Sigma Level: What It Is, How to Calculate DPMO, Minitab Use

What is the sigma level

The sigma level is a measure of how well a process meets specifications. The higher the level, the fewer defects per million opportunities. The Greek letter sigma represents the standard deviation, and the sigma level counts how many standard deviations fit between the process mean and the nearest specification limit.

The metric is the common language of Six Sigma. It allows very different processes, such as a machining line and an administrative process, to be compared on the same scale. For the context of capability with continuous data, also see Cp and Cpk.

How to calculate DPMO

DPMO stands for defects per million opportunities. Three concepts help before it. DPU is the number of defects per unit. DPO is the number of defects per defect opportunity. DPMO is DPO multiplied by one million.

DPO = D ÷ (U × O)    DPMO = DPO × 1,000,000

In this formula, D is the total number of defects found, U is the number of units inspected and O is the number of defect opportunities per unit. An opportunity is a point where a defect can occur and be checked independently. Defining opportunities well is the most delicate part: if they are invented to inflate the denominator, the result loses meaning.

Illustrative example with the math checked

Illustrative scenario: an assembly station inspects 2,000 boards. Each board has 5 defect opportunities. The inspection finds 38 defects in total.

1. Total opportunities: 2,000 × 5 = 10,000.
2. DPO = 38 ÷ 10,000 = 0.0038.
3. DPMO = 0.0038 × 1,000,000 = 3,800.
4. Yield (proportion without defects per opportunity) = 1 − 0.0038 = 0.9962, or 99.62%.
5. DPU = 38 ÷ 2,000 = 0.019 defects per board.

To get the sigma level, look for the z value that leaves 0.0038 of the area in the upper tail of the standard normal distribution. That value is approximately 2.67. Adding the conventional shift of 1.5, the sigma level is about 4.17. This places the process between levels 4 and 5 of the table.

Sigma level table

The usual sigma level table incorporates a 1.5 sigma shift of the process mean in the long term. This shift is a convention adopted in Six Sigma practice, not a law of nature. It assumes that, over time, the process mean drifts relative to the target, and for that reason the short-term level is expressed as 1.5 more than the calculated z.

Without the shift convention, six sigma would correspond to defects on the order of parts per billion. The famous 3.4 per million comes precisely from the 1.5. Always state which criterion you use when comparing results.

Spreadsheet formula

In spreadsheets, the sigma level comes from the inverse of the standard normal distribution. In Excel and Google Sheets, the function is NORM.S.INV (older Excel versions use NORMSINV). Check the name in your version.

Sigma level = NORM.S.INV(1 − DPMO ÷ 1,000,000) + 1.5

With the example above, the cell with a DPMO of 3,800 gives 1 − 0.0038 = 0.9962. The function returns about 2.67, and with the shift the result is about 4.17, matching the manual calculation. To work without the shift, just omit the 1.5 and state that in the report.

General procedure in Minitab

I do not describe exact menu paths, because they change between Minitab versions. The general procedure is this:

1. Organize the data in columns: number of defects and sample size in each subgroup.
2. Open the capability analysis for attribute data (binomial or Poisson, depending on the type of data) in the quality tools group.
3. Enter the defects column and the subgroup size column.
4. Run it and read the DPMO and the sigma level from the report. Check whether the report applies the 1.5 shift or not before comparing with the table.
5. Verify process stability on control charts, such as the p chart, before trusting the result.

A capability analysis is only valid if the process is under statistical control. See SPC and data reliability in MSA.

Precautions when interpreting

• Compare processes using the same definition of opportunity.
• Do not use DPMO with very small samples: one extra defect changes everything.
• Combine the indicator with the cost of the defect, as in cost of quality.
• For yield across several steps, see FPY and RTY.

Next step

Choose a process with inspection data available. Define the defect opportunities in writing, collect a representative sample, calculate the DPMO with the formula above and convert it to a sigma level in the spreadsheet. Record the date and the criterion used. Repeat the measurement after an improvement and compare using the same criterion. The result becomes the baseline of your improvement project.

Frequently asked questions

What is DPMO?

It is the number of defects per million opportunities, calculated as the total number of defects divided by the product of units and opportunities per unit, times one million.

Why does six sigma equal 3.4 defects per million?

Because the usual table assumes a conventional 1.5 sigma shift in the process mean. Without it, the value would be much lower.

What is the sigma level formula in a spreadsheet?

The inverse of the standard normal of (1 − DPMO ÷ 1,000,000), plus 1.5 when the conventional shift is adopted.

Does the sigma level work for administrative processes?

Yes. Just define the defect, such as a data entry error, and the error opportunities per document.

Does Minitab already apply the 1.5 shift?

It depends on the report and the version. Check the definition used in the output before comparing with the conventional table.

About the author

Vagner Soares

Lean Manufacturing & Behavioral Management Specialist

Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.

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