control chart and their types and other chart

control chr

1. LEARNING OBJECTIVES
•Statistical tools in quality improvement
•Understand how a control chart is used to detect
assignable causes
•Understand how to apply pattern analysis
•Construct and interpret control charts for variables
•Construct and interpret control charts for attributes
•Calculate and interpret process capability ratios X
•Construct and interpret a cumulative sum control
chart
•Use other statistical process control

2. Importance of Quality
•Quality of products and services has
become a major decision
•Consider quality of equal importance to
cost and schedule
•Quality improvement has become a major
concern to many U.S. corporations
•Statistical quality control, a collection of
tools that are essential in quality
improvement activities

3. Definition of Quality
•Quality means fitness for use
•Expect that the products we buy will meet our
requirements
•Requirements define fitness for use
•Determined through the interaction of quality of
design and quality of conformance
•Quality of design
–Different grades or levels of performance, reliability,
serviceability
•Quality of conformance
–Systematic reduction of variability and elimination of
defects

4. Quality Improvement
•Means the systematic elimination of waste
•Waste include
–Scrap and rework in manufacturing, inspection and
testing, warranty costs, and the time required to do
things over again
•Can eliminate much of this waste
–Lead to lower costs, higher productivity, increased
customer satisfaction, increased business reputation,
higher market share, and ultimately higher profits

5. History of Statistical Quality Control
•Defined as those statistical methods
•Dates back to the 1920s
•Shewhart was one of the early pioneers
•A modern control chart, one of the tools of SPC
•Dodge and Romig provided much of the leadership
•Work of these forms much of the basis of the SPC
•World War II saw the widespread of these methods
•Deming and Juran have been instrumental in
spreading SPC methods

6. Japanese and American Industry
•Japanese have been successful in deploying SPC
methods
•American industry suffered extensively from
Japanese competition
•Many U.S. companies have begun programs to
implement these methods

7. Online SPC
•Product must be built right the first time
•Manufacturing process must be capable of
operating with little variability
•Online SPC is a powerful tool
•Think of statistical process control (SPC) as a
set of problem-solving tools
•Major tools of SPC
–Histogram, Pareto chart, Cause-and-effect diagram,
Defect-concentration diagram, Control chart, Scatter
diagram, and Check sheet
•Control chart is the most powerful of the SPC tools

8. Natural Variability or Background
Noise
•Certain amount of natural variability always
exist
•Cumulative effect of many unavoidable
causes
•When it is small, we consider it acceptable
•Called a “stable system of chance causes.”
•Operating with only chance causes of
variation present is said to be in statistical
control
•Chance causes are an inherent part of the
process

9. Assignable Causes
•Other kinds of variability may be present
•Arises from three sources
–Unadjusted machines, operator errors, or
defective raw materials
•Such variability is large
•Refer to these as assignable causes
•Operating in the presence of assignable
causes is said to be out of control

10. Major Objectives
•Detect the occurrence of assignable causes
•Control chart is an online process-monitoring
technique
•Estimate the parameters of a production process
•Provide information that is useful in improving the
process
•Eliminate the variability in the process
•May not be possible to eliminate variability
completely
–Helps reduce it as much as possible

11. Control Chart
•A typical control chart
•Samples are selected
at periodic intervals
•Contains a center line
(CL)
•Upper control limit
(UCL) and the lower
control limit (LCL)
•Point that plots outside
of the control limits
–Evidence that the
process is out of control
•Sample points are
usually connected with
straight-line segments

12. Nonrandom Manner of the Plots
•If they behave in a nonrandom manner
–Indication that the process is out of control
•If 16 of the last 18 points plotted above the
center
–Very suspicious that something was wrong
•All points should have random pattern
•Methods can be applied to control charts
•Nonrandom pattern usually appears on a
control chart for a reason

13. Relationship Between Control
Charts and Hypothesis testing
•Close connection between control charts and
hypothesis testing
•Control chart is a test of the hypothesis
•A point within the control limits is equivalent
to failing to reject the hypothesis
•A point plotting outside the control limits is
equivalent to rejecting the hypothesis

14. General Model
•General model for a control chart
•Let W be a sample statistic
–Mean of W is µ
W

–Standard deviation of W is 
W
•CL, UCL, and LCL
•k is the distance from the CL
•Common choice is k=3
•First proposed by Shewhart
–Control charts are often called Shewhart control charts

15. Use of Control Charts
•Improve the process
•Identify assignable causes
•If they are eliminated from the process,
variability will be reduced
•May use the control chart as an estimating
device
•May estimate certain process parameters
•May then be used to determine the
capability of the process

16. Types of Control Charts
•Classified into two types
•Deals with quality characteristics that can be
measured on some continuous scale
•Called variables control charts
•May not able to measure the quality
characteristics on a continuous scale
•May judge each unit of product as either
conforming or nonconforming
–May count the number of nonconformities
(defects) on a unit of product
•Called attributes control charts

17. Designing Control Charts
•Must specify both the sample size and the
frequency of sampling
•Larger samples will make it easier to detect small
shifts
•Must also determine the frequency of sampling
•Desirable situation would be to take large samples
very frequently
•Is usually not economically feasible
•Take either small samples at short intervals or
larger samples at longer intervals
•Current industry practice tends to favor smaller,
more frequent samples
•Increased data will increase the effectiveness of
process control

18. Rational Groups
•Collect sample data according to the rational
subgroup
•Means that subgroups or samples
–Variability of the observations includes all the
chance
•Control limits will represent bounds for all the
chance variability
•Assignable causes will tend to generate
points that are outside of the control limits

