01 · Choose a meaningful grouping factor
Choose a meaningful grouping factor
One way analysis of variance compares the means of a continuous response across levels of one categorical factor. A warehouse might compare three packing bench layouts using completion time per standardized order. The null hypothesis states that every layout has the same population mean. The alternative says at least one mean differs; it does not say every pair differs or identify the best layout.
First ask whether observations represent independent experimental units. Four separate randomized orders per layout may support a simple model. Four cartons timed within one order do not provide four independently assigned orders. If each worker uses every layout, a blocked or repeated measures model can be better than ignoring worker identity. If layout was never randomized, differences may reflect workload, staffing, or location rather than layout itself.
02 · Understand the variance decomposition
Understand the variance decomposition
The model is yij = μ + τi + εij. Here yij is observation j from group i, μ is a baseline mean, τi is the group effect, and εij is unexplained error. Identifiability requires a constraint or reference coding for the effects. Classical fixed effect ANOVA assumes independent errors with common variance and, for exact small sample inference, normal distributions within groups.
Let k be the number of groups, N the total observations, ni the size of group i, ȳi its mean, and ȳ the overall mean. Between group sum of squares is SSB = Σ ni(ȳi−ȳ)². Within group sum of squares is SSE = ΣΣ(yij−ȳi)². Divide by k−1 and N−k to obtain mean squares. The statistic F = MSB/MSE compares signal attributed to groups with noise within groups.
03 · Worked packing layout example
Worked packing layout example
Fictional times in minutes are layout A: 8, 9, 10, 9; layout B: 10, 11, 12, 11; and layout C: 12, 13, 14, 13. Their means are 9, 11, and 13. The grand mean is 11. Every group contributes 2 to within group squared deviations, so SSE = 6. Between group variation is 4(9−11)² + 4(11−11)² + 4(13−11)² = 32.
With k = 3 and N = 12, between group degrees of freedom are 2 and error degrees of freedom are 9. Thus MSB = 16, MSE = 6/9 = 0.6667, and F = 24. The upper tail probability under F with 2 and 9 degrees of freedom is approximately 0.000247. Reject equal means at a prespecified 5% level. The result indicates differences among the layout populations if the model and sampling design hold.
The pooled residual standard deviation is √0.6667 = 0.8165 minutes. The A versus C estimate is −4 minutes, and its unadjusted standard error is √[0.6667(1/4+1/4)] = 0.5774. A prespecified single contrast has an ordinary t interval using nine error degrees of freedom. Comparing all three pairs requires a familywise procedure or a clearly stated exploratory interpretation; the omnibus F result alone does not license unlimited unadjusted claims.
04 · Move from an omnibus result to useful contrasts
Move from an omnibus result to useful contrasts
Plan comparisons around the decision. If A is the incumbent, the relevant questions may be B versus A and C versus A, rather than every possible pair. A control comparison procedure can align with that objective. Tukey comparisons address all pairwise differences; a Bonferroni adjustment can handle a small planned family. Report adjusted intervals as well as decisions, because an interval makes magnitude and precision visible.
A contrast is C = Σ ciμi, where the weights sum to zero. For A versus the average of B and C, weights are 1, −1/2, and −1/2. In the example its estimate is 9−(11+13)/2 = −3 minutes. Its standard error uses √[MSEΣ(ci²/ni)]. Specifying such a contrast beforehand can answer a practical question more directly than searching a table for whichever pair looks most significant.
05 · Check the residuals, not just group averages
Check the residuals, not just group averages
Calculate residuals by subtracting the fitted group mean from each observation. Plot residuals against fitted values and run order, and inspect their distribution. An increasing spread at slower layouts challenges constant variance. A time trend suggests drift or congestion that should have been blocked or modeled. One extreme order can dominate a small group, so investigate its mechanism and repeat the analysis transparently if a defensible correction is needed.
Welch ANOVA can be useful for unequal variances and unequal group sizes. A variance test can supplement inspection, but a nonsignificant variance test does not establish equality, especially with few observations. Transformations can address skew or a mean related spread, but the conclusion then concerns the transformed scale. Rank based comparisons ask different distributional questions and do not automatically become tests of equal original scale means.
06 · Separate treatment, composition, and practical value
Separate treatment, composition, and practical value
A statistically significant group effect can simply identify different order mixes. If layout A processes small orders and C handles bulky orders, the factor is entangled with complexity. Stratification, regression adjustment, or a randomized blocked comparison may be necessary. Adjustment helps only for measured variables represented with suitable models; it cannot erase an unmeasured difference in staff training or routing.
A strong application uses comparable work, balanced coverage, clear assignment, and a meaningful time saving threshold. A weak application treats different shift averages as evidence that one shift's workers are inferior. The study should distinguish process conditions from personal labels. Another weak application selects the smallest observed mean as the winner without acknowledging uncertainty or winner selection. A confirmation run helps check whether the chosen layout retains its advantage.
07 · Apply the evidence in DMAIC
Apply the evidence in DMAIC
Analyze uses ANOVA to test a specific grouped explanation for response differences. The output should identify which contrasts warrant deeper investigation, how large they are, and whether residual variation remains operationally unacceptable. The fraction SSB/SSTotal describes explained variation in this sample; it is not automatically the fraction of failures eliminated by changing layouts.
Improve turns the comparison into a controlled pilot with workload and safety constraints. Control checks that the chosen layout continues to work under new staff, volumes, and product mixes. Archive group sizes, assignment method, residual diagnostics, comparison family, and the rationale for any model changes. If no significant difference appears, judge the interval widths and power before concluding that all layouts are practically equivalent.
08 · Keep a reproducible comparison record
Keep a reproducible comparison record
For the fictional A versus C contrast, the unadjusted 95% interval is −4 ± 2.2622 × 0.5774, approximately −5.306 to −2.694 minutes. This interval is suitable for that single planned contrast under the shared variance model. If all pairwise comparisons were searched, it should be replaced with the relevant simultaneous interval. Report which layout is subtracted from which so the negative sign is clear. Also retain the actual layout definitions: a label such as A does not tell another team whether equipment position, reach distance, or replenishment arrangement changed. A meaningful effect estimate must remain attached to its intervention.
A group mean can also conceal a quality tradeoff. Collect damaged item or rework outcomes with consistent definitions during the layout experiment. The fastest layout may create more errors. Analyze each response using an appropriate model and preserve the planned decision rule instead of choosing a winner solely from the smallest time mean.
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.