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