HOW TO DEVELOP AND USE X BAR AND R CONTROL CHARTS?
AN EXAMPLE
Example: Control Charts for Variable DataSlip Ring Diameter (cm)
Sample 1 2 3 4 5 X R1 5.02 5.01 4.94 4.99 4.96 4.98 0.082 5.01 5.03 5.07 4.95 4.96 5.00 0.123 4.99 5.00 4.93 4.92 4.99 4.97 0.084 5.03 4.91 5.01 4.98 4.89 4.96 0.145 4.95 4.92 5.03 5.05 5.01 4.99 0.136 4.97 5.06 5.06 4.96 5.03 5.01 0.107 5.05 5.01 5.10 4.96 4.99 5.02 0.148 5.09 5.10 5.00 4.99 5.08 5.05 0.119 5.14 5.10 4.99 5.08 5.09 5.08 0.15
10 5.01 4.98 5.08 5.07 4.99 5.03 0.1050.09 1.15
DETERMINE CENTERLINE
• The centerline should be the population mean, µ• Since it is unknown, we use X Double bar, or the
grand average of the subgroup averages.
m
m
ii∑
== 1X
X
DETERMINE CONTROL LIMITSXbar chart
• The normal curve displays the distribution of the sample averages.
• A control chart is a time-dependent pictorial representation of a normal curve.
• Processes that are considered under control will have 99.73% of their graphed averages fall within 6σ.
UCL & LCL calculation
deviation standard3XLCL
3XUCL
=−=
+=
σσ
σ
DETERMINING THE VALUES OF THE CONTROL LIMITS USING RANGE
m
m
ii∑
== 1R
R
RAXUCL 2+=
RAXLCL 2−=
3-Sigma Control Chart Factors
Sample size X-chart R-chartn A2 D3 D4
2 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.86
DETERMINE CONTROL LIMITSR chart
• The range chart shows the spread or dispersion of the individual samples within the subgroup.– If the product shows a wide spread, then the
individuals within the subgroup are not similar to each other.
– Equal averages can be deceiving.• Calculated similar to x-bar charts;
– Use D3 and D4
CalculationFrom Table above:• Sigma X-bar = 50.09• Sigma R = 1.15• m = 10Thus;• X-Double bar = 50.09/10 = 5.009 cm• R-bar = 1.15/10 = 0.115 cm
Note: The control limits are only preliminary with 10 samples.It is desirable to have at least 25 samples.
CONTROL LIMITSCONTROL LIMITS
• UCLx-bar = X-D bar + A2 R-bar = 5.009 + (0.577)(0.115) = 5.075 cm
• LCLx-bar = X-D bar - A2 R-bar = 5.009 -(0.577)(0.115) = 4.943 cm
• UCLR = D4R-bar = (2.114)(0.115) = 0.243 cm
• LCLR = D3R-bar = (0)(0.115) = 0 cmFor A2, D3, D4: see Table
n = 5
X-bar Chart
4.94
4.96
4.98
5.00
5.02
5.04
5.06
5.08
5.10
0 1 2 3 4 5 6 7 8 9 10 11
Subgroup
X ba
r
LCL
CL
UCL
R Chart
0.00
0.05
0.10
0.15
0.20
0.25
0 1 2 3 4 5 6 7 8 9 10 11
Subgroup
Ran
ge
LCL
CL
UCL
WHAT DO YOU DO NOW ?
• Revise Control Limits ?
• The concept of Trial Control Limits
HOW TO REVISE CONTROL LIMITS ?
• One or two points outside ?
• Quite a few points outside ?
THREE CATEGORIES OF VARIATION
• Within-piece variation– One portion of surface is rougher than another
portion.
• Piece-to-piece variation– Variation among pieces produced at the same
time.
• Time-to-time variation– Service given early would be different from
that given later in the day.
SOURCES OF VARIATION
• Equipment– Tool wear, machine vibration, …
• Material– Raw material quality
• Environment– Temperature, pressure, humadity
• Operator– Operator performs- physical & emotional
CONTROL CHART VIEWPOINT
• Variation due to – Common or chance causes– Assignable causes
• Control chart may be used to discover “assignable causes”
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TYPICAL OUT-OF-CONTROL PATTERNS
• Point outside control limits• Sudden shift in process average• Cycles• Trends• Hugging the center line• Hugging the control limits• Instability
Shift in Process Average
Identifying Potential Shifts
Cycles
8-3 Introduction to Control Charts
8-3.4 Analysis of Patterns on Control Charts
8-3 Introduction to Control Charts
8-3.4 Analysis of Patterns on Control Charts
Trend
Process in Control
• When a process is in control, there occurs a natural pattern of variation.
• Natural pattern has: – About 34% of the plotted point in an imaginary
band between 1σ on both side CL.– About 13.5% in an imaginary band between 1σ
and 2σ on each side of CL.– About 2.5% of the plotted point in an imaginary
band between 2σ and 3σ on both side CL.
The NormalThe NormalDistributionDistribution
-3σ -2σ -1σ +1σ +2σ +3σMean
68.26%95.44%99.74%
σ = Standard deviation
LSL USL
-3σ +3σCL
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Control Chart Design Issues
• Basis for sampling• Sample size• Frequency of sampling• Location of control limits
Setting Control Limits
31
Pre-Control
nominalvalue
Green Zone
Yellow Zones
RedZone
RedZone
LTL UTL