1. ENGM 621: Statistical Process Control
Control Charts, Part I
Variables

2. Process Capability - Timing
Reduce Variability
Identify Special Causes - Good (Incorporate)
Improving Process Capability and Performance
Characterize Stable Process Capability
Head Off Shifts in Location, Spread
Identify Special Causes - Bad (Remove)
Continually Improve the System
Process Capability Analysis is performed when there are NO special causes of variability present – ie. when the process is in a state of statistical control, as illustrated at this point.
Time
Center the Process
LSL  USL

3. Shewhart’s Assumptions
The data generated by the process when it is in control:
Are normally distributed
Are independent
Have a mean and standard deviation that are fixed and unknown

4. Statistical Process Control
Originally developed by Walter Shewhart in 1924 at the Bell Telephone Laboratories
Late 1920s, Harold Dodge and Harry Romig developed statistically based acceptance sampling
Not recognized by industry until after World War II
Shewhart was a contemporary of Deming and Juran
Dodge and Romig also of Bell Labs
statistically based acceptance sampling is an alternative to 100% inspection

5. Common Causes
Special Causes

6. Histograms do not take into account changes over time.
Control charts can tell us when a process changes

7. Data on Quality Characteristics
Attribute data
Discrete
Often a count of some type
Variable data
Continuous
Often a measurement, such as length, voltage, or viscosity
We will use statistical methods for both types

8. Control Chart Concept Map
Quality Characteristic
Q-Sum Chart
n>10
n >1
Sm. shfts
X, Moving R
type of attribute
ni = n
p, np c
pvar u
X, S
X, R no
yes
variable
attribute
defective
defects

9. 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

10. TM 720: Statistical Process Control
Test of Hypothesis
A statistical hypothesis is a statement about the value of a parameter from a probability distribution.
Ex. Test of Hypothesis on the Mean
Say that a process is in-control if its’ mean is m0.
In a test of hypothesis, use a sample of data from the process to see if it has a mean of m0 .
Formally stated:
H0: m = m0 (Process is in-control)
HA: m ≠ m0 (Process is out-of-control)
(c) D.H. Jensen & R.C. Wurl

11. Test of Hypothesis on Mean (Variance Known)
TM 720: Statistical Process Control
Test of Hypothesis on Mean (Variance Known)
State the Hypothesis
H0: m = m0
H1: m ≠ m0
Take random sample from process and compute appropriate test statistic
Pick a Type I Error level (a) and find the critical value za/2
Reject H0 if |z0| > za/2
(c) D.H. Jensen & R.C. Wurl

12. UCL and LCL are Equivalent to the Test of Hypothesis
TM 720: Statistical Process Control
UCL and LCL are Equivalent to the Test of Hypothesis
Reject H0 if:
Case 1:
Case 2:
For 3-sigma limits za/2 = 3
(c) D.H. Jensen & R.C. Wurl

13. Two Types of Errors May Occur When Testing a Hypothesis
TM 720: Statistical Process Control
Two Types of Errors May Occur When Testing a Hypothesis
Type I Error - a
Reject H0 when we shouldn't
Analogous to false alarm on control chart, i.e.,
point lays outside control limits but process is truly in-control
Type II Error - b
Fail to reject H0 when we should
Analogous to insensitivity of control chart to problems, i.e.,
point does not lay outside control limits but process is never-the-less out-of-control
(c) D.H. Jensen & R.C. Wurl

14. Consequences of Incorrect Control Limits
TM 720: Statistical Process Control
Consequences of Incorrect Control Limits
Bad Thing 1:
A control chart that never finds anything wrong with the process, but the process produces bad product
Bad Thing 2:
Too many false alarms destroy the operating personnel’s confidence in the control chart, and they stop using it
(c) D.H. Jensen & R.C. Wurl

15. Variation and Control
A process that is operating with only common causes of variation is said to be in statistical control.
A process operating in the presence of special or assignable cause is said to be out of control.

16. Finding Trends and Special Causes
Inspection does not tell you about a problem until it becomes a problem
We need a mechanism to help us spot special causes when they occur
We need mechanism to help us determine when we have a trend in the data

17. Control Chart Applications
Establish state of statistical control
Monitor a process and signal when it goes out of control
Determine process capability
Note: Control charts will only detect the presence of assignable causes. Management, operator, and engineering action is necessary to eliminate the assignable cause.

18. Capability Versus Control
Capable
Not Capable
In Control Out of Control
IDEAL

19. Control Charts         x x  n
We assume that the underlying distribution is normal
with some mean  and some constant but unknown
standard deviation .
Let n x x   i n i  1

20. Distribution of x s s = n x  N (  ,  )
Recall that x is a function of random variables,
so it also is a random variable with its own
distribution. By the central limit theorem, we
know that
where, x  N (  ,  ) x s s = x x n

21. Control Charts x

      x
x 
x

22. Control Charts  3  UCL & LCL Set at
x 
x
LCL
UCL & LCL Set at
Problem: How do we estimate  &  ?  3  x

23. Control Charts m å x i x = i = 1 m m m å R R = i = 1 = f ( s ) s m

24. å Control Charts å x x = ® m m R R = = f ( s ) ® s m UCL = D R LCL = Di x = i = 1 m m m å R R = i = 1 = f ( s ) s m
UCL = x + A R
UCL = D R x 2 R 4
LCL = x - A R
LCL = D R x 2 R 3

26. Example
Suppose specialized o-rings are to be manufactured at .5 inches. Too big and they won’t provide the necessary seal. Too little and they won’t fit on the shaft. Twenty samples of 2 rings each are taken. Results follow.

