**Q:** I would appreciate some advice on how I can fairly assess process variation for metrics derived from “discrete” variables over time.

For example, I am looking at “unit iron/unit air” rates for a foundry cupola melt furnace in which the “unit air” rate is derived from the “continuous” air blast, while the unit iron rate is derived from input weights made at “discrete” points in time every 3 to 5 minutes.

The coefficient of variation (CV), for the air rate is exceedingly small (good) due to its “continuous’ nature” but the CV for iron rate is quite large because of its “discrete nature,” even when I use moving averages for extended periods of time. Hence, that seemingly large variation for iron rate then carries over when computing the unit iron/unit air rate.

I think the discrete nature of some process variables results in unfairly high assessments of process variation, so I would appreciate some advice on any statistical methods that would more fairly assess process variation for metrics derived from discrete variables.

**A:** I’m not sure I fully understand the problem, But I do have a few assumptions and possibly a reasonable answer for you. As you know, when making a measurement, using a discrete scale (red, blue, green; on/off, or similar), the item being measured is placed into one of the “discrete” buckets. For continuous measurements, we use some theoretically infinite scale to place the units location on that scale. For this latter type of measurement, we are often limited by the accuracy of the equipment to the level of precision the measurement can be accomplished.

In the question, you mention measurements of air from the “continuous” air blast. The air may be moving without interruption (continuously), yet the measurement is probably recorded periodically unless you are using a continuous chart recorder. Even so, matching up the reading with the unit iron readings every 3 to 5 minutes, does create individual readings for the air value. The unit iron reading is a “weights” based reading (not sure what is meant by derived, yet let’s assume the measurement is a weight scale of some sort.) Weight, like mass or length, is an infinite scale measurement, limited by the ability of the specific measurement system to differentiate between sufficiently small units.

I think you see where I’m heading with this line of thought. The variability with the unit iron reading may simply reflect the ability of the measurement process. I do not think either air rate or unit iron (weight based) is a discrete measurement, per se. Improve the ability to measure the unit iron and that may reduce some measurement error and subsequent variation. Or, it may confirm that the unit iron is variable to an unacceptable amount.

Another assumption I could make is that the unit iron is measured for the batch that then has unit air rates regularly measured. The issue here may just be the time scales involved. Not being familiar with the particular process involved, I’ll assume some manner of metal forming, where a batch of metal is created then formed over time where the unit air is important. And, furthermore, assume the batch of metal takes an hour for the processing. That means we would have about a dozen or so readings of unit air for the one reading of unit iron.

If you recall, the standard deviation formula is divided by square root of n (number of samples). In this case, there is about a 10 to 1 difference in n (10 for unit air to one for unit iron). Over many batches of metal, the ratio of readings remains at or about 10 to 1, thus impacting the relative stability of the two coefficient of variations. Get more readings for unit iron or reduce the unit air readings, and it may just even out. Or, again, you may discover the unit iron readings and underlying process is just more variable.

From the information provided, I think this provides two areas to conduct further exploration. Good luck.

Fred Schenkelberg

Voting member of U.S. TAG to ISO/TC 56

Voting member of U.S. TAG to ISO/TC 69

Reliability Engineering and Management Consultant

FMS Reliability

www.fmsreliability.com