Category Archives: Statistics

Xbar-s chart

Control charts: a lesson in variation

A manufacturing plant has two machine operators with different styles. At shift start, both would carefully set up the machine and begin making parts. Operator A would measure a sample of parts each half hour and tune the machine accordingly. If the mean diameter was .002” oversize, Operator A would adjust the machine to cut .002” smaller. Operator B measured and adjusted the machine only when it was restarted (e.g. after maintenance, breaks, and lunch). The plant manager noticed the different approaches and measured a sample of parts made by each operator. Operator A’s parts showed more variation. Yes, methodical Operator A was making poorer parts.

Understanding the sources of variation explains this surprising outcome. The X-bar chart  plots the average and standard deviation of the diameters of five consecutive parts taken each half hour.

Xbar-s chart

These charts represent a simple and elegant use of statistics and logic. Manufacturing processes have two primary sources of variation, common and special causes. Common variation is inherent in the system; the only way to improve it is to get a new system (e.g. buy a better machine). The rest of the variation is due to special causes, which can be controlled.

The insight at the heart of control charts is that variation within subgroups is due to common causes while variation between subgroups is special cause variation. Consecutive parts minimize special cause variation. They are made from adjacent sections of raw material, by the same operator in a similar frame of mind, at similar ambient and coolant temperatures and with the machine near the same maintenance level.

The control chart software plots the mean of each sample and uses the within subgroup variation as the estimate of common variation. These estimates are used to calculate control limits such that points outside the control limits are a reliable indication that the process should be studied and adjusted.. If a sample mean falls outside the control limits, adjust the machine. If it falls inside the limits, it is in the range of normal machine variation; leave the process alone. The chart indicates that the machine was “in control” the entire 10 hour run. No adjustments needed, yet Operator A adjusted the machine 20 times. Special cause variation is not reduced by adjusting the machine. In fact, unnecessary machine adjustments are another source of special cause variation. Operators need to know when a sample indicates that the machine is no longer running well. The control limits (0.995 and 1.005 for mean, 0.0078 for standard deviation) provide the needed screening.

It is not enough to do your best; you must know what to do, and then do your best. — W. Edwards Deming