's Example 7.3, the c chart of nonconformities on 26 samples of 100 printed circuit boards (c̄ = 19.85, UCL 33.22, LCL 6.48, samples 6 and 20 out of control; the page prints 33.21 and 6.48 because it keeps c̄ = 19.846 unrounded under the square root where the book rounds it to 19.85 first). The test script checks these values, the NIST/SEMATECH 6.3.3.1 wafer example (c̄ = 16, limits 4 and 28), Agresti's cross-tabs (X² 7.01 and 30.0 with their residuals), Agresti's aspirin comparison of two proportions, and the Poisson tests against scipy.

Limits of the tool

• The Pareto is descriptive: the order of the bars has no confidence interval, and categories with similar counts may swap places in the next period.
• The X² test needs expected counts of about 5 or more in most cells; it says that the strata differ, not which countermeasure to take.
• The rate tests assume Poisson counts with known exposures; clustered defects (several per unit from one cause) overdisperse the counts and make the tests, and the c chart limits, too narrow.
• The time chart uses 3σ limits only (no run rules), and a day without any record is absent rather than zero.
• The 5-Whys and the fishbone record reasoning; they do not verify it. A root cause is confirmed by data, by a stratified Pareto or a controlled change, not by the diagram.