2. No production process produces exactly identical
items.
Random (or natural) variation
Inherent in a process.
Dependent on equipment and machinery, engineering,
operator, and system of measurement
Can be reduced only through continuous improvements
in the system.
Assignable (or identifiable) variation
Due to identifiable factors.
These factors include equipment out of adjustment,
defective materials, changes in parts or materials, broken
machinery or equipment, operator fatigue or poor work
methods, or errors due to lack of training
Variations in Processes
3. Stable processes exhibit only random variations.
Unstable processes exhibit both random and
assignable variations.
Goal of SPC and control chats distinguish
between random and assignable variations in
output to identify if the process is functioning
correctly. If not, stop the process and take
corrective action.
Variations in Processes
4. What is Statistical Process Control?
• A statistical method that uses sampling and control charts
to monitor the production process to evaluate, detect and
prevent poor quality.
• Periodic samples from the production run are taken and
compared with control limits.
Process is in control if…
1. there are no sample points outside of the control limits
2. most points are near the process average (centre line)
3. there is approximately an equal number of points above
and below average
4. points appear to be randomly distributed around
average (i.e., no clear pattern)
5. Quality Measures
Attribute Measure
• Quality characteristic evaluated with a discrete
response
• good/bad; yes/no; correct/incorrect; Excellent/Very
Good/Good/Fair/Poor
Variable Measure
• Quality characteristic that is continuous and can be
measured
• Weight, length, voltage, volume, temperature, time
BU375 Chapter 3: Statistical Process Control 5
6. Control Charts
A control chart is a time-ordered plot of the sample statistic
of a quality characteristic of a product/service/process.
There are two major types of control charts:
• Attribute charts
• 𝑝-chart (e.g., the proportion of defects)
• 𝑐-chart (e.g., the number of defects per unit produced)
• Variable charts
• Mean chart or 𝑥-chart (sample means)
• Range chart or 𝑅-chart (sample range)
• In general, larger sample sizes are needed for attribute
charts because more observations are required to develop
a usable quality measure (e.g., proportion of defects of 5%
in sample of 100 is 5, whereas 5% is not even possible in a
sample of 10 (the minimum would be 10% for 1 defect)
7. One key idea of control charts is based on Normal Distribution:
3δ control limits
If a process is stable, only random variation exists and the
sample statistics of a quality characteristic will be Normal:
• 99.7% of its samples will be randomly located within ±3δ
control limits.
• It is very unlikely (0.3%) to have a sample outside ±3δ
control limits.
• If there is 1+ samples outside the ±3δ limits, it is a very
strong evidence that assignable variation exists in the
process. That is, the process is unstable.
8. One key idea of control charts is based on Normal Distribution:
2δ control limits
If a process is stable, only random variation exists and the
sample statistics of a quality characteristic will be Normal:
• 95.0% of its samples will be randomly located within ±2δ
control limits.
• It is unlikely (5.0%) to have a sample outside ±2δ control
limits.
• If there is 1+ samples outside the ±2δ limits, it is a strong
evidence that assignable variation exists in the process.
That is, the process is unstable.
9. 2δ vs. 3δ control limits
• Type I error (false alarm): A control chart
determines a process to be unstable when it is
actually stable.
• Type II error: A control chart determines a
process to be stable when it is actually unstable.
• Which type of control limits is more sensitive to
process change, 2δ or 3δ?
• Give an example of a product for which 2δ control
limits should be used for its quality control ?
10. Where to use Control Charts
• At critical points in process:
• Where process has tendency to go out of control
• Where particularly harmful and costly if out of control
• Examples:
• At the beginning of a process because it’s a waste to begin a
production process with bad input
• Before a costly or irreversible point, after which product is
difficult to rework or correct
• Before and after assembly or painting operations that might
cover defects
• Before the outgoing final product or service is delivered
BU375 Chapter 3: Statistical Process Control 10
11. How to Use Control Charts?
1. Take samples from the process and compute the
appropriate sample statistic
2. Use the sample statistic to calculate control limits
and draw the control chart
3. Plot sample results on the control chart and
determine the state of the process (in or out of
control; stable or unstable)
4. Investigate possible assignable causes and take
any indicated actions
5. Continue sampling from the process and reset the
control limits when necessary
12. When a process is out of control?
BU375 Chapter 3: Statistical Process Control 12
1). 1+ samples out of control limits
13. When a process is out of control?
BU375 Chapter 3: Statistical Process Control 13
2). Samples display a non-random pattern
14. 𝒑-chart (proportion of defects follows
Normal distribution)
• Take a sample of 𝑛 items periodically from production or
service process
• Calculate the proportion of defective items in each sample
• Does this proportion fall within the acceptable limits?
