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Central Limit Theorem -- Dickeson

November 2001
Production standards are predictions of the amount of time and materials required for a form or job for purchasing, estimating, pricing and scheduling. Predictability is essential for process management of printing. Is the process of printing so nearly chaotic that time and materials can be forecast no better than next week's weather? Or can production time and materials be predictable—at least within ranges?

The Central Limit Theorem (CLT) of statistics tells us that the averages of a group of individual samples will be approximately "normally" distributed. (CLT is the theory used by Deming/Shewhart in charting control limits for product variances.) For printing it can be applied to activities that constrain production throughput, such as time required for makereadies (machine changeovers) and net machine speeds. It can, of course, be applied to other activities, as well. Isn't this SPC (Statistical Process Control)? No, not as we usually think of that system. We're not measuring properties of physical products but, instead, "activities" that consume time and materials. We're adapting a technique from CLT and SPC for print production needs.

CLT involves statistical "sampling" and "probability." Of course there are a few statistical terms that need to be mastered. A "sample" is a set of data "records." The "mean" (average) of a sample is called the "X Bar." The mean of a group of sample means is the X Bar Bar. The standard deviation of X Bars from the X Bar Bar is termed a "sigma."

Ideally, a sample should consist of 6-10 consecutive records, and there should be 20 or more sample sets. If we had sample sizes of six for 20 sample-sets we'd have 120 records in total represented by 20 X Bars and one X Bar Bar. The spreadsheet application on our computer could then calculate a standard deviation—a sigma.

The X Bar Bar plus three sigmas establishes the Upper Control Limit (UCL) and X Bar Bar minus three sigmas the Lower Control Limit (LCL). The three sigmas above and three below—six sigmas total—is the control range. It'll include 99.7 percent of all variations due to chance. This assumes a "normal" distribution of X Bars. (When charted, the X Bars should roughly form a bell-shaped normal curve.) Slightly more than 95 percent of all variations will be within plus or minus two sigmas, and 68 percent within plus or minus one sigma.



 

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