Central Limit Theorem -- Dickeson
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.
Looking at Web Speeds
Above is a chart of net running speed samples of a web offset press applying Central Limit Theory. It has the six-sigma control range plotted in as the UCL and LCL. We predict, with 99.7 percent probability, that the net speed of all runs on this press will fall between 28.86 and 20.16 thousand impressions per hour (iph). Any run faster or slower than these limits is due to special causes that need investigation.
Why is this significant for our industry? We simply have to get a better grip on managing variation. Right now, it manages us.
W. Edwards Deming taught us: Reduce variance to manage productivity and quality. In the chart the difference between the upper and lower control limits—the range—is 8,700 impressions per hour. Great golly, Molly! To predict with 99.7 percent validity, the best we can say is that this press will produce a job at a net speed somewhere between 20 and 29,000 iph. If you're satisfied with 95 percent predictive accuracy you can narrow the range to 5,800 iph. Cut to 68 percent accuracy and you're within 2,900 iph.
How accurate do you want to be? Pick a number from the statistical table. If we want to manage the variance, we must continuously narrow that range of variance.