01 · Describe association between paired measurements
Describe association between paired measurements
Pearson's correlation coefficient describes direction and strength of a linear relationship between two numeric variables. Warehouse pairs might be travel distance and picking time for the same order. Join values on the same independent unit. Correlating unrelated columns after sorting them can manufacture an association. Define the population and time window, and preserve identifiers for groups and run order.
The coefficient r ranges from −1 to +1. Positive values mean larger x accompanies larger y; negative values mean an opposite tendency. Magnitudes near one indicate close alignment around a straight line. A value near zero means little linear association, not necessarily no relationship. A symmetric curve can have near zero correlation despite a strong deterministic relationship.
02 · Understand the calculation
Understand the calculation
Define Sxx = Σ(xi−x̄)², Syy = Σ(yi−ȳ)², and Sxy = Σ(xi−x̄)(yi−ȳ). Then r = Sxy/√(SxxSyy). The index i identifies paired observations, and x̄ and ȳ are their sample means. Scaling covariance by both spreads makes correlation unitless. If either variable is constant, the denominator is zero and correlation is undefined.
Converting meters to feet does not change r. Swapping x and y also does not change it, so correlation has no outcome direction. A regression slope carries units and changes when variables are reversed. Avoid universal thresholds such as calling every r above 0.7 strong. Operational importance depends on measurement noise, predictor range, mechanism, intervention choices, and consequences of prediction errors.
03 · Worked order distance example
Worked order distance example
For six fictional orders, distances x in hundreds of meters are 1, 2, 3, 4, 5, and 6. Times y in minutes are 9, 11, 10, 14, 13, and 15. Their means are 3.5 and 12. Centered sums are Sxx = 17.5, Syy = 28, and Sxy = 20. Hence r = 20/√(17.5 × 28) = 0.9035.
The coefficient describes a clear positive linear tendency in this small sample. Its square is 0.8163. In a simple least squares regression with an intercept on the same observations, that equals R². It is not the percentage of picking time caused by distance. Order size, congestion, and experience can still explain part of the association.
For independent pairs from a bivariate normal population, Fisher's transformation z = atanh(r) = 0.5 ln[(1+r)/(1−r)] has approximate standard error 1/√(n−3). With n = 6, transform z ± 1.96/√3 back using tanh. The approximate 95% population correlation interval is 0.345 to 0.990. Its width shows why a high point estimate from six observations should not be treated as precise process knowledge.
04 · Inspect points and groups
Inspect points and groups
A scatterplot should show every actual pair with units and, where relevant, distinct shift or order class markings. Look for curved patterns, clusters, restricted ranges, unequal spread, and isolated extreme points. One distant large order can make r high while ordinary orders have little relationship. Investigate the order's mechanism and measurement. Report sensitivity to its inclusion when both scopes are defensible.
A pooled coefficient may mainly reflect differences between groups. Busy shifts could have longer distances and longer times even when the within shift relationship is weak. Conversely, aggregation can hide a consistent conditional relationship. Compare within group plots and use an appropriate regression model. Correlations can differ simply because groups have different predictor ranges, so inspect slopes and residual spreads rather than interpreting every r difference as a changed physical mechanism.
05 · Association is not an intervention effect
Association is not an intervention effect
Correlation alone cannot show that deliberately shortening a route will reduce time. Complex orders may increase both travel and item handling. Congestion can affect route choice and speed simultaneously. Slow picking can also cause a worker to choose a different path, reversing an assumed causal direction. Temporal order, process observation, measurement quality, and interventions strengthen the causal case.
A useful Analyze conclusion is a testable mechanism: relocating frequently co-picked stock could shorten travel and reduce time for comparable orders. A planned pilot can randomize suitable route or stock assignment while recording item count and congestion. A coefficient helps prioritize that investigation; it does not justify staffing changes or a savings forecast without further evidence about the mechanism and feasible intervention.
06 · Choose alternatives and quantify uncertainty
Choose alternatives and quantify uncertainty
Spearman correlation applies a correlation calculation to ranks and describes monotonic association. It can help with ordinal scales or outlier sensitivity, but it does not detect every nonlinear pattern. Kendall's coefficient describes pairwise concordance. Select a measure for the scientific question and measurement scale, then report it explicitly. Choosing whichever measure looks strongest after examining outcomes creates a selection problem.
Under a normal independent pair model, a zero Pearson correlation test uses t = r√[(n−2)/(1−r²)] with n−2 degrees of freedom. Its p value does not quantify practical strength or validate linearity. Repeated orders from one picker, route clusters, and time series require dependence aware inference. Predictor and response measurement error can weaken observed association. Restricted range can lower r while the underlying process slope remains unchanged.
07 · Use correlation well through DMAIC
Use correlation well through DMAIC
A strong application reports the plot, coefficient, independent sample count, interval, observed range, and relevant group checks. It names plausible alternative explanations and a follow-up design. A weak application screens hundreds of coefficients, selects the largest, and declares a root cause without confirmation. Another weak use treats r near zero as proof that an input does not matter despite a visible curve.
Analyze uses correlation to narrow candidate drivers and specify variables for more complete modeling. Improve tests whether manipulating a controllable driver changes the customer outcome. Control monitors inputs and outcomes under the new conditions. If correlation falls after improvement, inspect range restriction before claiming the relationship disappeared. Removing most distance variation can reduce r even when distance still affects time.
08 · Read the plotted points and a nonlinear counterexample
Read the plotted points and a nonlinear counterexample
The six plotted orders match the worked calculation exactly. The third order takes only ten minutes at three hundred meters while the fourth takes fourteen at four hundred meters. Such differences are useful prompts for process investigation. A high r does not mean every successive order must be slower, and it does not quantify the residual error in minutes. Retain the raw observations when moving from this summary to a prediction model.
For a separate fictional counterexample, centered route deviations x = −2, −1, 0, 1, 2 have delay scores y = 4, 1, 0, 1, 4. The mean x is zero and symmetric products sum to zero, so Pearson correlation is exactly zero. Yet y = x² determines every point. That example shows why plotting is essential before declaring an input irrelevant. It also shows that converting to ranks does not automatically resolve every nonmonotonic relationship. A curved model and a credible explanation are needed when both unusually short and unusually long release intervals create delays.
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.