Center Line: 𝐶𝐿 = 𝑝
Upper Control Limit: 𝑈𝐶𝐿 = 𝑝 + 𝑧𝜎𝑝
Lower Control Limit: 𝐿𝐶𝐿 = 𝑝 − 𝑧𝜎𝑝
𝑧 = # of SDs from the process average
𝑝 = average sample proportion defective (estimate of process average)
𝜎𝑝 = SD of sample proportion, and
𝜎𝑝 =
𝑝 1 − 𝑝
𝑛
15. 𝒑-chart Example
A quality control manager counted the
number of jeans parts made in a garment
factory by taking samples of 100 jeans on
an hourly basis for a period of 20 hours.
a) Construct a 3𝜎 control chart
b) The process is in control?
17. 𝒄-chart (number of defects follows Poisson
distribution)
• Designed to control the number of defects per unit of product
• No sample size, but can count the number of defects for a unit
of product
𝐶𝐿 = 𝑐
𝑈𝐶𝐿 = 𝑐 + 𝑧 ∗ 𝜎𝑐 = 𝑐 + 𝑧 ∗ 𝑐
𝐿𝐶𝐿 = 𝑐 − 𝑧 ∗ 𝜎𝑐 = 𝑐 − 𝑧 ∗ 𝑐
𝑧 = # of SD from the process average
𝑐 = mean number of defects per unit of product
𝜎𝑐 = sample SD, 𝜎𝑐 = 𝑐
18. 𝒄-chart Example
Hotel management conducts random inspections of
individual rooms to check on housekeeping defects.
a) Using 3𝜎 control limits, comment on whether or
not the process is in control?
b) A room is examined and found to have 7 defects. Is
the process still in control?
18
20. Mean ( 𝒙 ) chart
• It measures the process average or mean.
• Each time a sample is taken from the process, the
sample mean is calculated and plotted on the chart.
• It is often used together with Range (𝑹-) chart to
determine whether a process is in control.
• It can be constructed in two ways depending on if
the population standard deviation (𝜎) is known.
21. Mean ( 𝒙 ) chart when σ is known
𝐶𝐿 = 𝑥
𝑈𝐶𝐿 = 𝑥 + 𝑧𝜎𝑥
𝐿𝐶𝐿 = 𝑥 − 𝑧𝜎𝑥
where
• 𝜎𝑥 =
𝜎
𝑛
= standard deviation of the distribution of sample means
• 𝜎 = process standard deviation
• 𝑛 = sample size
• k = number of samples
• 𝑧 = number of standard deviations from the process mean
• 𝑥 = average of the sample means (process mean)
𝑥 =
𝑥1 + 𝑥2 + ⋯ + 𝑥𝑘
𝑘
22. Mean ( 𝒙 ) chart when σ is unknown
𝐶𝐿 = 𝑥
𝑈𝐶𝐿 = 𝑥 + 𝐴2𝑅
𝐿𝐶𝐿 = 𝑥 − 𝐴2𝑅
where:
• 𝐴2 is obtained from Control Factors Table (Table 3.1). They were
developed specifically for determining the control limits for 𝒙 -charts
and are comparable with 3σ limits.
• 𝑅 = average of sample ranges
• 𝑥 = average of the sample means
23.
24. Range (𝑹-) chart
• Range is the simplest measure of process variability.
• Sample Range = difference between the largest and
smallest values of a sample.