19. Two Approaches-First Approach
•Two approaches to constructing rational subgroups
are used
•First approach
–Consists of units that were produced at the same time
•Used when the primary purpose is to detect
process shifts
•Minimizes variability within a sample
•Maximizes variability between samples
•Also provides better estimates of the standard
deviation of the process

20. Two Approaches-Second Approach
•Second approach
–Consists of units of product that are
representative of all units
•Each subgroup is a random sample of all
process output
•Used when the objective is to make
decisions about the acceptance of all units
of product

21. X AND R CONTROL CHARTS
•X chart is the most widely used chart for monitoring
central tendency
–Charts based on either the sample range or the sample
standard deviation are used to control process variability
•µ and  are known
–Quality characteristic has a normal distribution
•Use µ as the center line for the control chart
–Place the upper and lower 3-sigma limits
•Width is inversely related to the sample size n

22. The 3-sigma Limits
•Utilizes the sample mean to monitor the
process mean
•Choose the constant z
α/2 to be 3
•Use of 3-sigma limits implies that
α=0.0027
•Point plots outside the control limits when
the process is in control is 0.0027

23. Unknown µ and 
•Estimate them on the basis of preliminary
samples
•CL, UCL, and LCL
•Constant A
2
is tabulated for various sample sizes

24. R Chart
•R … charts can be used to control process
variability
•Can easily determine the parameters of the R …
chart
•CL will obviously be the average range and the
control limits are
•r … is the sample average range, and D
3 and D
4
are tabulated for various sample sizes in Appendix
Table X

25. Example
•Twenty-five samples of size 5 are drawn from a
process at one-hour intervals, and the following
data are obtained:
•Find trial control limits for and R chartsX

26. Solution
0)344.0(0
344.0
728.0)344.0(115.2
3
4



rDLCL
CL
rDUCL
rx and
344.0
25
60.8
510.14
25
75.362
 rx
312.14)344.0(577.0510.14
510.14
708.14)344.0(577.0510.14
2
2



rACLLCL
CL
rACLUCL
• Calculations for
• The trial control limits for
chart are
x
• For the R chart, the trial control limits are

27. CONTROL CHARTS FOR
INDIVIDUAL MEASUREMENTS
•Sample size used for process control is n=1
•Examples
–Automated inspection and measurement technology is used
–Production rate is very slow
–Repeat measurements on the process differ
•In such situations, the individuals control chart is useful
•Uses the moving range of two successive observations to
estimate the process variability.
•Moving range is defined as
•Estimate of 

28. Control Limits
•CL, UCL, and LCL
•CL, UCL, and LCL for moving ranges

29. Example
•Twenty successive
hardness measurements
are made on a metal alloy,
and the data are shown in
the following table
•(a) Using all the data,
compute trial control limits
for individual observations
and moving-range charts
•Construct the chart and
plot the data
•Determine whether the
process is in statistical
control. If not, assume
assignable causes can
be found to eliminate
these samples and
revise the control limits.

30. Solution
Individuals and MR(2) - Initial Study
--------------------------------------------------------------------------------

•Control limits for chart I Control limits for R chart
Ind.x | MR(2)
----- | -----
UCL: + 3.0 sigma = 60.8887 | UCL: + 3.0 sigma = 9.63382
Centerline = 53.05 | Centerline = 2.94737
LCL: - 3.0 sigma = 45.2113 | LCL: - 3.0 sigma = 0
|
out of limits = 0 | out of limits = 0
--------------------------------------------------------------------------------
•Chart: Both Normalize: No

20 subgroups, size 1 0 subgroups excluded

Estimated
process mean = 53.05
process sigma = 2.61292
mean MR(2) = 2.94737
X

31. Control Chart

0 4 8 12 16 20
45
49
53
57
61
Ind.x
53.05
60.8887
45.2113
0 4 8 12 16 20
subgroup
0
2
4
6
8
10
MR(2)
2.94737
9.63382
0
• Control chart and the moving range chart

32. ATTRIBUTE CONTROL CHARTS
•Classify a product as either defective or
nondefective
•Achieve economy and simplicity in the
inspection operation
•May judge each unit of product as either
conforming or nonconforming
•Discuss the fraction-defective control
chart, or P chart

33. P chart
•D is the number of defective units
•D is a binomial random variable with unknown parameter p
•Fraction defective of each sample is plotted on the chart
•Variance
•CL, UCL, LCL for the P chart
•… is the observed value of the average fraction defective
•Based on the normal approximation to the binomial
distribution

34. U Chart
•Monitor the number of defects in a unit of product
•Control the number of defects
•May use the control chart for defects per unit, or
the U chart
•Modeled by the Poisson distribution
•Sample consists of n units and there are C total
defects

35. Control Chart
•CL, UCL, and LCL
• is average number of defects per unit
•When λ is small, the normal approximation may
not be adequate
u

36. Example
•Suppose the following
fraction defective has been
found in successive
samples of size 100 (read
down):
•Using all the data, compute
trial control limits for a
fraction-defective control
chart, construct the chart,
and plot the data.

37. Solution

•UCL: + 3.0 sigma = 0.237957
•Centerline = 0.135333
•LCL: - 3.0 sigma = 0.0327096
•Estimated

–mean P = 0.135333
–sigma = 0.0342079
•out of limits = 2

38. U chart
•U chart for fraction defective

0 5 10 15 20 25 30
subgroup
0
0.2
0.4
0.6
0.8
1
P
0.135333
0.237957
0.0327096