What it means
The NIST/SEMATECH e-Handbook of Statistical Methods explains that a control chart has a centre line for the mean of an in-control process, plus an upper control limit and a lower control limit. The limits are chosen so almost all points fall inside them while the process stays in control.
The handbook says the usual multiple is 3, which is why they are called 3-sigma limits. It adds that in the US it is an acceptable practice to base limits on a multiple of the standard deviation, whether or not the data are normally distributed.
The term is used whether the standard deviation is a population value, an estimate or a standard value. For normal data the link to probability is simple.
NIST notes that three standard deviations in one direction leaves 0.00135 in the tail, or 0.0027 in both directions, which makes 3-sigma limits the practical equivalent of 0.001 probability limits. A point outside them signals that an assignable cause may be at work.
The handbook calls the choice of limits a judgment about risk. It says two in a thousand is a purely arbitrary number, and the right level depends on how much risk the quality program will accept.
It also notes that statisticians in the UK generally prefer probability limits. Skewed data change the picture.
NIST gives a Poisson example with np = 0.8, where the upper 3-sigma limit is 0.8 + 3 x sqrt(0.8) = 3.48 and the lower limit is 0. The chance of exceeding the upper limit is then about 0.009, not 0.001.
Staying inside the limits does not prove control. NIST says that if points show a pattern, such as 25 of 30 above the centre line, something may still be wrong, because in control means all points are between the limits and form a random pattern.
Treat a signal outside the limits as a prompt to investigate, not as proof of a cause.
In practice
Real-world examples.
Example
A fictional bottling line has a mean fill of 500 ml and a standard deviation of 2 ml. The 3-sigma limits are 500 + 6 = 506 and 500 - 6 = 494. A bottle measured at 508 ml falls outside the limits and triggers an investigation of the filling nozzle and the supply pressure.
Example
A fictional clerk tracks daily payment errors with a mean of 0.8 and treats them as Poisson counts. The upper limit is 0.8 + 3 x sqrt(0.8) = 3.48, and the lower limit is 0. A day with 4 errors is outside the limit, so the clerk reviews that day's batch for a common cause such as a changed file format.
Example
A fictional process has 25 of its last 30 points above the centre line, though none is outside the limits. The pattern is not random, so the team investigates even though no single point crossed a limit. They find that a new supplier's material has quietly shifted the average upwards.
Formula
Calculation
UCL = mean + 3 x standard deviation.
LCL = mean - 3 x standard deviation.
Example: mean 100 and standard deviation 2 give UCL = 100 + 6 = 106 and LCL = 100 - 6 = 94. For normal data the chance of a point outside both limits is about 0.0027.
A second check shows how the limits are used. A fictional invoice-processing team has a mean cycle time of 250 minutes and a standard deviation of 5 minutes, so UCL = 250 + 15 = 265 and LCL = 250 - 15 = 235. A reading of 262 sits inside the limits and is treated as normal variation, while a reading of 268 falls outside and prompts a review. With a 0.0027 chance per point, a false alarm would appear roughly once in every 370 points, because 1 / 0.0027 is about 370.Case study
Seen in the real world.
This case study is fictional and illustrative. A small plant packs bolts into boxes. Long run data show a mean box weight of 100 and a standard deviation of 2, in the same units. The quality lead sets 3-sigma limits at 106 and 94 on a control chart.
For weeks the points scatter between the limits. One day a box weighs 108, so the lead looks for a cause and finds a worn scale. Later the points stay inside the limits but 12 in a row fall above the centre line. The lead treats the run as a warning of a drift.
A fix to the filling machine restores the pattern. The lesson is that the limits catch large shifts, while patterns catch slow ones. Use both and investigate rather than assume.
Watch out
Common mistakes.
- Assuming all points inside the limits means the process is fine, when a non-random pattern can still signal a problem.
- Assuming 3-sigma always means 0.0027, when that holds for normal data and skewed data give different risks.
- Treating the tail risk as fixed, when NIST says where to put the limits is a judgment about how much risk the program will accept.
Questions
People also ask.
What are three-sigma limits?
They are control limits set three standard deviations above and below the centre line of a control chart.
Why three sigma?
It is the customary multiple. For normal data it gives about a 0.0027 chance of a point outside by chance, which keeps false alarms rare.
Do they work for skewed data?
Less well. NIST's Poisson example shows the upper limit risk rising to about 0.009.
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