Introduction to SPC

1
Introduction to SPC

• Review of normal (Gaussian) distributions. (VERY
  IMPORTANT for SPC)
• For a certain population of people, the average
  male height is 68 inches, with a standard
  deviation of 3 inches.
• What is the probability that the next male from
  this population has a height greater than 70
  inches? (what information here is missing?)

2
Normal Review (continued)

• What is the probability that the average height
  of the next nine males is greater than 70 inches?
• Central Limit Theorem (see next slide)
• If the mean of the population shifted to 72
  inches, what is the probability that the next
  male is less than 70 inches (assume normal
  distribution)?
• If a sample of nine came from the new population,
  what is the probability the average of the sample
  will be less than 70 inches?

3
Central Limit Theorem Example of the Roll of
the Dice
Theoretical Frequency Distribution based on 216
rolls...
Average of Two Dice m3.5, s1.23
Average of Three Dice m3.5, s0.99
One Die m3.5, s1.87
4
Statistical Process Control

• Sample output of process and
• make inferences about its state
• Demonstrate that the
• distribution of process output
• is known and unchanging
• Plot and monitor over time
• Use statistical tests to detect shifts
• and anomalies and react to them quickly
• Use statistical evidence to guide and
• confirm process improvements

5
Evolution from Inspection to SPC
SPC.
SPC.
SPC.
6
Statistical Process Control Topics

• Introduction to Variability
• Control Charts
• General info
• x-bar Charts, R Charts
• p Charts, np Charts
• Type 1 and Type 2 Errors
• Process Capability Analysis

7
Variation
The less variation, the better off we are.
Improve Capabilities
Common cause variation Inherent in the
system Assignable cause variation
Event-related, special (assignable special)
Analyze and Act
8
Analyze and Act React to Assignable Causes

• Note unusual variation diagnosed by using a
  common test to evaluate individual data points
• Identify cause by noting what change in the
  process occurred at that point in time
• Eliminate cause or build in the cause
• Monitor performance to verify the effect of the
  fix
• Generally, assignable causes cause points outside
  of control limits!

9
Improve Process Capabilities Drive Out Common
Causes

• Variation is inherent in the system
• Dont react to individual points (this is
  tampering)
• Analyze possible factors affecting variation (use
  Cause and Effect Diagram, Pareto Analysis)
• Work to reduce variation Make an improvement,
  that is, introduce a special cause
• Monitor performance to verify the effect of the
  intended improvement

10
X-bar and R charts

• Sample output of process - parameter
• of interest is continuously variable
• Plot one chart to track sample means and
• another one to track sample ranges (variation)
• Use statistical evidence to detect changes
• and improve the process to better position
• the mean and to reduce variation

11
Underlying Assumptions

• process mean m and standard deviation s when the
  process is in control
• process may go out of control in two possible
  ways
• mean shifts to m1, with standard deviation
  unchanged
• standard deviation shifts to s1, with mean
  unchanged
• sample means are normally distributed (when in or
  out of control, because either
• process output measurements on individual units
  are normally distributed when in or out of
  control
• OR Central Limit Theorem applies
• n gt 30 OR
• If distribution unimodal or symmetric, then much
  smaller ns are acceptable to assume normality (n
  on the order of 4).

12
Basic Probabilities Concerning the Distribution
of Sample Means
Std. dev. of the sample means
13
Estimation of Mean and Std. Dev. of the
Underlying Process

• use historical data taken from the process when
  it was known to be in control
• usually data is in the form of samples
  (preferably with fixed sample size) taken at
  regular intervals
• process mean m estimated as the average of the
  sample means (the grand mean)
• process standard deviation s estimated by
• standard deviation of all individual samples
• OR mean of sample range R/d2, where
• sample range R max. in sample minus min. in
  sample
• and d2 value from look-up table (appendix
  A-7)

