1. Statistical Process Control
Operations Management
Dr. Ron Lembke

2. Designed Size

3. Natural Variation

4. Theoretical Basis of Control Charts
Properties of normal distribution
95.5% of allX fall within ± 2

5. Theoretical Basis of Control Charts
Properties of normal distribution
99.7% of allX fall within ± 3

6. Skewness Lack of symmetry Pearson’s coefficient of skewness:
Positive Skew > 0
Skewness = 0
Negative Skew < 0

7. Kurtosis Amount of peakedness or flatness Kurtosis = 0 Kurtosis < 0

8. Heteroskedasticity
Sub-groups with different variances

9. Design Tolerances Design tolerance:
Determined by users’ needs
USL -- Upper Specification Limit
LSL -- Lower Specification Limit
Eg: specified size +/ inches
No connection between tolerance and 
completely unrelated to natural variation.

10. Process Capability
LSL
USL
Capable
LSL
USL
Not Capable
LSL
USL
LSL
USL

11. Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002
Are we in trouble?

12. Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002
LSL = 1.5 – 0.01 = 1.49
USL = = 1.51
Mean: Std. Dev. = 0.002
LCL = *0.002 = 1.499
UCL = = 1.511
Process
Specs
1.49
1.499
1.51
1.511

13. Capability Index
Capability Index (Cp) will tell the position of the control limits relative to the design specifications.
Cp>= 1.0, process is capable
Cp< 1.0, process is not capable

14. Process Capability, Cp Tells how well parts produced fit into specs3 3
LSL
USL

15. Process Capability Tells how well parts produced fit into specs
For our example:
Cp=0.02/0.012 = 1.667
1.667>1.0 Process not capable

16. Packaged Goods What are the Tolerance Levels?
What we have to do to measure capability?
What are the sources of variability?

17. Production Process Mix % Wrong wt. Wrong wt. Candy irregularity
Make Candy
Make Candy
Make Candy
Mix
Package
Put in big bags
Make Candy
Mix %
Wrong wt.
Wrong wt.
Make Candy
Make Candy
Candy irregularity

18. Processes Involved Candy Manufacturing: Mixing: Individual packages:
Are M&Ms uniform size & weight?
Should be easier with plain than peanut
Percentage of broken items (probably from printing)
Mixing:
Is proper color mix in each bag?
Individual packages:
Are same # put in each package?
Is same weight put in each package?
Large bags:
Are same number of packages put in each bag?
Is same weight put in each bag?

19. Weighing Package and all candies
Before placing candy on scale, press “ON/TARE” button
Wait for 0.00 to appear
If it doesn’t say “g”, press Cal/Mode button a few times
Write weight down on form

20. Candy colors Write Name on form Write weight on form
Write Package # on form
Count # of each color and write on form
Count total # of candies and write on form
(Advanced only): Eat candies
Turn in forms and complete wrappers

22. Peanut Candy Weights
Avg. 2.18, stdv 0.242, c.v. = 0.111

23. Plain Candy Weights
Avg 0.858, StDev 0.035, C.V

24. Peanut Color Mix website Brown 17.7% 20% Yellow 8.2% 20% Red 9.5% 20%
Blue 15.4% 20%
Orange 26.4% 10%
Green 22.7% 10%

25. Plain Color Mix Class website Brown 12.1% 30% Yellow 14.7% 20%
Red 11.4% 20%
Blue 19.5% 10%
Orange 21.2% 10%
Green 21.2% 10%

26. So who cares? Dept. of Commerce
National Institutes of Standards & Technology
NIST Handbook 133
Fair Packaging and Labeling Act

27. Acceptable?

29. Package Weight “Not Labeled for Individual Retail Sale”
If individual is 18g
MAV is 10% = 1.8g
Nothing can be below 18g – 1.8g = 16.2g

30. Goal of Control Charts See if process is “in control”
Process should show random values
No trends or unlikely patterns
Visual representation much easier to interpret
Tables of data – any patterns?
Spot trends, unlikely patterns easily

