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Math Details

This page gives the exact formulas Quantum XL uses to build a Capability Scorecard. Each equation lists what it computes and where it appears in the output.

Notation

Term Description
\(U_i\) number of units for row \(i\) (must be a positive number, or the row is skipped)
\(D_i\) number of defects for row \(i\) (must be a positive number, or the row is skipped)
\(O_i\) opportunities per unit for row \(i\) (defaults to 1 when not supplied or not a positive number)
\(TO_i\) total opportunities for row \(i\)

Per-row metrics

\[ TO_i = U_i \times O_i, \qquad DPU_i = \frac{D_i}{U_i}, \qquad DPO_i = \frac{D_i}{TO_i}, \qquad DPMO_i = DPO_i \times 10^6 \]

These fill the Total Opportunities, DPU, DPO, and DPMO columns of the scorecard table, one row per part.

Totals row

\[ U_{\text{tot}} = \sum_i U_i, \qquad D_{\text{tot}} = \sum_i D_i, \qquad TO_{\text{tot}} = \sum_i TO_i \]
\[ DPU_{\text{tot}} = \sum_i DPU_i, \qquad DPO_{\text{tot}} = \frac{D_{\text{tot}}}{TO_{\text{tot}}}, \qquad DPMO_{\text{tot}} = DPO_{\text{tot}} \times 10^6 \]

Note the asymmetry: the totals row's DPU is the sum of the per-row DPU values, while the totals DPO and DPMO are recomputed from the summed defects and summed total opportunities (an opportunity-weighted rate, not a sum or average of the per-row rates).

Pareto charts

Two Pareto charts rank the rows: one by \(DPU_i\) and one by \(DPMO_i\). In each chart the rows are sorted descending by value, with ties broken by name in case-sensitive ordinal order, and the cumulative percentage after the \(j\)-th bar is

\[ \text{Cum}\%_j = \frac{\sum_{r=1}^{j} v_r}{\sum_{r} v_r} \times 100 \]

where \(v_r\) is the ranked value (\(DPU\) or \(DPMO\)). All rows are shown; there is no "Others" grouping.

See Also

References

  1. Montgomery, D. C. (2013). Introduction to Statistical Quality Control (7th ed.). Wiley.
  2. Breyfogle, F. W. (2003). Implementing Six Sigma: Smarter Solutions Using Statistical Methods (2nd ed.). Wiley.
  3. Pyzdek, T., and Keller, P. (2018). The Six Sigma Handbook (5th ed.). McGraw-Hill.
  4. Juran, J. M., and De Feo, J. A. (2010). Juran's Quality Handbook (6th ed.). McGraw-Hill.