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Statistical Process Control
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Statistical Process Control
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Why do we Need SPC?
To Help Ensure Quality
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Quality means fitness for use
- quality of design
- quality of conformance
Quality is inversely proportional to variability.
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What is Quality?
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• Transcendent definition: excellence
– Excellence in every aspect
• PERFORMANCE
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How well the output does what it is supposed to do.
• RELIABILITY
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The ability of the output (and its provider) to
function as promised
What is Quality
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• CONVENIENCE and ACCESSIBILITY
How easy it is for a customer to use the product or
service.
• FEATURES
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The characteristics of the output that exceed the output’s
basic functions.
• EMPATHY
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The demonstration of caring and individual attention to
customers.
What is Quality
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• CONFORMANCE
The degree to which an output meets specifications or
requirements.
• SERVICEABILITY
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How easy it is for you or the customer to fix the output
with minimum downtime or cost.
• DURABILITY
How long the output lasts.
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What is Quality
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• AESTHETICS
How a product looks, feels, tastes, etc.
• CONSISTENCY
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The degree to which the performance changes over
time.
• ASSURANCE
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The knowledge and courtesy of the employees and
their ability to elicit trust and confidence.
What is Quality
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• RESPONSIVENESS
Willingness and ability of employees to help
customers and provide proper services.
• PERCEIVED QUALITY
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The relative quality level of the output in the eyes of
the customers.
What is Quality
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• Product-based definition: quantities of
product attributes
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– Attributes are non-measurable types of data
• What are all the different features
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What is Quality
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• User-based definition: fitness for intended
use
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– How well does the product meet or exceed the
expected use as seen by the user
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What is Quality
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• Value-based definition: quality vs. price
– How much is a product or service going to
cost and then how much attention to quality
can we afford to spend.
– Cheap product, little quality
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What is Quality
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• Manufacturing-based definition:
conformance to specifications
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– The product has both variable specifications
(measurable) and attributable specifications
(non-measurable) that manufacturing monitors
and ensures conformance
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What is Statistical Process
Control?
What is SPC
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• SPC, Statistical Process Control, is a
process that was designed in the 1930’s to
characterize changes in process variation
from a standard.
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– It can be used for both attributes and variables
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• The basic tool used in SPC is the control
chart
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– There are various types of control charts
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•
•
•
•
•
Mean chart
Range chart
Median chart
Mean and range chart (X and R)
c chart
p chart, etc.
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• Control charts
– a graphical method for detecting if the
underlying distribution of variation of some
measurable characteristic of the product seems
to have undergone a shift
– monitor a process in real time
– map the output of a production process over
time and signals when a change in the
probability distribution generating observations
seems to have occurred
– are based on the Central Limit Theory
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• Central Limit Theorem says that the
distribution of sums of Independent and
Identically Distributed (IID) random
variables approaches the normal
distribution as the number of terms in the
sum increases.
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– Things tend to gather around a center point
– As they gather they form a bell shaped curve
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• The center of things are described in various ways
– Geographical center
– Center of gravity
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• In statistics, when we look at groups of numbers,
they are centered in three different ways
– Mode
– Median
– Mean
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Mode
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• Mode
– Mode is the number that occurs the most
frequently in a group of numbers
• 7, 9, 11, 6, 13, 6, 6, 3,11
– Put them in order
– 3, 6, 6, 6, 7, 9, 11, 11, 13
– 3, 6, 6, 6, 7, 9, 11, 11, 13
• The mode is 6
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Median
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~ is like the geographical center,
• Median (X)
it would be the middle number
• 7, 9, 11, 6, 13, 6, 6, 3,11
– Put them in order
– 3, 6, 6, 6, 7, 9, 11, 11, 13
– 3, 6, 6, 6, 7, 9, 11, 11, 13
• 7 is the median
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Mean
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• Mean is the average of all the numbers and
_
is designated
by the symbol μ for
X
population mean and for sample mean
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– The mean is derived by adding all the numbers
and then dividing by the quantity of numbers
– X1 + X2 + X3 + X4 + X5 + X6 + X7 +…+Xn
n
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… to the nth number …
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… multiplied by 1 over n
The mean …
The sum of …
n
_
X
=
… is equal to …
… from the first number …
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n
Σ
i=1
Xi
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… all the numbers …
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If we had the numbers, 1,2,3,6 and 8, you can
see below that they “balance” the scale. The
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mean is not geometric center but like the center
of gravity
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6
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8
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• As the numbers are accumulated, they are
put in order, smallest to largest, and the
number or each number can then be put into
a graph called a Histogram
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• A normal curve is considered “normal” if
the following things occur
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– The shape is symmetrical about the mean
– The mean, the mode and the median are the
same
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X
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Variation
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X
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Workshop I
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Central Tendency
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Variation
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• The numbers that were not exactly on the
mean are considered “variation”
– When weighing the candy, the manufacturer is
targeting a specific weight
– Those that do not hit the specific weight are
variations.