14
Example Estimation of Mean and Std. Dev. of the
Underlying Process
Estimate of the process mean m 2.3 Estimate
of the process std. dev. (1) Combined std.
dev. of all 30 points s 1.1 OR (2) s
R/d2 (n5) 2.7/2.326 1.2
15
Determination of Control Limits

• For the x-bar chart
• - Center Line grand mean
• - Control Limits Co's usually use
• - Can analyze process capability based on the
  specification limits
• For the R chart
• - Center Line average range
• - Control Limits
• Alternative Use an Economic Approach
• - Consider the cost impact of out-of-control
  detection delay (Type 2 error), false alarm (Type
  1 error) and sampling costs
• - Difficult to estimate costs

16
Ex. Two Machines -Process Capability Analysis and
x-bar and R-charts
17
X-bar vs. R charts

• R charts monitor variability Is the variability
  of the process stable over time? Do the items
  come from one distribution?
• X-bar charts monitor centering (once the R chart
  is in control) Is the mean stable over time?
• gtgt Bring the R-chart under control, then look
• at the x-bar chart

18
How to Construct a Control Chart

• 1. Take samples and measure them.
• 2. For each subgroup, calculate the sample
  average and range.
• 3. Set trial center line and control limits.
• 4. Plot the R chart. Remove out-of-control
  points and revise control limits.
• 5. Plot x-bar chart. Remove out-of-control
  points and revise control limits.
• 6. Implement - sample and plot points at standard
  intervals. Monitor the chart.

19
X-bar and R chart example

• Look at handout R Chart.
• R-bar sum( R )/num. samples 87/25 3.48.
• UCL D4R-bar 2.1143.48 7.357.
• LCL D3R-bar 0
• Review samples, eliminate sample 3.
• Do over! New R-bar 3.29, UCL 6.95
• R-bar chart now in control, proceed to X-bar!

20
X-bar chart

• Grand mean, X-bar 500.6/24 20.86.
• Control limits 20.86 /- A2R-bar 20.86 -
  (.5777)3.29
• UCL 20.86 1.9 22.76
• LCL 20.86 1.9 18.96
• Bring X-bar chart under controleliminate points
  15, 22, 23.

21
Conclusion of problem

• Redo R chart without samples 15, 22, and 23 (and
  3 is out as well).
• R-bar 3.24
• Control limits (repeat previous procedure 0,
  6.845.
• Grand mean (center line for x-bar) 20.77
• Control limits 20.77 /- (.5777)(3.24)
  18.90, 22.64
• Our control limits for both charts are now set.

22
Type 1 and Type 2 Error
23
Common Tests to Determine if the Process is Out
of Control

• One point outside of either control limit
• 2 out of 3 points beyond UCL - 2 sigma
• 7 successive points on same side of the central
  line
• of 11 successive points, at least 10 on the same
  side of the central line
• of 20 successive points, at least 16 on the same
  side of the central line

24
Type 1 Errors for these Tests
Test Probability Type 1 Error
1/1
2(0.00135)
0.0027
2/3
0.00052
7/7
(0.5)7
0.0078
10/11
0.00586
16/20
0.0059
25
Type 2 Error

• Suppose m1 gt m
• Type 2 Error

• This is the probability of a sample average
  being below the upper control limit. We have not
  examined possibility of being below LCL, why?
• Power 1- Type 2 Error. Power increases as
  
• n increases, as (m1-m) increases, and as s
  decreases.
• Extension to m1 lt m is straightforward

26
Sensitivity of Type I and Type II Errors

• To (UCL-LCL)/s
• To n
• To s
• To m1 - m

27
Example of Type 1 and 2 Errors
Suppose m 100 m1 102 s 4 n
9 assume 3 sigma control limits Find Type 2
Error
28
Example of Type 1 and 2 Errors (cont.)
Type 2 Error Type 1 Error
ProbShift detected in third sample after shift
occurred Average number of samples taken
before the shift is detected Probno false
alarm for first 32 samples, but then false
alarm occurs in 33rd sample Average number
of samples before a false alarm (ARL)