01 · Make spread a customer problem
Make spread a customer problem
Variation reduction aims to make outcomes more predictable, not merely improve their average. A dispatch process with a ten minute average but occasional thirty minute delays can miss customer promises. Define which spread matters: within a stable shift, across days, across order types, or in the upper tail. Different sources need different interventions.
Specify the response and sampling structure. An overall standard deviation can combine ordinary within condition noise with systematic differences between conditions. Stratify by relevant mechanisms, such as carrier, dock, order complexity, and release policy. Establish that the measurement system can distinguish process variation. A noisy timestamp system can make a stable operation appear inconsistent and obscure whether a change actually reduced spread.
02 · Quantify spread on a relevant scale
Quantify spread on a relevant scale
Sample variance is s² = Σ(yi−ȳ)²/(n−1), where y are observations, ȳ their mean, and n the independent sample count. Standard deviation s is its square root and has the response's units. Variance emphasizes large deviations. Range is easy to understand but highly sample size dependent and sensitive to extreme values.
Interquartile range describes the middle half and can help for skewed times, but may hide costly tail delays. Coefficient of variation s/ȳ compares spread relative to a positive mean and requires a meaningful ratio scale. It can mislead near zero or when changing the mean alone changes the ratio. Report a measure aligned with the customer problem, often including a percentile or exceedance rate alongside a central spread summary.
03 · Worked before and after spread comparison
Worked before and after spread comparison
Suppose fictional independent normal dispatch batches before a release rule change have n = 20, mean ten minutes, and standard deviation four. Twenty independently sampled batches after the change have the same mean and standard deviation two. Variances are sixteen and four, so the before to after ratio is F = 16/4 = 4. The estimated standard deviation fell by 50%, while estimated variance fell by 75%. Those percentages are not interchangeable.
Under independent normal samples and equal population variances, the ratio follows an F distribution with nineteen and nineteen degrees of freedom. The two sided p value is approximately 0.00404. Here Fq,19,19 is the lower tail q quantile, defined by cumulative probability q. A 95% interval for the population variance ratio is F/F0.975,19,19 to F/F0.025,19,19, approximately 1.583 to 10.106. The estimate favors reduced variation but the amount is uncertain. Square rooting the interval gives a standard deviation ratio interval of approximately 1.258 to 3.179.
If the dispatch specification is at most sixteen minutes, the normal model estimates an upper tail probability of 1−Φ[(16−10)/4] ≈ 0.0668 before and 1−Φ[(16−10)/2] ≈ 0.00135 after. Φ is the standard normal cumulative distribution function. These are model based probabilities, not observed defect rates. Validate tail fit and stability before translating them into customer savings.
04 · Separate within and between sources
Separate within and between sources
The law of total variance is Var(Y) = E[Var(Y|G)] + Var[E(Y|G)], where G identifies a condition or group. It separates average within group variation from variation between group means. A dock standardization can reduce differences between docks without changing within dock noise. Equipment maintenance may reduce within dock noise. Pooling everything hides that distinction.
Use rational subgroups and suitable control charts to assess stability. An unstable process with special causes should not be summarized as one enduring noise distribution. Identify and investigate changes in time order. A before and after histogram loses sequence information. A seemingly narrower after distribution may reflect a quiet week or exclusion of complex orders rather than a process improvement.
05 · Choose variance comparisons carefully
Choose variance comparisons carefully
The classical F variance test is sensitive to departures from normality. Levene or median based Brown–Forsythe comparisons can be more robust. The median based version works with absolute deviations from group medians and tests whether those deviation distributions show different average spread under its framework. It still requires independent units and an appropriate comparison design.
If means and variances are linked, consider transformation or a model for conditional spread. A log scale can suit positive multiplicative times, but interpret back transformed summaries correctly. For repeated batches and clustered orders, variance components models can separate worker, day, batch, and residual components. Estimate uncertainty for these components; a component at zero can indicate limited information rather than proof that the source never matters.
06 · Design a reduction that survives noise conditions
Design a reduction that survives noise conditions
A robust experiment evaluates controllable settings across deliberately varied nuisance conditions. A release rule that gives low spread only at light volume may fail during peak demand. Replicate at treatment combinations so spread can be estimated, and preserve the experimental unit. Compare both mean and variation, because reducing spread while shifting the mean toward a specification can worsen customer risk.
Investigate mechanisms: synchronized arrivals, replenishment shortages, queue discipline, inconsistent work methods, equipment reliability, and measurement ambiguity. Standardization should target verified sources rather than demanding every operator work at the same pace. Removing a long delay requires understanding why it occurred. If it is a real customer relevant failure, deleting it from the dataset makes the variation metric improve without changing the process.
07 · Judge evidence and connect to DMAIC
Judge evidence and connect to DMAIC
A strong application separates stable and unstable periods, compares similar workload populations, estimates uncertainty, and demonstrates an intervention that reduces a relevant source. It reports mean, spread, tail behavior, and measurement conditions. A weak application compares ranges from samples of different size or credits variation reduction to a changed product mix. A nonsignificant variance test is not proof of equal spread.
Analyze locates the sources and mechanisms of variation. Improve tests practical changes across important noise conditions. Control monitors stability and tail outcomes with rational sampling and response plans. Preserve subgroup definitions, comparison assumptions, workload coverage, and the effect on customer risk. A reduction in one summary statistic is successful only when it reflects more reliable performance in the operation customers actually experience.
08 · Connect variation to stability and economic decisions
Connect variation to stability and economic decisions
In the worked variance interval, Fq,19,19 denotes the lower tail q quantile: the value with cumulative probability q. This convention makes the larger 0.975 quantile the lower interval divisor and the smaller 0.025 quantile the upper divisor. Some references use an upper tail convention, so the subscript must be defined before copying formulas or critical values.
A narrower distribution changes the reliability of planning even when mean time stays constant. However, the normal tail predictions depend strongly on stable conditions and tail shape. A mixture of ordinary batches and rare equipment outages can have the same standard deviation as a normal distribution while producing very different missed cutoff risks. Inspect actual exceedances and failure modes. If the primary cost comes from outages, model or prevent those mechanisms directly rather than optimizing a pooled standard deviation.
A control plan should specify how changes in spread will be detected and investigated. Rational sampling must include busy periods without combining fundamentally different populations indiscriminately. Review the mean whenever spread changes, because an apparently tighter process can drift toward the specification boundary. Sustainable variation reduction keeps the customer outcome and process mechanism visible together.
Connections
Related methods and phase use
Placed under Analyze because of the outcome it supports. Use the method in other phases when its purpose and assumptions fit the work; the phase list is a guide to selection, not a compulsory sequence.
Sources
Professional foundations and further reading
The explanation and fictional teaching examples are written for LeanTrainingLab. Linked primary sources provide professional foundations; example values and local conventions are stated in the guide. The accepted toolset draws on Six Sigma+Lean Toolset, second edition (Lunau, editor; Springer, 2013). No book text or examples are reproduced here.