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• Will there always be variation?
• There are two types of variation
– Common cause variation
– Special cause variation
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Common Cause
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• Common cause variation is that normal
variation that exists in a process when it is
running exactly as it should
– eg. In the production of that candy
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• When the operator is running the machine properly
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Within cycle time allotted for each drop of candy
Candy is properly placed on trays
Temperatures are where they need to be
Mixture is correct
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• When the machine is running properly
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–
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Tooling is sharp and aligned correctly
All components are properly maintained
Voltage is correct
Safety interlocks are properly set
• When the material is correct
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–
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Hardness
Size
Thickness
Blend
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• When the method is correct
– Right tonnage machine
– Proper timing
• When the environment is correct
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Ambient temperature
Ambient humidity
Dust and dirt
Corrosives
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• When the original measurements are correct
– Die opening dimensions
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• As we have just reviewed, common cause
variation cannot be defined by one
particular characteristic
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– It is the inherent variation of all the parts of the
operation together
• Voltage fluctuation
• Looseness or tightness of bearings
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• Common cause variation must be optimized
and run at a reasonable cost
– Don’t spend a dollar to save a penny
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Special Cause
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• Special cause is when one or more of the
process specifications/conditions change
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Temperatures
Tools dull
Voltage drops drastically
Material change
Stops move
Bearings are failing
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• Special cause variations are the variations
that need to be corrected
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– But how do we know when these problems
begin to happen?
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Statistical Process Collection and Control!
Workshop II
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Range and Mean Variation
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• Variation in the process occurs two major
ways.
– The range changes
– The mean changes
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• Range is the smallest data point subtracted
from the largest data point
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– It represents the total data spread that has been
sampled
– If the range gets smaller, or more significantly
if it gets larger, something has changed in the
process.
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X
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Changed Process
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Normal Process
Changed Range
Normal Range
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X
Changed Process
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Normal Process
Changed Range
Normal Range
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• Give me some examples that you would
think would cause the range to tighten up
• How about loosen up?
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• If you remember, the mean is the point
around which the data is centered
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– If the mean changes, then it would mean that
the central point has changed
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• Now lets see what the curve would look like
if the mean changed
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X
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• What might cause the mean to shift?
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• Most of the time we see both happen to
some degree. In the previous example, you
may have noticed that the range also
became smaller
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X’
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X
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• It is important that when parts are being
sampled from an entire population of
production, that they are randomly selected
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– This is called statistical collection of data
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• Why do you think the sampling should be
done randomly?
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Workshop III
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Black beads and White beads
Statistical Process Control
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Gathering Meaningful Data
Why do we need Data?
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• Assessment
– Assessing the effectiveness of specific
techniques or corrective actions
• Evaluation
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– Determine the quality of a process or product
• Improvement
– Help us understand where improvement is
needed
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• Control
– To help control a process and to ensure it does
not move out of control
• Prediction
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– Provide information and trends that enables us
to predict when an activity will fail in the future
• Characterization
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– Help us understand weaknesses and strengths
of products and processes
Deciding on the Data
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• Before data is collected, a plan needs to be
put in place
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– If there is no clear objective, then the data is
very likely not going to be useful to anyone
– Do not collect data for just because you are
supposed to
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Deciding on the Data
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• Base the data being collected on the
hypothesis of the process to be examined
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– Know what the expected results should be
– Know what parts of the process can give
meaningful data
– Understand the process as a whole
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Deciding on the Data
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• Consider the impact of the data collection
process on the whole organization
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– Data collection can be very expensive and time
consuming
– Data collection imparts a “Big Brother is
Watching” feeling on employees
– Data collection causes employees to work
harder and more focused than they really would
– Data gathering is tedious and requires a
commitment
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Deciding on the Data
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• Data gathering must have complete, total,
100% support from management
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– Data gathering inherently takes time, analysis
even more time, and positive results even more
time than that.
