01 · Define what the line is meant to do
Define what the line is meant to do
Simple linear regression estimates the average value of a continuous outcome as a function of one predictor. A warehouse may use order travel distance to explain picking time or estimate expected work within a familiar distance range. Specify whether the aim is description, prediction, or causal estimation. A useful predictor can have a slope that does not represent the effect of changing it.
Use pairs from independent units and preserve run order. Distance often was observed, not assigned. Large orders may require more travel and more handling, so a single predictor cannot separate those pathways. Record potential confounders such as item count, experience, and congestion. A controlled experiment or a suitable multivariable model can investigate them.
02 · Fit and define the model
Fit and define the model
The model is yi = β0 + β1xi + εi. Here y is the outcome, x the predictor, β0 the mean at x = 0, β1 the mean change per x unit, and ε unexplained error. Least squares chooses coefficients that minimize SSE = Σ(yi−ŷi)². A useful fit still requires an appropriate process model.
The fitted slope is b1 = Sxy/Sxx, with centered sums Sxy = Σ(x−x̄)(y−ȳ) and Sxx = Σ(x−x̄)². The intercept is b0 = ȳ−b1x̄. A residual is ei = yi−ŷi. Residual standard deviation s = √[SSE/(n−2)] estimates unexplained spread under the constant variance linear model. A constant predictor cannot identify a slope.
03 · Worked picking example
Worked picking example
For six fictional orders, x in hundreds of meters is 1, 2, 3, 4, 5, 6; y in minutes is 9, 11, 10, 14, 13, 15. The means are 3.5 and 12, while Sxx = 17.5 and Sxy = 20. Thus b1 = 1.1429 minutes per hundred meters and b0 = 8. The equation is ŷ = 8 + 1.1429x.
Predictions are 9.1429, 10.2857, 11.4286, 12.5714, 13.7143, and 14.8571. Residuals are −0.1429, 0.7143, −1.4286, 1.4286, −0.7143, and 0.1429. SSE = 5.1429, so s = √(5.1429/4) = 1.1339 minutes. The total centered response sum of squares is 28, and R² = 1−5.1429/28 = 0.8163.
Slope standard error is s/√17.5 = 0.2711. With t0.975,4 = 2.7764, the 95% slope interval is approximately 0.3903 to 1.8954. The conventional zero slope test gives p ≈ 0.0135. These results support association under the model; they do not show that every arbitrary route reduction causes the fitted saving.
04 · Distinguish mean and individual prediction
Distinguish mean and individual prediction
At x0, the estimated mean is b0 + b1x0. Its standard error is s√[1/n + (x0−x̄)²/Sxx]. For a new individual observation, add process noise: s√[1 + 1/n + (x0−x̄)²/Sxx]. Multiply by the appropriate t critical value for intervals under the conventional assumptions.
At x0 = 3.5, predicted mean time is 12. The 95% mean interval is approximately 10.715 to 13.285 minutes. The prediction interval is about 8.600 to 15.400. The latter concerns one new comparable order. Both widen away from the observed predictor center. A prediction at x = 12 extrapolates far beyond the data, and the ordinary interval formula does not account for a changed mechanism outside the studied range.
05 · Validate the residuals
Validate the residuals
Plot residuals against x and fitted values. Curvature suggests omitted structure; a fan suggests changing variance. Plot against run order to find drift or dependence. Inspect a normal probability plot when small sample normal based inference is central. Six observations provide weak diagnostic evidence; new observations across the range are valuable even when the current fit looks acceptable.
Check leverage and influence. An extreme x value can determine the slope with only a small residual. Influence combines predictor position and disagreement with the fitted line. Investigate the order before excluding it. High R² does not validate form, independence, constant variance, or representativeness. Weighted least squares, transformation, or a nonlinear model may improve fit, but the interpretation changes and the revision needs independent validation.
06 · Interpret units, scope, and limitations
Interpret units, scope, and limitations
The slope describes a conditional mean association in this model. The intercept might represent handling and setup, but zero distance was not observed, so calling eight minutes a measured fixed setup time overstates the evidence. Ordinary least squares treats x as measured without consequential error. Significant distance measurement error can distort the slope and may require a measurement error model.
A strong use reports units, range, slope uncertainty, residual diagnostics, and prediction errors relevant to the decision. A weak use converts the line into a universal labor standard, says R² means percentage prediction accuracy, or ignores item count. A relationship can be statistically clear while individual predictions remain too uncertain for assigning a tight service deadline.
07 · Carry the learning through DMAIC
Carry the learning through DMAIC
Analyze quantifies a candidate relationship and reveals unexplained patterns. Residual outliers may identify separate failure modes. Multiple regression can investigate whether distance remains associated after accounting for item count or congestion. Randomized route changes offer stronger evidence for a controllable cause.
Improve confirms the proposed change under comparable workloads and evaluates quality as well as time. Control monitors whether predictor ranges, residual spread, and customer outcomes remain similar. Refit when conditions materially change. Preserve raw pairings, equation, intended range, diagnostics, and uncertainty. A slope copied into a project slide without these conditions cannot be responsibly reused for another site.
08 · Read diagnostics as process questions
Read diagnostics as process questions
The residual plot below uses the exact six residuals from the worked line. Its alternating deviations illustrate that a fitted equation does not reproduce individual orders exactly. With only six observations, a pattern can be accidental, and the sample is too small for a reliable diagnosis of every possible failure. Use the plot to ask about handling complexity, aisle interruptions, and measurement, then obtain additional independent orders rather than fitting many extra terms immediately.
Consider the difference between an unusual response and an unusual predictor. An order near the middle distance with a large residual may reveal a scanner outage. An extremely distant order with a small residual may have high leverage and strongly determine the estimated slope. Both warrant investigation, but for different reasons. Preserve factual corrections in the data log and keep a sensitivity analysis separate from the primary prespecified analysis.
If normal based errors are inappropriate, the remedy should follow the defect in the model. A variance related fan may suggest weighted fitting or a transformed response. A curve suggests changing the mean function. Serial residuals suggest a dependence model or a different sampling design. Merely using a robust standard error does not make every one of these problems disappear. Any revised model should improve the intended operational use and be checked on new data.
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.