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Free sample · ASQ CSSGB

Try a full lesson: Capability and DPMO to sigma level on a transactional process

This is a complete lesson from the course (module 3 of 6, Outline section III.A, III.B, III.C, III.D, III.E, III.F), followed by 10 practice questions with full explanations. Nothing to sign up for.

What this lesson covers

Capability and sigma level questions sit in two places on the ASQ CSSGB exam: process performance and capability in the Measure Phase (20 of the 100 scored questions) and the business results of projects in the Define Phase (also 20). Both reward the same few formulas, used in the right order. Many candidates learn them on machined parts, but the exam's scenarios are just as likely to be about invoices, claims or calls. This free lesson works one office process from raw summaries to a capability index, a defect rate and a sigma level, and then lists the traps.

The two questions a capability study answers

  1. For a measured characteristic (days, minutes, dollars): how does the spread and center of the process compare with the customer's limit? That is an index such as Cpk or Ppk, and from it an expected share outside the limit.
  2. For counted defects (wrong codes, missing fields): how often does a defect occur per opportunity? That is DPMO, and from it a sigma level.

The two routes meet at the same scale: a z-value, the number of standard deviations between the process and the limit.

The worked example (invented)

An accounts payable team processes supplier invoices. The customer for this process is the supplier, who has payment terms of 10 days from receipt. There is no lower limit: paying early is never a defect.

The team collected one month of data:

  • Days from receipt to payment: mean 6.4 days. The within-subgroup standard deviation, estimated from the moving ranges of a stable individuals chart, is 0.9 days. The overall standard deviation of all the invoices is 1.2 days.
  • Errors: 4,000 invoices were audited. Each has 5 opportunities for an error (vendor ID, purchase order match, amount, tax code, due date). The audit found 86 errors, spread over 79 invoices.

Step 1: capability for the time measure

With an upper limit only, Cp and Pp cannot be computed (they need both limits). Use the one-sided indices.

  • Cpk (Cpu) = (USL - mean) / (3 x sigma_within) = (10 - 6.4) / (3 x 0.9) = 3.6 / 2.7 = 1.33.
  • Ppk (Ppu) = (USL - mean) / (3 x sigma_overall) = 3.6 / (3 x 1.2) = 3.6 / 3.6 = 1.00.
  • Check through z: the limit sits 3.6 / 0.9 = 4.0 within-sigmas above the mean (4.0 / 3 = 1.33) and 3.6 / 1.2 = 3.0 overall sigmas above it (3.0 / 3 = 1.00).

Reading it: in any short stretch the process looks comfortable, but over the month the mean moves enough to pull performance down to 1.00. That gap is the clue to look for what changes between days or weeks (month-end peaks, staffing).

Step 2: the share of late payments

The customer feels the long-term performance, so use the overall sigma. z = 3.0, and the area beyond z = 3.00 is given as 0.00135.

  • Late payments are expected at about 1,350 per million invoices (0.135%).
  • Check: 0.00135 x 1,000,000 = 1,350, and 6.4 + 3.0 x 1.2 = 10.0, the limit.

Reported with the 1.5 shift convention, a long-term z of 3.0 is a sigma level of 3.0 + 1.5 = 4.5. The measured short-term z is 4.0, so this process drifts by about 1.0 sigma, not 1.5. Say which figure you report.

Step 3: DPU, DPMO and first-pass yield for the errors

  • DPU = 86 / 4,000 = 0.0215 errors per invoice.
  • DPO = 86 / (4,000 x 5) = 86 / 20,000 = 0.0043. DPMO = 4,300.
  • Check: DPU / 5 = 0.0215 / 5 = 0.0043, and 4,300 x 0.02 = 86 (20,000 opportunities is 0.02 million).
  • First-pass yield (invoices with no error) = (4,000 - 79) / 4,000 = 3,921 / 4,000 = 98.0%.
  • The Poisson estimate e^(-DPU) = e^(-0.0215), given as 0.9787, is 97.9%. The real yield is a little higher because some invoices held two errors: errors clustered instead of falling at random.

