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DMS Statistics

This topic explains how key DMS statistics are calculated.

See the list of available statistics.

Definitions

  • A Unit is a single item produced by a process. A sample size of 100 indicates that there were 100 units.
  • A Non-Conforming Unit is an item that was scrapped by a process. This is often abbreviated as NCU (plural NCUs).
  • \(k\) = Number of data records.
  • \(n_k\) = Sample size of the \(k\)th record.
  • \(D_k\) = Defect count of the \(k\)th record.
  • \({NCU}_k\) = Count of NCUs in the \(k\)th record.

\(\text{Sum Sample Size}\) and \(\text{Sum Defects}\) are calculated by summing the related statistics from all records.

\[ \begin{aligned} \text{Sum Sample Size} &= \sum_{i=1}^{k} n_i \\ \text{Sum Defects} &= \sum_{i=1}^{k} D_i \\ \end{aligned} \]

The following statistics are calculated based on \(\text{Sum Sample Size}\) and \(\text{Sum Defects}\).

\[ \begin{aligned} \text{Sum Good} &= \text{Sum Sample Size} - \text{Sum Defects} \\ \text{Percent Defects} &= \frac{\text{Sum Defects}}{\text{Sum Sample Size}} \times 100 \\ \text{DPK} &= \frac{\text{Sum Defects}}{\text{Sum Sample Size}} \times \text{1 thousand} \\ \text{DPM} &= \frac{\text{Sum Defects}}{\text{Sum Sample Size}} \times \text{1 million} \\ \text{DPB} &= \frac{\text{Sum Defects}}{\text{Sum Sample Size}} \times \text{1 billion} \\ \end{aligned} \]

\(\text{Sum NCU}\) is calculated by summing the related statistic from all records.

\[ \text{Sum NCU} = \sum_{i=1}^{k} {NCU}_i \\ \]

The following statistics are calculated based on \(\text{Sum NCU}\)

\[ \begin{aligned} \text{Percent NCU} &= \frac{\text{Sum NCU}}{\text{Sum Sample Size}} \times 100 \\ \text{PPM} &= \frac{\text{Sum NCU}}{\text{Sum Sample Size}} \times \text{1 thousand} \\ \text{PPK} &= \frac{\text{Sum NCU}}{\text{Sum Sample Size}} \times \text{1 million} \\ \text{PPB} &= \frac{\text{Sum NCU}}{\text{Sum Sample Size}} \times \text{1 billion} \\ \text{Yield} &= \frac{\text{Sum Sample Size} - \text{Sum NCU}}{\text{Sum Sample Size}} \times 100 \\ \end{aligned} \]

Costs

\(\text{Sum Defect Cost}\) is calculated by summing the product of the count of each Defect by its Defect Cost. The related \(\text{Cost Per Unit}\) statistic is based on this summed cost.

\[ \begin{aligned} \text{Sum Defect Cost} &= \sum_{i=1}^{k} (D_i \times \text{Defect's cost}) \\ \text{Cost Per Unit} &= \frac{\text{Sum Defect Cost}}{\text{Sum Sample Size}} \end{aligned} \]

\(\text{Sum NCU Cost}\) is calculated by summing the product of the count of each record's NCUs by the NCU Cost of that record's Process.

\[ \text{Sum NCU Cost} = \sum_{i=1}^{k} ({NCU}_i \times \text{Process's NCU cost}) \]

\(\text{Sum Total Cost}\) is the cost to produce the entire sample. That is, the \(\text{Sum NCU Cost}\) if \(\text{Sum NCU}\) were equal to \(\text{Sum Sample Size}\).

\[ \text{Sum Total Cost} = \sum_{i=1}^{k} (n_i \times \text{Process's NCU cost}) \]

Sigma

\(\text{Defect Sigma}\) and \(\text{NCU Sigma}\) are calculated based on the population's \(\text{DPM}\) (for \(\text{Defect Sigma}\)) or \(\text{PPM}\) (for \(\text{NCU Sigma}\)) and the following lookup table.

