1. X-bar and R Control Charts
Chapter 6 Part 3
X-bar and R Control Charts

2. Attribute Data Data that is discrete
Discrete data is based on “counts.”
Assumes integer values
Number of defective units
Number of customers who are “very satisfied”
Number of defects

3. Variables Data
X-bar and R chart is used to monitor mean and variance of a process when quality characteristic is continuous.
Continuous values (variables data) can theoretically assume an infinite number of values in some interval.
Time
Weight
Ounces
Diameter

4. X-bar and R Chart
X-bar chart monitors the process mean by using the means of small samples taken frequently
R chart monitors the process variation by using the sample ranges as the measure of variability
Range = Maximum value – Minimum value

5. Notation

6. Notation

7. Notation

8. Example of Notation
A company monitors the time (in minutes) it takes to assemble a product.
The company decides to sample 3 units of the product at three different times tomorrow:
9 AM
12 Noon
3 PM
What is the sample size, n?
What is k, the number of samples?

9. Assembly Time (minutes)
Suppose the following data are obtained.
How would you compute
X-bar? R
R-bar
X double bar
Hour
Assembly Time (minutes) X1 X2 X3
9:00 AM 5 12 4
12 Noon 6 8 10
3:00 PM 9 2

10. Assembly Time (minutes)
Example of Notation
Hour
Assembly Time (minutes)
Sample
Mean,
Sample Range, R X1 X2 X3
9:00 AM 5 12 4
21/3 = 7
12 – 4 = 8
12 Noon 6 8 10
24/3 = 8
10 – 6 = 4
3:00 PM 9 2
15/3 = 5
9 – 2 = 7
= 20/3 = 6.7
=19/3 =6.3

11. Sample Means
First Sample

12. Sample Means
Second Sample
Third Sample

13. Estimated Process Mean

14. Sample Ranges

15. Mean of R

16. Underlying Distributions
When constructing an X-bar chart, we actually have two distributions to consider:
The distribution of the sample means , and
The process distribution, the distribution of the quality characteristic itself, X.
The distribution of is a distribution of averages.
The distribution of X is a distribution of ???

17. Underlying Distributions
These distributions have the same mean
Their variances (or standard deviations) are different.
Which distribution has the bigger variance?
Would you expect more variability among averages or among individual values?
The variability among the individual values is ???

18. Underlying Distributions
The standard deviation among the sample means is smaller by a factor of
Therefore,

19. Underlying Distributions
Sampling distribution of
Distribution of X

20. Distribution of X X M m M m
LCL
UCL M m
The distribution of X is assumed to be normal. This assumption needs to be tested in practice.

21. Distribution of X-bar M m M m
LCL
UCL M m
If the distribution of X is normal, the distribution of X-bar will be normal for any sample size.

22. Control Limits for X-bar Chart
Since we are plotting sample means on the X-bar chart, the control limits are based on the distribution of the sample means.
The control limits are therefore

23. Control Limits for X-bar Chart
Distribution of
LCL
UCL

24. Control Limits for X-bar Chart

25. Control Limits for X-bar Chart
A2 is a factor that depends on the n, the sample size,
and will be given in a table.

26. Example of X-bar Chart
A company that makes soft drinks wants to monitor the sugar content of its drinks.
The sugar content (X) is normally distributed, but the means and variances are unknown.
The target sugar level for one of its drinks is 15 grams.
The lower spec limit is 10 grams.
The upper spec limit is 20 grams.

27. Example of X-bar Chart
The company wants to know how much sugar on average is being put into this soft drink and how much variability there is in the sugar content in each bottle.
The company also wants to know if the mean sugar content is on target.
Lastly, the company wants to know the percentage of drinks that are too sweet and the percentage that are not sweet enough. (Next section)

28. Example of X-bar Chart
To obtain this information, the company decides to sample 3 bottles of the soft drink at 3 different time each day:
10 A.M,
1:00 P.M. and
4:00 P.M.
The company will use this data to construct an X-bar and R chart. (In practice, you need samples to construct the control limits.)
For the past two days, the following data were collected:

29. Example of X-bar Chart Day Hour X1 X2 X3 1 10 am 17 13 6 1 pm 15 12 2421 2 18 10
What is n?
What is the k?
What is the next step?