– Management can be very impatient some times
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Guidelines for Useful Data
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• Data must contain information that allows for
identification of types of errors or changes made
• Data MUST include cost to making changes
• Data must be compared in some way to
expectation or specification
• Benchmarks must be available from historical data
if possible
• Data must be clear enough to allow for
repeatability in the future, or on other projects
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• Data should cover the following
–
–
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What, by whom and for what purpose
What are the specifications
Who will gather the data
Who will support the data gathering
• And have they agreed
– How will the data be gathered
– How will the data be validated
– How will the data be managed
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Tips on data collection
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• Establish goals and the data gathering and
define and be prepared for questions that
may occur during gathering
• Involve all the people who are going to be
affected by the gathering, analysis and
corrective/preventive actions
• Keep goals small to start with, because the
data could be almost overwhelming
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• Design the data collection form to be
simple, so that comparison is easy
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– Provide training in the process to be followed
• Include any validation criteria (check
against known calibrated gauge, etc.) and
record results.
• Automate as much as possible
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• The data will be collected onto a collection
form that will then be transferred to a
control chart
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– In many cases the two forms are the same
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• These control charts will help us to decide if
the process is running as we want, and
hopefully tell us that it is beginning to
change so we can avoid producing bad parts
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• These charts can be grouped into two major
categories
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– They are grouped based on the type of data
being collected
• Attributes data
• Variables data
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Attributes
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• Attributes data
– Non-measurable characteristics
• Can be very subjective
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– We must develop specific descriptions for each
attribute
• Called operational definitions
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Blush
Splay
Scratched
Color
Presence, etc.
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Attributes
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Attributes
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– Observations are counted
• Yes/No
• Present/Absent
• Meets/Doesn’t meet
– Visually inspected
• Go/no-go gauges
• Pre-control gauges
– Discreet scale (has limits)
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Attributes
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• If color happens to be an attribute that is being
inspected for
– Typically, the expected color sample is given
– Maybe a light and dark sample is given
• The acceptable range is in between.
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– A reject is not measured, just counted as one
• Scratches might be given by location, length of
scratch, depth of scratch, width of scratch, visible
at a certain distance, etc.
Attributes
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• The description becomes more detailed as
the quality becomes more critical.
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– It is very important that inspection is not used
to sort out chronic problems
– Inspection is used to collect data
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• Inspection should be as close to the process as
possible and by the operators
Attributes
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• On an attributes chart, the characteristic
name is used
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– Scratched, wrong color, specks, bubbles, etc.
– Every time an unacceptable characteristic is
found, it is marked on the data collection sheet
using a method such as counting marks ( IIII )
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Variables
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• Developed through measuring
– Is very objective
– Can be temperature, length, width, weight,
force, volts, amps, etc.
• Uses a measuring tool
– Scale
– Meter
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• Scale increments are specified by design
engineers, customer, etc.
Variables
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Variables
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– As with attributes, variables inspection should
be done as close to process as possible,
preferably by the operator.
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• Inspection should not be done to sort but for data
collection and correction of the process
• This will allow for quick response and rapid
correction, minimizing defect quantities
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Variables
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• Important considerations for data collection
– The measuring tools must be accurate and
precise
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• Accurate means that it is can produce similar
measurements as a standard
• Precise means repeatability
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Variables
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– Sampling program used
• Sampling involves removing a representative
quantity of components from the normal production
and inspecting them
• The sample size and frequency is very important in
remaining confident that any negative process
changes are being uncovered.
• Sampling is also dependent on the type of charting
you will be doing, and will be discussed with each
chart type.
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Variables
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• After the variables are being collected, they
can then be visualized as a Normal Curve
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– The mean can be calculated and another
characteristic of the normal curve, the standard
deviation, can also be calculated
– There are many low priced and free programs
to allow input and automatic calculation
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Standard Deviation of Variables
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• Standard deviation is characteristic of all normal
curves
– One standard deviation on each side of the mean would
represent 68.26% of the area beneath the curve (or
68.26% of all data)
– Two standard deviations on each side of the mean
would represent 95.44% of the area beneath the curve
– Three standard deviations on each side of the mean
would represent 99.74% of the area.
