Estimation of Population σ from Sample Data
Control Limits versus Specification Limits
The 6σ Spread versus Specification Limits
Calculating Process Capability Indices
specification limits and control limits
Learn to calculate and interpret the process
capability indices: Cp, Cr and Cpk
Process capability refers to the ability of a process to
produce products or provide services capable of
meeting the specifications set by the customer or
Knowing the process capability gives insight into
whether or not the process will be able to meet future
demands place on it
Determining the process capability aids industry in
meeting their customer demands – a customer may ask
for part tolerances so fine that the machines are not
capable of producing to that level of exactness
An undersized part A may not mate correctly with an
oversized part B
Sample values and their averages provide insight into the
behavior of an entire population
ẋ becomes a more reliable estimate of µ as the sample size
is increased
If the process can be assumed to be normal, the
population standard deviation can be estimated from
either the standard deviation associated with the sample
standard deviation (s) or the range (R) :
It is important to note that a process in statistical control
will not necessarily meet specifications as established by
the customer
There is a difference between a process conforming to
specifications and a process performing within statistical
Specifications communicate what the customers expect,
want or need from the process – considered the voice of
the customer
Control limits are the voice of the process – a prediction
of the variation that the process will exhibit in the near
Difference : specifications relay wishes and control limits
tell of reality
The spread of the individuals in a process, 6σ, is the
measure used to compare the realities of production with
the desires of the customers
Case I : 6σ < USL - LSL
This allows for more room for
process shifts while staying
within the specifications.
Notice that even if the process
drifts out of control, the
change must be dramatic
before the parts are
considered out of
Case II : 6σ = USL - LSL
A shift in the process
mean or an increase in
the variation present in
the process will creates
an out of specification
Case III : 6σ > USL - LSL
Process is incapable of
meeting the specification set
by the customer. To correct
this problem, management
intervention will be
necessary. The capability of
the process cannot be
improved without changing
the existing process
Capability Index, Cp
Cp > 1.00 : Case I situation exists. The greater this value, the better
Cp = 1.00 : Case II situation exists. This is not optimal, but it is feasible
Cp < 1.00 : Case III situation exists. Value of less than 1 are undesirable
and reflect the process’s inability to meet the specification
Capability Ratio, Cr
Cr < 1.00 : Case I
Cr = 1.00 : Case II
Cr > 1.00 : Case III
Centering of the Process, Cpk
Where Z (min) is the smaller of :
Cpk = negative number
Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1
Note : Cpk is the ratio that reflects how the process is performing in relation to a nominal, center or
target value.
When Cp has a value of 1.0 or greater, the process is producing
product capable of meeting specifications
The Cp value does not reflect process centering
When the process is centered Cp = Cpk
Cpk is always less than or equal to Cp
When Cp is greater than or equal to 1.0 and Cpk has a value of 1.00 or
more, it indicates the process is producing product that conforms to
When Cpk has a value less than 1.00, it indicates the process is
producing product that does not conform to specifications
A Cp value of less than 1.00 indicates that the process is not capable
A Cpk value of zero indicates the process average is equal to one of
the specification limits
A negative Cpk value indicates that the average is outside the
specification limits
Hotels use statistical information and control charts to
track their performance on a variety of indicators.
Recently a hotel manager has been asked whether or not
his team is capable of maintaining scores between 8 and
10 (on a scale of 1 to 10) for “overall cleanliness of room”.
The most recent data has a mean of 8.624, a standard
deviation of 1.446 and n = 10. calculate and interpret Cp
and Cpk
State of Process Control
 A process is considered to be in a state of control or
under control when the performance of the process
falls within the statistically calculated control limits and
exhibits only chance / common causes
 When a process is under control, it is considered stable
and the amount of future variation is predictable
 Several benefits to a stable process:
There is a rational basis for planning
 Quality levels are predictable
 When improvements are made, the effects of the changes
can be determined quickly and reliably
Control Chart Interpretation
 Correct interpretation of control charts is essential to
managing a process
 Understanding the sources and potential causes of
variation is critical to good management decisions
 Misinterpretation can lead to a variety of losses,
Blaming people for problems that they cannot control
 Spending time and money looking for problems that do
not exist
 Spending time and money on process adjustments or new
equipment that are not necessary
 Taking action where no action is warranted
There are two circumstances under which the control
chart is revised and new limits calculated
 If a charts exhibits good control and any changes
made to improve the process are permanent
 When the new operating conditions become routine
and no out-of-control signals have been seen
The revisions provide a better estimate of the
population standard deviation – a better understanding
of the entire process can be gained
Control limits are also revised if pattern exist – provided
that the patterns have been identified and eliminated
The new limits will reflect the changes and
improvements made to the process – used to judge the
process behavior in the future
Steps taken to revise the charts:
1. Interpret the original chart
2. Isolate the cause
3. Take corrective action
4. Revise the chart
It is necessary to remove any undesirable points or
groups of points, the causes of which have been
determined and corrected
If no cause can be found and corrected, then the points
cannot be removed from the chart
The points removed will equal zero and the
calculations will continue from there

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