(No.6) Measurement and Error
Printer prices are falling nearly daily while printing speed continues to rise. This trend has turned the printer into a kind of display device. I paid about 20,000 yen for my printer eight years ago. I have paid several times over that for ink over the years. Ink is a precious material, running in the hundreds of yen per gram.
Ink and toner come packaged in their own containers. Net weight of the contents must be equal to or more than the listed weight. If not, the maker can expect user claims. In the manufacturing process, these powder or liquid contents are packed and weighed. Powder contents are poured from a hopper into small package containers (Fig.1). When the measured weight reaches the expected mean value the system stops pouring.
With products made in this way, statistical variance of measurement reveals that half of the containers packed will fall below the expected weight μ. To avoid possible user claims, the contents are adjusted to μ+kσ, where σ is the standard deviation, and k is a number. When k=2, for example, about 98% of the containers have more than µ. Even so, a small number of users will complain to the ink maker. In handling these cases, the maker may swap a new container for the old one, in hopes of satisfying the customer. In any case, the maker ends up paying the additional cost of kσ for the additional content. The value of σ is proportional to the additional cost.
How can we reduce σ? It is well known that standard deviation is inversely proportional to the root of the number of measurements n. If n=9, for example, the additive cost will be 1/3 the original cost.
About twenty years ago, a scale manufacturer asked my advice on a patent application for their new high precision scale procedure.
Following is the procedure:
- Prepare n+m number of small-scale containers.
- After weighing every small scale container, take out n number of scale containers whose total weight is the same or closest to the expected weight Then pour all the selected containers into a packaging container cassette.
In the above procedure, the number of possible conbinations is expressed as m+nCn, which is usually a very large number. The expected error will be reduced to the inverse of the root of the combination, if all the combinations are independent of each other. This procedure requires many computations, but with today's computers, computation cost is almost negligible.
When powder is weighed and packed into a cartridge, standard deviation , σ , will be reduced by splitting the weighing process into n number of small scales.;
, where σ0 is the original standard deviation.
As we saw above, statistics offers elaborate ideas for overcoming possible errors with a limited number of samples. The Bootstrap methods introduced recently and often used for Pattern Classification or Learning Theory closely resemble this method. For more information, refer to Pattern Classification by Duda, Hart and Stork; Wiley Interscience, 1999.
Ink cartridges on the market do not indicate net weight. This may be a clever business model to avoid user claims.