VariableVariable - a single quality characteristic - a single quality characteristic that can be measured on a numerical scale.that can be measured on a numerical scale.

When working with variables, we should When working with variables, we should monitor both the monitor both the meanmean value of the value of the characteristic and the characteristic and the variabilityvariability associated associated with the characteristic.with the characteristic.

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

Types of variationTypes of variation Within the pieceWithin the piece Piece to piecePiece to piece Time to timeTime to time

Types of variation causes:Types of variation causes: Chance (Natural/Common) causes.Chance (Natural/Common) causes. Assignable (Special) cause.Assignable (Special) cause.

Chance causesChance causes NaturalNatural ExpectedExpected NumerousNumerous Small importanceSmall importance Difficult to detect or identifyDifficult to detect or identify

In a state of statistical control.In a state of statistical control.

Assignable causesAssignable causes Large in magnitudeLarge in magnitude Easy to detect or identifyEasy to detect or identify

Out of control.Out of control.

Control Charts Procedures:Control Charts Procedures:1.1. Select quality characteristicSelect quality characteristic

2.2. Choose rational subgroupChoose rational subgroup

3.3. Collect dataCollect data

4.4. Determine trial central line and control Determine trial central line and control limitslimits

5.5. Establish the revised central line and Establish the revised central line and control limitscontrol limits

6.6. Achieve objectiveAchieve objective

X bar chart monitors the X bar chart monitors the betweenbetween sample sample variabilityvariability

R chart monitors the R chart monitors the withinwithin sample sample variability.variability.

Guidelines on size:Guidelines on size: With larger subgroups, the control chart With larger subgroups, the control chart

becomes more sensitive to small variationbecomes more sensitive to small variation With larger subgroups, the inspection cost With larger subgroups, the inspection cost

per subgroup increasesper subgroup increases If destructive testing is required, a minimal If destructive testing is required, a minimal

number is beneficialnumber is beneficial Statistically, subgroups of 4 or more will Statistically, subgroups of 4 or more will

have their averages normally distributed have their averages normally distributed regardless of their population distributionregardless of their population distribution

Subgroup of 5 are widely used in industrySubgroup of 5 are widely used in industry

Control Limits for the X-bar chartControl Limits for the X-bar chart

AA22 is found in Appendix VI for various values of is found in Appendix VI for various values of n.n.

RAxLCL

xLineCenter

RAxUCL

2

2

Control Limits for the R chartControl Limits for the R chart

DD33 and D and D44 are found in Appendix VI for various are found in Appendix VI for various values of values of nn..

UCL D R

Center Line R

LCL D R

4

3

Estimating the Process Standard DeviationEstimating the Process Standard Deviation The process standard deviation can be estimated The process standard deviation can be estimated

using a function of the sample average range.using a function of the sample average range.

This is an unbiased estimator of This is an unbiased estimator of

R

d2

Trial Control LimitsTrial Control Limits If the process is in control for the m If the process is in control for the m

samples collected, then the system was in samples collected, then the system was in control in the past.control in the past.

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

Control Limits, Specification Limits, and Natural Control Limits, Specification Limits, and Natural Tolerance LimitsTolerance Limits

Control limitsControl limits are functions of the natural are functions of the natural variability of the processvariability of the process

Natural tolerance limits Natural tolerance limits represent the represent the natural variability of the process (usually natural variability of the process (usually set at 3-sigma from the mean)set at 3-sigma from the mean)

Specification limits Specification limits are determined by are determined by developers/designers.developers/designers.