Step 4: DPMO to sigma level

Treat the DPO as a long-term tail area. The upper-tail area 0.0043 is given as matching z = 2.63.

  • Long-term z = 2.63. Sigma level with the 1.5 shift = 2.63 + 1.5 = 4.13, about 4.1 sigma.
  • Check the direction: a smaller DPMO must give a larger z. A DPMO of 1,350 matched z = 3.0 in step 2, and 4,300 is worse, so its z (2.63) is lower. It is.

Now the two halves of the process can be compared on one scale: timeliness at about 4.5 sigma (by convention) and accuracy at about 4.1. Accuracy is the weaker link, so the next project should target the invoice errors, starting with a Pareto chart of the five opportunity types.

Common traps

  • Computing Cp with one limit. No lower limit means no tolerance width. Report Cpu or Ppu.
  • Using within sigma for the customer's defect rate. The customer lives with the long-term spread; use overall sigma for expected late payments.
  • Mixing up DPU and DPMO. DPU is per invoice; DPMO divides by opportunities and scales to a million. 0.0215 errors per invoice is not 21,500 DPMO.
  • Counting defective invoices as defects. 79 invoices held 86 errors. DPU and DPMO use 86; first-pass yield uses 79.
  • Adding the shift twice, or subtracting it. Long-term z plus 1.5 gives the reported short-term sigma level, once.
  • Inventing opportunities. Count only opportunities that are real, checked and agreed in advance. Padding the count makes DPMO look better without changing a single error.

Try it

The free practice set has 10 questions on this topic: DPU, DPMO and sigma level for office processes, Poisson yield, one-sided and two-sided capability, the share beyond a limit, and what a gap between Cp and Pp means. Each answer explanation works the calculation twice.

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Practice set

Print these questions for a class (PDF) Answer key on page 2. Free, no signup.

Pick an answer, then check it. Every wrong option is explained too.

1. Define Phase

A payroll team processes 2,500 expense claims. Each claim has 5 opportunities for a defect, and the team finds 75 defects. What is the DPU?

2. Define Phase

A purchasing team issues 1,800 purchase orders. Each order has 6 opportunities for a defect, and an audit finds 27 defects. What is the DPMO?

3. Define Phase

A process runs at a long-term 2,500 DPMO. An upper-tail area of 0.0025 corresponds to z = 2.81. Using the convention of adding a 1.5 sigma shift, what short-term sigma level is reported?

4. Define Phase

A claims process runs at 0.12 defects per unit, with defects spread at random (Poisson). Given e^(-0.12) = 0.8869, about what share of claims have no defects at all?

5. Measure Phase

A lender promises a decision within 5.0 days (an upper limit only). Decision times have a mean of 3.2 days and an overall standard deviation of 0.5 days. What is Ppk?

6. Measure Phase

A team computes capability for days to pay a supplier invoice, which has an upper limit of 10 days and no lower limit. Why can it report Cpk but not Cp?

7. Measure Phase

A payroll file must reach the bank between 24 and 48 hours before payday. Send times have a mean of 40 hours before payday and a within-subgroup standard deviation of 2.0 hours. What is Cpk?

8. Measure Phase

Time to answer a customer email has a mean of 15.0 minutes and an overall standard deviation of 2.5 minutes, with an upper limit of 22.5 minutes. The area beyond z = 3.00 is 0.00135. About how many emails per million exceed the limit?

9. Measure Phase

An invoice process has a Ppk of 1.10, set by its nearer limit, and almost nothing falls outside the far limit. Using the convention that adds a 1.5 sigma shift, what sigma level does it report?

10. Measure Phase

For a loan-processing time, Cp is 1.50 but Pp is only 0.95. What does the gap most likely mean?

0 of 10 checkedCheck each answer to see your score.

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