Six Sigma explained: what the sigma counts

Sigma as a distance

Every process has a target and tolerance. Say a machine cuts rods with a target length of 10.00 mm, and anything between 9.94 mm and 10.06 mm is acceptable. Those are the lower and upper specification limits (LSL and USL). The rods it actually produces vary around a mean with some standard deviation σ. The sigma level asks: how many of those σ fit between the mean and the nearest limit?

If σ = 0.01 mm and the process is centred on 10.00, the upper limit is 0.06 mm away, which is 6σ. That is a six-sigma process. If σ were 0.02 mm, the same limit would be only 3σ away. Tightening the process, not widening the tolerance, is what raises the sigma level. It is the same idea as a z-score: the distance from the mean to the limit, measured in standard deviations.

Defects per million opportunities

Six Sigma programmes, which grew out of work at Motorola in the 1980s, measure quality in defects per million opportunities (DPMO). An opportunity is any place a defect could occur, so a unit with four critical features has four opportunities.

DPMO = defects ÷ (units × opportunities per unit) × 1,000,000

For example, an inspection of 500 assembled units, each with 4 opportunities, finds 7 defects. There were 500 × 4 = 2,000 opportunities, so DPMO = 7 ÷ 2,000 × 1,000,000 = 3,500. The yield per opportunity is 1 − 0.0035 = 99.65%.

The definition of an opportunity changes the answer, so it has to be fixed before counting. Had the same 7 defects been counted against 10 opportunities per unit instead of 4, the DPMO would fall to 1,400 with no change to the product at all. Defects per unit (here 7 ÷ 500 = 0.014) avoids that problem but cannot compare a simple part with a complex one. Teams that report DPMO usually publish their opportunity counts alongside it for this reason.

The 1.5σ shift and the sigma table

A perfectly centred, normally distributed process at 6σ would put almost nothing outside its limits: about 0.002 defects per million across both tails. The number everybody quotes, 3.4, is far larger, because the standard table assumes the process mean will drift over the long run by up to 1.5σ toward one limit. A "6σ" process is therefore treated as being only 4.5σ from the nearer limit in the long term, and the tail of a normal distribution beyond 4.5σ is 3.4 per million.

In general, a sigma level of k is converted to long-term DPMO using the one-sided tail beyond k − 1.5:

DPMO = [1 − Φ(k − 1.5)] × 1,000,000

The two right-hand columns differ by orders of magnitude, and a lot of confusion comes from mixing them. The first column is the Six Sigma convention; the last is what the normal curve says for a process that never moves. At 3σ, for instance, the centred figure of 2,700 per million is the familiar 99.73% of the empirical rule, while the shifted figure is 66,807.

Going the other way turns a measured DPMO into a sigma level. For the 3,500 DPMO in the example above, the z-value that leaves 0.35% in the upper tail is 2.70. Adding the conventional 1.5 gives a sigma level of 4.20. The process is between four and five sigma.

How sigma level connects to Cp and Cpk

Process capability indices express the same distances as ratios. Cp compares the width of the tolerance with the natural spread of the process, taken as 6σ. Cpk does the same for the nearer side only, so it penalises a process that is off centre.

Cp = (USL − LSL) / 6σ Cpk = min(USL − μ, μ − LSL) / 3σ

For the rod example, with a tolerance of 9.94 to 10.06 mm and σ = 0.01 mm:

Cp is 0.12 ÷ 0.06 = 2.0 in both rows, because Cp ignores where the mean sits. In the shifted row the nearer limit is 10.06 − 10.015 = 0.045 mm away, so Cpk = 0.045 ÷ 0.03 = 1.5. That shifted process leaves 3.4 per million above the upper limit, which is exactly the 6σ row of the sigma table. The rule of thumb connecting the two scales is sigma level = 3 × Cpk measured on the nearer side; with the shift convention it is 3 × Cpk + 1.5. You can run your own numbers through the Cpk calculator.

Try it: process capability calculator

The calculator below opens with a sample run of 30 rod measurements against limits of 9.93 and 10.07 mm; enter your own measurements and specification limits to get Cp, Cpk, Pp, Ppk and the expected parts per million outside the limits.

Open the full process capability index calculator for the within and overall SD estimates explained.

Where the arithmetic can mislead

All of these tables assume normally distributed output. A process with a skewed distribution or occasional gross errors will have a very different tail, and tail probabilities in the parts-per-million range are exactly where the normal assumption is least trustworthy. The σ also needs to be estimated from enough data collected while the process is stable; a Cpk worked out from 20 parts carries a lot of uncertainty.

Finally, a sigma level is a statement about the process, not about the product. A process can be six sigma on a feature that customers do not care about and three sigma on the one that causes returns. Choosing which opportunities to count matters as much as counting them correctly.