 Variable - a single quality characteristic that can be measured on a numerical scale.  When working with variables, we should monitor both the mean value of the characteristic and the variability associated with the characteristic.

“One of the axioms or truisms of manufacturing is that no two objects are ever made exactly alike”

Types of variation Types of variation  Within the piece  Piece to piece  Time to time Types of variation causes: Types of variation causes:  Chance (Natural/Common) causes.  Assignable (Special) cause.

Chance causes Chance causes  Natural  Expected  Numerous  Small importance  Difficult to detect or identify In a state of statistical control. In a state of statistical control.

Assignable causes Assignable causes  Large in magnitude  Easy to detect or identify Out of control. Out of control.

Control Charts Procedures: 1. Select quality characteristic 2. Choose rational subgroup 3. Collect data 4. Determine trial central line and control limits 5. Establish the revised central line and control limits 6. Achieve objective

X bar chart monitors the between sample variability X bar chart monitors the between sample variability R chart monitors the within sample variability. R chart monitors the within sample variability.

Guidelines on size: Guidelines on size:  With larger subgroups, the control chart becomes more sensitive to small variation  With larger subgroups, the inspection cost per subgroup increases  If destructive testing is required, a minimal number is beneficial  Statistically, subgroups of 4 or more will have their averages normally distributed regardless of their population distribution  Subgroup of 5 are widely used in industry

Control Limits for the X-bar chart A 2 is found in Appendix VI for various values of n. A 2 is found in Appendix VI for various values of n.

Control Limits for the R chart D 3 and D 4 are found in Appendix VI for various values of n. D 3 and D 4 are found in Appendix VI for various values of n.

Estimating the Process Standard Deviation The process standard deviation can be estimated using a function of the sample average range. The process standard deviation can be estimated using a function of the sample average range. This is an unbiased estimator of  This is an unbiased estimator of 

Trial Control Limits If the process is in control for the m samples collected, then the system was in control in the past. If the process is in control for the m samples collected, then the system was in control in the past. If all points plot inside the control limits and no systematic behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production. If all points plot inside the control limits and no systematic behavior is identified, then the process was in control in the past, and the trial control limits are suitable for controlling current or future production.

Control Limits, Specification Limits, and Natural Tolerance Limits Control limits are functions of the natural variability of the process Control limits are functions of the natural variability of the process Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean) Natural tolerance limits represent the natural variability of the process (usually set at 3-sigma from the mean) Specification limits are determined by developers/designers. Specification limits are determined by developers/designers.