30. Example of X-bar Chart Day Hour X1 X2 X3 R 1 10 am 17 13 6 36/3 =12 11
1 pm 15 12 24
51/3 =17
4 pm 21
48/3 =16 9 2
42/3 =14 5 18
54/3 =18 10
45/3 =15 8
= 92/6
= 15.33
= 51/6
= 8.5 X

31. X-bar Chart Control Limits

32. Table A: X-bar Chart Factor, A2n A2 2
1.88 3
1.02 4
0.73 5
0.58

33. X-bar Chart Control Limits

34. X-bar Chart Control Limits

36. Interpretation of X-bar Chart
The X-bar chart is in control because ????
This means that the only source of variation among the sample mean is due to random causes.
The process mean is therefore stable and predictable and, consequently, we can estimate it.

37. Interpretation of X-bar Chart
Our best estimate of the mean is the center line on the control chart, which is the overall mean (X-double bar) of grams.
If the process remains in control, the company can predict that all bottles of this soft drink produced in the future will have a sugar content of, on average, grams.

38. Interpretation of X-bar Chart
This prediction, however, indicates that there is a problem with the location of the mean.
The process mean is off target by 0.33 grams ( ).
The process mean, although stable and predictable, is at the wrong level and should be corrected to the target.

39. Interpretation of X-bar Chart
Since the process mean is in control, there are no special causes of variation that may be responsible for the mean being off target.
Since the operators are responsible for correcting problems due to special causes and management is responsible for correcting problems due to random causes of variation, management action is required to fix this problem.

40. Interpretation of X-bar Chart
The reason is that, because the process is in control, the filling machines require more than a simple adjustments (typically due to special causes) which can be made by the operators.
The machines may require
new parts,
a complete overhaul, or
they may simply not be capable of operating on target, in which case a new machine is required.

41. Interpretation of X-bar Chart
Expecting the operators to adjust the mean to the target when the process is in control is analogous to requiring that you produce zero heads (head = defective unit) if you are hired to toss a fair coin 100 times each day.
Why?

42. R Chart Monitors the process variability (the variability of X)
Tells us when the process variability has changed or is about to change.
R chart must be in control before we can use the X-bar chart.

43. R Chart Rules for detecting changes in variance:
If at least one sample range falls above the upper control limit, or there is an upward trend within the control limits, process variability has increased.
If at least one sample range falls on or below the lower control limit, or there is a downward trend within the control limits, process variability has decreased.

44. R Chart Control Limits

45. Table B: Factors for R Chartn D3 D4 2
3.27 3
2.57 4
2.28 5
2.11

46. R Chart Control Limits

48. Interpretation of R Chart
Since all of the sample ranges fall within the control limits, the R chart is in control.
The standard deviation is stable and predictable and can be estimated—done in next section.
This does not necessarily mean that the amount of variation in the process is acceptable.

49. Interpretation of R Chart
Continuous improvement means the company should continuously reduce the variance.
Since the process variation is in control, management action is required to reduce the variation.

50. Expected Pattern in a Stable Process
X-bar Chart
UCL
LCL
Time
Expected pattern is a normal distribution

51. How Non-Random Patterns Show Up
Sampling
Distribution
(process variability is increasing)
UCL
Does not reveal increase
x-Chart
LCL
UCL
R-chart
Reveals increase
LCL

52. How Non-Random Patterns Show Up
(process mean is
shifting upward)
Sampling
Distribution
UCL
x-Chart
Detects shift
LCL
UCL
Does not detect shift
R-chart
LCL

53. Is a Stable Process a Good Process?
“In control” indicates that the process mean is stable and hence predictable.
A stable process, however, is not necessary a “good” (defect free) process.
The process mean, although stable, may be far off target, resulting in the production of defective product.
In this case, we have, as Deming puts it, “A stable process for the production of defective product.”

54. Control Limits vs. Spec. Limits
Control limits apply to sample means, not individual values.
Mean diameter of sample of 5 parts, X-bar
Spec limits apply to individual values
Diameter of an individual part, X

55. Control Limits vs. Spec. Limits
Sampling distribution, X-bar
Process distribution, X
Mean=
Target
USL
LSL
Lower control limit
Upper control limit

56. Underlying Distributions
Sampling distribution of
Distribution of X
LSL
USL
LCL
UCL
Control limits are put on distribution of X-bar
Spec limits apply to the distribution of X

57. Responsibility for Corrective Action
Special Causes
(Process out of control)
Random Variation
(Process in Control)
Operators
(workers)
Management

58. Benefits of Control Charts
Control charts prevent unnecessary adjustments.
If process is in control, do not adjust it.
Adjustments will increase the variance.
Management action is required to improve process.
Adjustments should be made only when special causes occur.

59. Benefits of Control Charts
Control charts assign responsibility for corrective action.
Control charts are the only statistical valid way to estimate the mean and variance of a process or product.
Control charts make it possible to predict future performance of a process and thereby take early corrective action.