31. NFL Control Chart?

32. Control Charts
Values
UCL
avg
LCL
Sample Number

33. Definitions of Out of Control
No points outside control limits
Same number above & below center line
Points seem to fall randomly above and below center line
Most are near the center line, only a few are close to control limits
8 Consecutive pts on one side of centerline
2 of 3 points in outer third
4 of 5 in outer two-thirds region

34. Control Charts Normal Too Low Too high 5 above, or below Run of 5
Extreme variability

35. Control Charts
UCL 2σ 1σ
avg 1σ 2σ
LCL

36. Control Charts
2 out of 3 in the outer third

37. Out of Control Point? Is there an “assignable cause?”
Or day-to-day variability?
If not usual variability, GET IT OUT
Remove data point from data set, and recalculate control limits
If it is regular, day-to-day variability, LEAVE IT IN
Include it when calculating control limits

38. Attributes vs. Variables
Good / bad, works / doesn’t
count % bad (P chart)
count # defects / item (C chart)
Variables:
measure length, weight, temperature (x-bar chart)
measure variability in length (R chart)

39. Normality

40. R Chart Type of variables control chart Shows sample ranges over time
Interval or ratio scaled numerical data
Shows sample ranges over time
Difference between smallest & largest values in inspection sample
Monitors variability in process
Example: Weigh samples of coffee & compute ranges of samples; Plot

41. Hotel Example
You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?

42. Hotel Data Day Delivery Time

43. Mean and Range - Hotel Data
Sample
Day Delivery Time Mean Range
Sample Mean =

44. R &X Chart Hotel Data Sample Day Delivery Time Mean Range
Largest
Smallest
Sample Range =

45. Hotel Data – Mean and Range
Sample
Day Delivery Time Mean Range

46. X Chart Control Limits
Sample Mean at Time i
Sample Range at Time i
# Samples

47. X Chart Control Limits
A2 from Figure 13.10

48. Figure 13.10 Limits Sample Size (n) A2 D4 D5 2 1.88 3.27 3 1.02 2.57 4
3.27 3
1.02
2.57 4
0.73
2.28 5
0.58
2.11 6
0.48
2.00 7
0.42
0.08
1.92 8
0.37
0.14
1.86 9
0.34
0.18
1.82 10
0.31
0.22
1.78 11
0.29
0.26
1.74

49. R &X Chart Hotel Data Sample Day Delivery Time Mean Range

50. X Chart Control Limits

51. X Chart Solution* `
X, Minutes 8
UCL 6 4 2
LCL 1 2 3 4 5 6 7
Day

52. R Chart Control Limits Figure 13.10, p.402 Sample Range at Time i
# Samples

53. Figure 13.10 Limits Sample Size (n) A2 D4 D5 2 1.88 3.27 3 1.02 2.57 4
3.27 3
1.02
2.57 4
0.73
2.28 5
0.58
2.11 6
0.48
2.00 7
0.42
0.08
1.92 8
0.37
0.14
1.86 9
0.34
0.18
1.82 10
0.31
0.22
1.78 11
0.29
0.26
1.74

54. R Chart Control Limits

55. R Chart Solution
UCL

56. Attribute Control Charts
Tell us whether points in tolerance or not
p chart: percentage with given characteristic (usually whether defective or not)
np chart: number of units with characteristic
c chart: count # of occurrences in a fixed area of opportunity (defects per car)
u chart: # of events in a changeable area of opportunity (sq. yards of paper drawn from a machine)

57. p Chart Control Limits # Defective Items in Sample i Sample i Size
z = 2 for 95.5% limits; z = 3 for 99.7% limits
# Samples

58. p Chart Example
You’re manager of a 1,700 room hotel. For 7 days, you collect data on the readiness of all of the rooms that someone checked out of. Is the process in control (use z = 3)?
© 1995 Corel Corp.

59. p Chart Hotel Data # Rooms No. Not Proportion Day n Ready p
1 1, /1,300 =

60. p Chart Control Limits

61. p Chart Solution

62. Hotel Room Readiness P-Bar