DPM or PPM Sigma
> 501350 < 1.5
501350 1.5
308770 2
158687 2.5
66810 3
22750 3.5
6210 4
1350 4.5
232.9 5
31.8 5.5
3.4 6
< 3.4 > 6.0

Insights

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Definitions

  • Insights can be generated for the same fields that can be used to group a Pareto chart. For brevity, these fields are referred to as Traceability.
  • Traceability with a type of Integer, Decimal, or Currency are treated as continuous. All other types are referred to as discrete.
  • Insights are automatically calculated for all Traceability that have not been excluded by the Retrieval or the Traceability itself.

Pareto Insights

When a Pareto chart is drawn in a Dashboard, Insights are automatically calculated. The calculations are based on the type of Traceability.

Discrete Traceability

For each discrete Traceability, the data is grouped by that field. Then a one-sided ANOM (Analysis of Means) for Poisson distributions is used to determine whether the largest bar in the chart is significantly larger than the mean of all bars. This is done for both the DPM and Cost per Unit bar methods. The ANOM's upper decision limit determines which groups contain significantly more defects or higher costs per unit than the mean of all other groups. If the bar with the highest percent defects or cost per unit is greater than the upper decision limit, that grouping and bar label are reported.

See The Analysis of Means: A Graphical Method for Comparing Means, Rates, and Proportions (Nelson, Wludyka, and Copeland 2005) for a complete mathematical discussion of ANOMs.

Continuous Traceability

For each continuous Traceability, a linear regression is run on the data records with the Traceability value as the x-axis and the record’s percent defects as the y-axis. If the regression's p-value is less than the Significance Level Global Setting and its \(R^2\) is greater than the Minimum Impact Global Setting, there is a significant linear relationship between the Traceability and the percent defects.

The same analysis is performed for each record’s cost per unit to determine whether continuous Traceability values have an impact on the cost per unit.

Notes

  • Insights are not calculated for Traceability that has been used to drill down into the current state of the Pareto. This is because these fields will only have one bar.
  • Insights are not calculated for the Traceability that is currently being used to group the Pareto. Which Tracaebility value results in the largest bar for that grouping should be obvious.
  • Traceability fields that appear on fewer than three data records cannot be analyzed.

Statistical Listing

Stat Description
Cost Per Unit The average cost of a single unit in the data set.
Count Records Total number of records.
Count Records With NCU Number of records with any NCUs.
Count Records With Real Time Failures Number of records with at least one real-time failure.
Count Records With Zero NCU Number of records with zero NCUs.
Count Total Real Time Failures Total number of real-time failures in the retrieval.
Cumulative Yield Product of the yield of each individual process.
Defect Sigma Calculated based on DPM and a lookup table.
DPB Defects per billion units.
DPK Defects per thousand units.
DPM Defects per million units.
Most Recent Date Retrieved Date/Time of the most recent record.
NCU Sigma Calculated based on PPM and a lookup table.
Oldest Date Retrieved Date/Time of the least recent record.
Percent Defects The percentage of sample size made up of defects.
Percent NCU The percentage of the sample size made up of NCUs.
Percent Records With NCU The percentage of records with NCU > 0.
Percent Records with Real Time Failures The percentage of the number of records with at least one real-time failure, out of the total number of records.
Percent Records With Zero Defects The percentage of records with zero defects.
PPB NCUs per billion units.
PPK NCUs per thousand units.
PPM NCUs per million units.
Sum Defect Cost Sum of all Defect costs.
Sum Defects Sum of all Defect counts.
Sum Good Sum of all non-defects (i.e. Sum Sample Size minus Sum Defects).
Sum NCU Sum of all NCUs counts.
Sum NCU Cost Sum of all NCU costs.
Sum Sample Size Sum of all sample sizes.
Sum Total Cost Cost to create the entire sample; the cost if the entire sample were NCUs.
Yield The percentage of the sample that are not NCUs.