Quality Calculations

How to Calculate Quality and Inspection Metrics: FPY, RTY, Sigma, and Sampling

A worked walkthrough of the five core quality formulas: yield and first pass yield, defect rate as DPMO, rolled throughput yield, sigma level, and sampling size.

Start with yield, the simplest ratio. Yield equals good units divided by units started. If you start 1,000 parts and 940 pass final inspection, yield is 940 divided by 1,000, or 94.0 percent. First pass yield is stricter: it counts only units that passed the first time with no rework. If 60 of those 940 good parts needed a touch-up, first pass yield is 880 divided by 1,000, or 88.0 percent. Use the Yield and First Pass Yield calculators to keep the two separate, because reporting rework-inflated yield as FPY is the most common error in quality dashboards.

Defect rate is best expressed as defects per million opportunities, or DPMO, so parts of different complexity compare fairly. DPMO equals defects divided by (units times opportunities per unit), times 1,000,000. Suppose you inspect 500 circuit boards, each with 200 solder joints (opportunities), and find 15 defects. That is 15 divided by (500 times 200), which is 15 divided by 100,000, times 1,000,000, giving 150 DPMO. The Defect Rate calculator handles this. Note the difference between defects and defectives: one board with three bad joints is three defects but one defective unit.

Rolled throughput yield captures what a multi-step process actually delivers end to end. RTY is the product of each step's first pass yield, not the average. For a four-step line running FPYs of 0.98, 0.95, 0.97, and 0.99, RTY equals 0.98 times 0.95 times 0.97 times 0.99, which is 0.894, or 89.4 percent. Averaging those four numbers gives a misleading 97.25 percent. The Rolled Throughput Yield calculator multiplies the chain for you. RTY falls fast as steps multiply, which is why a 20-step process with 99 percent per step still only yields about 81.8 percent.

Sigma level converts DPMO into a standard capability score, assuming the common 1.5 sigma long-term shift. A process at 6,210 DPMO sits near 4.0 sigma; 233 DPMO is about 5.0 sigma; and the famous 3.4 DPMO is 6.0 sigma. To go the other way, first compute yield per opportunity: 1 minus (DPMO divided by 1,000,000). For 6,210 DPMO that is 0.99379 yield, which maps to roughly 4.0 sigma on the shifted table. The Sigma Level calculator does the inverse-normal lookup so you do not need Z-tables. Track this monthly, not per shift, since small samples make sigma jump.

Inspection sampling tells you how many parts to check and when to accept a lot. Under an acceptance sampling plan, you draw a sample of size n from a lot of size N and accept if defects are at or below the acceptance number c. For a lot of 5,000 at general inspection level II and an AQL of 1.0 percent, a typical plan is n equals 200 with c equals 5: accept the lot if 5 or fewer sample parts are defective, reject at 6. The Inspection Sampling calculator returns n and c from lot size and AQL so you are not guessing sample counts.

Warranty return rate closes the loop with field data. It equals returned units divided by units shipped in a defined window, times 100. Ship 40,000 units, take 260 warranty returns in 12 months, and your rate is 260 divided by 40,000, or 0.65 percent. Always fix the denominator to a cohort by ship date, because comparing this month's returns against this month's shipments understates the rate for products with a lag to failure. The Warranty Return Rate calculator anchors returns to the correct shipment cohort so early-life and wear-out failures are not blended.

Watch your units and opportunity definitions, because they silently change every downstream number. Opportunities per unit must be fixed and documented: if one team counts 200 solder joints and another counts 50 component placements, their DPMO figures are not comparable even on identical boards. Yield percentages should carry the same time base and the same pass criteria. When you feed FPY into RTY, confirm each step's FPY excludes rework, or the rolled number will read high. A quick sanity check: RTY should always be lower than the lowest single-step FPY in the chain.

Tie the formulas together with one part flow to see how they stack. Start 1,000 units, scrap 30 at incoming, rework 50 mid-line, and reject 20 at final. Yield is 950 divided by 1,000, or 95.0 percent, but first pass yield is 900 divided by 1,000, or 90.0 percent because the 50 reworked units do not count. If those 1,000 units carry 100 opportunities each and you logged 140 defects total, DPMO is 140 divided by 100,000 times 1,000,000, or 1,400 DPMO, roughly 4.5 sigma. Run each number through its calculator and the chain stays consistent.

Published 2026-07-01.