28. Control Charts

30. Control Charts

32. X-Bar Control Charts X-bar charts can identify special causes of
variation, but they are only useful if the process
is stable (common cause variation).

33. Control Limits for Range
UCL = D4R = 3.268*.002 = .0065
LCL = D3 R = 0

34. Why Monitor Both Process Mean and Process Variability?
TM 720: Statistical Process Control
Why Monitor Both Process Mean and Process Variability?
Process Over Time
Control Charts
X-bar R
X-bar R
X-bar R
(c) D.H. Jensen & R.C. Wurl

35. Teminology Causes of Variation: Meaning of Control: Assignable Causes
Keep the process from operating predictably
Things that we can do something about
Common / Chance Causes
Random, inherent variation in the process
Meaning of Control:
In Specification
Meets customer constraints on product
In Statistical Control
No Assignable Causes of variation present in the process

36. Statistical Basis for Control Charts

37. Statistical Basis of x Chart
TM 720: Statistical Process Control
Statistical Basis of x Chart
Suppose a quality characteristic is x ~ N(m, s)
and we know m and s
If x1, x2, …, xn is a random sample of size n then:
and
(c) D.H. Jensen & R.C. Wurl

38. Statistical Basis of x Chart Cont'd
TM 720: Statistical Process Control
Statistical Basis of x Chart Cont'd
A 1-a confidence interval then is given by
Which is equivalent to
Where LCL and UCL are the lower and upper control limits, respectively
(c) D.H. Jensen & R.C. Wurl

39. Statistical Basis of X-bar
TM 720: Statistical Process Control
Statistical Basis of X-bar
R – the range – is a sample statistic
If x1, x2, …, xn is a random sample of size n from a normal distribution then one can estimate  using the sample standard deviation (if n is larger) or the average range for small samples:
where d2 is a function of n and can be found in Appendix VI
(c) D.H. Jensen & R.C. Wurl

40. Statistical Basis of x-bar
If we replace 3 sigma hat of estimate for sigma hat, we get

41. For n=2,

42. TM 720: Statistical Process Control
For R-Chart
General model for R chart
(c) D.H. Jensen & R.C. Wurl

43. For n=2,

44. Identifying Shifts, Trends

45. Shift in Process Average

46. Identifying Potential Shifts

47. Cycles

48. Trend

49. Western Electric Sensitizing Rules:
One point plots outside the 3-sigma control limits
Two of three consecutive points plot outside the 2-sigma warning limits
Four of five consecutive points plot beyond the 1-sigma limits
A run of eight consecutive points plot on one side of the center line
Western Electric Handbook, 1956

50. Additional sensitizing rules:
Six points in a row are steadily increasing or decreasing
Fifteen points in a row with 1-sigma limits (both above and below the center line)
Fourteen points in a row alternating up and down
Eight points in a row in both sides of the center line with none within the 1-sigma limits
An unusual or nonrandom pattern in the data
One of more points near a warning or control limit

51. Special Variables Control Charts
x-bar and s charts
x-chart for individuals

52. X-bar and S charts
Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R
S chart limits:
UCL = B6σ = B4*S-bar
Center Line = c4σ = S-bar
LCL = B5σ = B3*S-bar
X-bar chart limits
UCL = X-doublebar +A3S-bar
Center line = X-doublebar
LCL = X-doublebar -A3S-bar

54. Operating Characteristic (OC) Curve
TM 720: Statistical Process Control
Operating Characteristic (OC) Curve
Ability of the x and R charts to detect shifts (sensitivity) is described by OC curves
For x chart; say we know s
Mean shifts from m0 (in-control value) to m1 = m0 +ks (out-of-control value)
The probability of NOT detecting the shift on the first sample after shift is
(c) D.H. Jensen & R.C. Wurl

55. TM 720: Statistical Process Control
OC Curve for x Chart
Plot of b vs. shift size (in std dev units) for various sample sizes n
x chart not effective for small shift sizes, i.e., k  1.5s
Performance gets better for larger n and larger shifts (k)
(c) D.H. Jensen & R.C. Wurl

56. Developing Control Charts
Prepare
Choose measurement
Determine how to collect data, sample size, and frequency of sampling
Set up an initial control chart
Collect Data
Record data
Calculate appropriate statistics
Plot statistics on chart

57. Next Steps Determine trial control limits
Center line (process average)
Compute UCL, LCL
Analyze and interpret results
Determine if in control
Eliminate out-of-control points
Re-compute control limits as necessary

58. Final Steps Use as a problem-solving tool Compute process capability
Continue to collect and plot data
Take corrective action when necessary
Compute process capability