𝐶𝐿 = 𝑅
𝑈𝐶𝐿 = 𝐷4 ∗ 𝑅
𝐿𝐶𝐿 = 𝐷3 ∗ 𝑅
where
• 𝑅 = average range for samples 𝑅 =
∑𝑅
𝑘
• 𝑅 = range for each sample (maximum – minimum value)
• 𝑘 = number of samples (subgroups)
• 𝐷3 and 𝐷4 obtained from Control Factors Table (Table 3.1). They are
table values like 𝐴2 for determining control limits that have been
developed based on range values rather than standard deviations
25. Example
The Goliath Tool Company produces slip-ring bearings, which
look like flat donuts or washers. They fit around shafts or rods
(e.g., drive shafts in machinery or motors). At an early stage in the
production process for a particular slip-ring bearing, the outside
diameter of the bearing is measured. Employees have taken 10
samples (1 per day for 10 days) of five slip-ring bearings and
measured the diameter of the bearings. The individual
observations from each sample (or subgroup) are shown in the
table on the next slide. The company wants you to develop
control charts with 3-sigma limits to monitor the process and
determine if it is capable of producing within specification
(tolerance) limits.
1. If from historical data, σ is known to be 0.08
2. If σ is unknown
31. The need to use 𝒙 and R Charts together
(a)
These
sampling
distributions
result in the
charts below
(Sampling mean is
shifting upward but
range is consistent)
R-chart
(R-chart does not
detect change in
mean)
UCL
LCL
x-chart
(x-chart detects
shift in central
tendency)
UCL
LCL
32. The need to use 𝒙 and R Charts together
R-chart
(R-chart detects
increase in
dispersion)
UCL
LCL
(b)
These
sampling
distributions
result in the
charts below
(Sampling mean
is constant but
dispersion is
increasing)
x-chart
(x-chart does not
detect the increase
in dispersion)
UCL
LCL
33. Process Capability
The ability of a process to produce acceptable outputs that
conform to design specifications (i.e., no defects)
Design Specification:
• Lower and upper specification limits (LSL and USL) established by
engineering design or customer requirements
• E.g., Net weight = 9.0 oz ± 0.5 oz
Process Performance or Process’s Natural Variation:
• Process center
• Process variability
• E.g., Process mean = 8.80 oz; Process SD = 0.12 oz
34. Process Capability Ratio, 𝑪𝒑
𝐶𝑝 =
Design specification tolerance range
Process variation range
=
USL −𝐿𝑆𝐿
6σ
• Use 𝑪𝒑 when the process is centered between the design
specification limits.
• The greater the 𝐶𝑝, the greater the probability that the
process output will fall within the design specifications.
35. Process Capability Ratio, 𝑪𝒑
What follows is an illustration of a centered 3-sigma
process. What is the 𝑪𝒑?
36. Process Capability Ratio, 𝑪𝒑
What follows is an illustration of a centered 6-sigma
process. What is the 𝑪𝒑?
37. Process Capability Ratio, 𝑪𝒑
Use 𝑪𝒑 when the process is centered between the design
specification limits.
Cp > 1: process range is less than or equal to design
specification range; therefore, the process is capable of
meeting design specs
Cp < 1: process range is greater than design specification
range; the process is therefore NOT capable of meeting
design specs
38. Process Capability Index, 𝑪𝒑𝒌
𝐶𝑝𝑘 = min
𝑥 − LSL
3𝜎
,
USL − 𝑥
3𝜎
Use 𝑪𝒑𝒌 when the process may not be centered between the
design specification limits.
BU375 Chapter 3: Statistical Process Control 38
39. Process Capability Index, 𝑪𝒑𝒌
Use 𝑪𝒑𝒌 no matter if the process is centered between the
design specification limits.
𝑪𝒑𝒌 > 1: process range is less than or equal to design
specification range; therefore, the process is capable of
meeting design specs
𝑪𝒑𝒌 < 1: process range is greater than design
specification range; the process is therefore NOT capable
of meeting design specs
40. Example: We are the maker of this cereal. Consumer Reports has just
published an article that shows that we frequently have less than 16
ounces of cereal in a box. Let’s assume that the government says that we
must be within ± 5 percent of the weight advertised on the box. We go out
and buy 1,000 boxes of cereal and find that they weigh an average of
15.875 ounces with a standard deviation of .529 ounces. Is our cereal
production process capable of meeting the government specifications?
41. Approaches to Managing Process Quality
The most progressive companies
design quality into the process, thereby
reducing the need for inspection/tests.
