Statistical Process Control with formulas

Statistical Process Control with formulas with examples

1. Rev. 04/16/06 SJSU Bus. 142 - David Bentley 1
Chapter 14 – Statistical
Process Control (SPC)
Control charts for variables and
for attributes, special control
charts, control vs. process
capability

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Statistical Process Control
(SPC)

A methodology for monitoring a
process to identify special causes of
variation and signal the need to take
corrective action when appropriate

SPC relies on control charts
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Variation (from Chapter 11)
Many sources of uncontrollable variation
exist (common = chance = random causes)
Special (assignable = special = non-random)
causes of variation can be recognized and
controlled
Failure to understand these differences can
increase variation in a system

“Tweaking” a process that is in control (common)

Ignoring a process that is out of control (special)
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Common
Causes
Special
Causes
Common Causes vs.
Special Causes

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Histograms do not
take into account
changes over time.
Control charts can
tell us when a
process changes
Histograms
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Control Chart Applications

Establish state of statistical
control

Monitor a process and signal
when it goes out of control

Determine process capability
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Capability Versus Control

Process capability (C
p
or C
pk
)

Measure of variability against design
specifications

Specs set by customer or design engineer

Spec width: USL & LSL (or UTL & LTL)
Statistical process control (SPC)

Measure of variability against control limits

Control limits calculated from sample data

UCL and LCL

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Capability Versus Control
Control
Capability
Capable
Not Capable
In Control Out of Control
IDEAL
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Commonly Used Control
Charts
Variables data

x-bar (mean) and R- (range) charts

x-bar and s- (standard deviation) charts

Charts for individuals

(x-charts)

Moving range charts
Attribute data

For “defectives” (p-chart, np-chart)

For “defects” (c-chart, u-chart)
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Developing Control Charts
1.Prepare

Choose measurement

Determine how to collect data, sample
size, and frequency of sampling

Set up an initial control chart
2.Collect Data

Record data

Calculate appropriate statistics
Plot statistics on chart
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Next Steps
3.Determine trial control limits

Center line (process average)

Compute UCL, LCL
4.Analyze and interpret results

Determine if in control

Eliminate out-of-control points

Recompute control limits as necessary
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Mean & Range Control
Charts
Take required number of samples
Mean (X-bar) charts (see Appendix B)

Calculate mean (X-bar) for each sample

Calculate grand mean (X-double-bar)

Calculate range (R) for each sample

Calculate mean of all sample ranges (R-bar)

Calculate UCL and LCL for means

Plot grand mean and control limits on X-bar
chart

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Mean & Range Control
Charts

Range (R) charts (see Appendix B)

Calculate UCL and LCL for ranges

Plot range mean and control limits on R-
chart

Plot additional samples and determine if
within range limits

Note: factors based on the size of each
sample, not the number of samples!

15. 04/17/06 SJSU Bus 142- David Bentley 15
Mean (X-bar) Chart Control
Limits
UCL
X-bar = X-double-bar + A
2 (R-bar)
LCL
X-bar = X-double-bar - A
2 (R-bar)
Where X-double-bar = the grand mean,
And R-bar = the mean of the sample ranges
And A
2 = the value in Appendix B for n

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Range (R) Chart Control
Limits
UCL
R = D
4 (R-bar)
LCL
R = D
3 (R-bar)
Where R-bar = the mean of the sample
ranges, and D
4 and D
3 = the values in
Appendix B for n

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Typical Out-of-Control
Patterns
Point outside control limits

Stop the process!
Sudden shift in process average
Cycles
Trends
Hugging the center line
Hugging the control limits
Instability
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Shift in Process Average
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Identifying Potential Shifts
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Cycles
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Trend
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Final Steps
5.Use as a problem-solving tool

Continue to collect and plot data

Take corrective action when necessary
6.Compute process capability
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SPC Control Limit
Calculations
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Excel Template
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Special Variables Control
Charts
x-bar and s-charts

Use standard deviation (s) chart in place of R-
chart

For each sample: s = ((x – x-bar)
2
/n-1)

For k samples: s-bar = s/k

UCL
s = B
4(s-bar), LCL
s = B
3(s-bar)

Use s-bar rather than r-bar for x-bar calculations
x-chart, moving range charts for individuals

x-chart: automated 100% inspection, n = 1

Moving range: better process variability measure
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(Mod. 10/14/02 DAB)

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Charts for Attributes
Fraction nonconforming (defective)

p-chart (proportion or percentage chart)
Fixed sample size

Variable sample size (units produced varies)
np-chart for number nonconforming

Fraction x sample size
Charts for defects

c-chart (total number of defects/unit)

u-chart (average number of defects/unit)
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Control Chart Formulas

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Control Chart Selection
Quality Characteristic
variable attribute
n>1?
n>=10 or
computer?
x and MR
no
yes
x and s
x and R
no
yes
defective defect
constant
sample
size?
p-chart with
variable sample
size
no
p or
np
yes
constant
sampling
unit?
c u
yes no
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48. Rev. 10/14/02 SJSU Bus. 142 - David Bentley 48
Control Chart Design Issues

Basis for sampling (homogeneous samples)

Sample size (n usually = 3 to 10)

Frequency of sampling (cost vs. risk)

Location of control limits

Type I vs. Type II risk analysis

Cost vs. risk again!

Number of samples (k usually = 25 to 30)
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(Mod. 10/14/02)
Sample size and detecting
shift

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Pre-Control
nominal
value
Green Zone
Yellow Zones
Red
Zone
Red
Zone
LTL UTL
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SPC Implementation
Requirements

Top management commitment

Project champion

Initial workable project

Employee education and training

Accurate measurement system
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