1. Basic Training for Statistical Process Control
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Example Problems
Hypothesis Testing & Relationship to Continuous Variable Control Charts
Continuous Variable Control Charts

2. Continuous Variable Control Charts
Outline
Anthropomorphic Data Hypothesis Test Results
Ishikawa’s (Magnificent Seven) Tools
Hypothesis Testing to Control Charts
X-Bar & R-Charts
X-Bar & Sigma-Charts
Development of Trial Control Limits
Sensitizing Rules for Control Charts
Practice With Control Charts
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Continuous Variable Control Charts

3. Grip Strength Data Results
R-L Side, Equal Variance Dom. Means Comparison:
L = x1 = 129.4, S12 = 2788,
n1 = 34 people
R = x2 = 104.0, S22 = 1225,
n2 = 20 people
Sp = 47.1, v = 52
Two-Sided Test at  = .05
HA: There is a difference
Test: Is | t0 | > t.025, 52?
|1.91| > NO!
Keep the Null Hypothesis:
There is NOT a difference btwn L & R !
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Continuous Variable Control Charts

4. Grip Strength Data Results
R-L Side, No Assumptions Means Comparison:
L = x1 = 129.4, S12 = 2788,
n1 = 34 people
R = x2 = 104.0, S22 = 1225,
n2 = 20 people
v = 51
Two-Sided Test at  = .05
HA: There is a difference
Test: Is | t0 | > t.025, 51?
|2.12| > YES!
Reject the Null Hypothesis:
There IS a difference btwn L & R!
Why is this wimpy test significant when the other wasn’t?
ANS: Check the equal variance assumption!
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Continuous Variable Control Charts

5. Grip Strength Data Results
Unknown 0
Variances Comparison:
S12 = 2788
n1 = 34, v1 = 33
S22 = 1225
n2 = 20, v2 = 19
Two-Sided Test at  = .10
HA: There is a difference
Test: Is F0 > F.05, 33, 19?
2.276 > YES!
(Should also check F1– /2, 33, 19)
Reject the Null Hypothesis:
There IS a difference in variance!
At  = .05, this test is just barely not significant
(Should also have checked for Normality with Normal Prob. Plot)
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Continuous Variable Control Charts

6. Moving from Hypothesis Testing to Control Charts
A control chart is like a sideways hypothesis test
Detects a shift in the process
Heads-off costly errors by detecting trends 0  2
2-Sided Hypothesis Test
Shewhart Control Chart
Sideways Hypothesis Test CL
LCL
UCL
Sample Number
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Continuous Variable Control Charts

7. Shape, Location, & Spread
Shape is governed by Central Limit Theorem
Normally distributed if sample size is large enough
Usually pick n = 4, 5, or 6
Location is measured by X-Bar Chart
Measures variation between sample locations
Spread is measured by either R-Chart or -Chart
Measures variation changes within a sample
Requires rational sampling - e.g. avoid a taking a single sample across two shifts - difference in location could show up as change in variation.
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Continuous Variable Control Charts

8. Continuous Variable Control Charts
X-Bar & R-Charts
Most common of all control charts
Can approximate non-Normal processes
X-Bar Control Limits:
Approximate 3 limits are found from range & table
R-Chart Control Limits:
Approximate, asymmetric 3 limits from range & table
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Continuous Variable Control Charts

9. Continuous Variable Control Charts
X-Bar & R-Charts
Limits can also be generated from historical data:
X-Bar Control Limits:
Approximate 3 limits are found from known 0 & table
R-Chart Control Limits:
Approximate, asymmetric 3 limits from 0 & table
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Continuous Variable Control Charts

10. R - (Range) Control Chart X-Bar (Means) Control Chart
X-Bar & R-Charts
The X-Bar Chart checks variability in location between samples
The R-Chart checks for changes in sample variation
R - (Range) Control Chart
LCL
UCL
Sample Number R
X-Bar (Means) Control Chart x
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Continuous Variable Control Charts

11. Continuous Variable Control Charts
X-Bar & Sigma-Charts
Used when sample size is greater than 10
X-Bar Control Limits:
Approximate 3 limits are found from S & table
Sigma-Chart Control Limits:
Approximate, asymmetric 3 limits from S & table
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Continuous Variable Control Charts

12. Continuous Variable Control Charts
X-Bar & Sigma-Charts
Limits can also be generated from historical data:
X-Bar Control Limits:
Approximate 3 limits are found from known 0 & table
Sigma-Chart Control Limits:
Approximate, asymmetric 3 limits from 0 & table
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Continuous Variable Control Charts

13. Steps for Trial Control Limits
Start with 20 to 25 samples
Use all data to calculate initial control limits
Plot each sample in time-order on chart.
Check for out of control sample points
If one (or more) found, then:
Investigate the process;
Remove the special cause; and
Remove the special cause point and recalculate control limits.
If can’t find special cause - drop point & recalculate anyway
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Continuous Variable Control Charts

14. Control Chart Sensitizing Rules
Western Electric Rules:
One point plots outside the three-sigma limits;
Eight consecutive points plot on one side of the center line;
Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line; or
Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line.
If chart shows lack of control, investigate for special cause
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Continuous Variable Control Charts

15. Control Chart Sensitizing Rules
Additional Sensitizing Rules:
One or more points very near a control limit.
Six points in a row steadily increasing or decreasing.
Eight points in a row on both sides of the center line, but none in-between the one-sigma warning limits on both sides of the center line.
Fourteen points in a row alternating above and below the center line.
Fifteen points in a row anywhere between the one-sigma warning limits (including either side of the center line).
Any unusual or non-random pattern to the plotted points.
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Continuous Variable Control Charts

16. Practice With Control Charts
X-Bar & R-Chart use for Anthropomorphic Data
Setting Trial Limits
Utilization for Control
Problems from CH 4:
# 5, 9, 11, 13, 23, 24, 32
Little calculation, more interpretation & thinking
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Continuous Variable Control Charts