Sigma Level Calculator

How to use the sigma level calculator

The Sigma tab accepts as much or as little as you know. The more you enter, the more it reports:

• DPMO only: sigma level, yield and percent defects.
• Defects and total opportunities: the same, with the DPMO worked out for you and shown.
• Defects, number of units and opportunities per unit: the full set, including rolled throughput yield (RTY), percent of defective units and defects per million units (DPM).
• DPMO and opportunities per unit: sigma level, yield, RTY, percent of defective units and DPM, without needing the raw counts.

Units and opportunities per unit are counted in whole numbers, so 2.9 opportunities is read as 2. If you fill in defects, units and opportunities per unit, those counts decide the DPMO and any value typed in the DPMO box is ignored.

The Sample size tab answers a different question: how many measurements are needed to estimate a process to a chosen precision. See the last section below.

What is Six Sigma in process control?

Every process varies, and quality control compares that variation with a specification. A process runs at a sigma level of k when the specification limits sit k standard deviations from the process average. The higher k is, the smaller the share of output that falls outside the limits, which means fewer defects, less scrap and rework, and fewer unhappy customers.

Six Sigma is also the name of a quality-improvement approach that began at Motorola in the 1980s and was later adopted widely in manufacturing and services. It uses the sigma level as a common scale for comparing very different processes: a call centre, a bottling line and an invoicing department can all be put on the same scale once their defects are counted.

Formulas used

total opportunities = units × opportunities per unit

DPMO = defects ÷ total opportunities × 1,000,000

yield % = 100 − DPMO ÷ 10,000

sigma level = z such that 2 × (1 − Φ(z)) = DPMO ÷ 1,000,000Φ is the standard normal cumulative distribution function. The tail is counted on both sides of the average.

RTY = (1 − DPMO ÷ 1,000,000)^{opportunities per unit}

defective units % = (1 − RTY) × 100; DPM = (1 − RTY) × 1,000,000

Yield describes opportunities that came out right. Rolled throughput yield describes whole units that came out right, and it is always lower once a unit has more than one opportunity to fail, because the individual yields multiply.

Worked example

An inspection finds 42 defects in 1,200 invoices, and each invoice has 5 fields that can be wrong. Total opportunities are 1,200 × 5 = 6,000, so DPMO = 42 ÷ 6,000 × 1,000,000 = 7,000.

The yield per opportunity is 100 − 7,000 ÷ 10,000 = 99.30% and the sigma level is about 2.70. For a whole invoice to be error-free, all five fields must be right: RTY = 0.993⁵ ≈ 96.55%, so about 3.45% of invoices contain at least one error, or roughly 34,513 per million.

Note how a 99.3% field accuracy sounds good but still spoils one invoice in thirty. That gap between yield and RTY is why multi-step processes need much tighter control at each step.

Sigma level and defect rate

These values come from the normal distribution with both tails counted and no extra adjustment:

Published Six Sigma charts frequently quote 3.4 DPMO for "six sigma". That figure includes a conventional 1.5 sigma allowance for drift of the process average, so it belongs to a 4.5 sigma process measured on one side only. This calculator does not apply any such shift: 3.4 DPMO is reported as a sigma level of about 4.65. Keep the convention in mind when you compare results with other sources.

Yield across several steps

When a unit passes through several steps, the chance it survives all of them is the product of the step yields. Three steps with yields of 98%, 99% and 97% give 0.98 × 0.99 × 0.97 = 0.9411, an RTY of 94.11%, lower than any single step.

Parts count matters as much. Ten parts that are each made at 3 sigma (99.73% good) yield about 97.33% good assemblies, and a hundred such parts yield only about 76.31%. The same hundred parts at a 99.99966% yield, which corresponds to 3.4 DPMO, give about 99.97%. Enter the DPMO of one part together with the number of parts as opportunities per unit to run these cases.

Sample size for process control

Choosing a sample size trades cost against certainty. For a measurement with a known or assumed standard deviation σ, the number of samples needed to estimate its average to within a given precision is:

n = (2 × z × σ ÷ E)², rounded upz is the critical value for the confidence level; E is the maximum error, the full width of the confidence interval.

Example: with σ = 10 and a maximum error of 2 (an interval of ±1), the 95% level uses z ≈ 1.96, giving (2 × 1.96 × 10 ÷ 2)² ≈ 384.15, so 385 samples are needed. Raising the confidence to 99% pushes this to 664, and halving the allowed error quadruples the requirement.

The standard deviation is usually unknown in advance, so use a value from a pilot run or earlier batches. Related tools: the standard deviation calculator, the confidence interval calculator and the z-score calculator.