– Six standard deviations on each side of the mean would
represent 99.9999966% of the area (3.5 defects per
million).
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X
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68.26%
.240
.242
-3σ
-2σ
.244
.246
-σ 95.44%
99.74%
.248
.250
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4
σ
2σ
3σ
.252
.254
.256
.258
.260
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σ 2σ 3σ
-3σ -2σ -σ
.240
.242
.244
.246
-3
-2
.248
-1
0
.250
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.252
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.254
.256
.258
.260
Standard Deviation
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• Whenever we talk about standard deviation,
there are two types
– Population standard deviation
– Sample standard deviation
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Population Standard Deviation
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• The population is considered the greater lot
that the sample is taken from
– A shipment
– A days production
– A shifts production
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• The symbol for the population standard
deviation is the Greek letter sigma (σ)
Sample Standard Deviation
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• The sample standard deviation is taken from
the specific sample data, which is
considerably smaller than the sample that
would be taken if sampling the entire
population
• The symbol for the sample standard
deviation is the lower case “s”
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• The formula for the standard deviation is
σ =
√
Σ (X – μ )2
p-1
s =
√
Σ (X – X )2
n-1
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s =
√
Σ (X – X )2
n-1
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Workshop IV
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Sampling Plans
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Statistical Process Control
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Pareto Diagrams
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• Vilfredo Pareto
– Italy’s wealth
• 80% held by 20% of people
• Used when analyzing attributes
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– Based on results of tally numbers in specific
categories
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• What is a Pareto Chart used for?
– To display the relative importance of data
– To direct efforts to the biggest improvement
opportunity by highlighting the vital few in
contrast to the useful many
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Constructing a Pareto Chart
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• Determine the categories and the units for
comparison of the data, such as frequency, cost, or
time.
• Total the raw data in each category, then determine
the grand total by adding the totals of each
category.
• Re-order the categories from largest to smallest.
• Determine the cumulative percent of each
category (i.e., the sum of each category plus all
categories that precede it in the rank order, divided
by the grand total and multiplied by 100).
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Constructing a Pareto Chart
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• Draw and label the left-hand vertical axis with the
unit of comparison, such as frequency, cost or
time.
• Draw and label the horizontal axis with the
categories. List from left to right in rank order.
• Draw and label the right-hand vertical axis from 0
to 100 percent. The 100 percent should line up
with the grand total on the left-hand vertical axis.
• Beginning with the largest category, draw in bars
for each category representing the total for that
category.
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Constructing a Pareto Chart
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• Draw a line graph beginning at the righthand corner of the first bar to represent the
cumulative percent for each category as
measured on the right-hand axis.
• Analyze the chart. Usually the top 20% of
the categories will comprise roughly 80% of
the cumulative total.
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• Lets assume we are listing all the rejected products
that are removed from a candy manufacturing line
in one week
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– First we put the rejects in specific categories
•
•
•
•
•
•
•
No wrapper
No center
Wrong shape
Short shot
Wrapper open
Underweight
Overweight
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– Then we tally how many of each category we
have
•
•
•
•
•
•
•
No wrapper - 10
No center - 37
Wrong shape - 53
Short shot - 6
Wrapper open - 132
Underweight - 4
Overweight – 17
– Get the total rejects - 259
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– Develop a percentage for each category
•
•
•
•
•
•
•
No wrapper – 10/259 = 3.9%
No center – 37/259 = 14.3%
Wrong shape – 53/259 = 20.5%
Short shot – 6/259 = 2.3%
Wrapper open – 132/259 = 51%
Underweight – 4/259 = 1.5%
Overweight – 17/259 = 6.6%
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• Now place the counts in a histogram, largest to
smallest
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70
60
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51%
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40
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20.5%
14.3%
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6.6%
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0
Wrapper
Open
No Center
3.9%
No Wrapper
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2.3%
1.5%
Underweight
• Finally, add up each and plot as a line diagram
100
92.4%
96.3%
98.6%
100.1%
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85.8%
0011 0010
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71.5%
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51%
50
40
30
20.5%
14.3%
20
6.6%
10
0
Wrapper
Open
No Center
3.9%
No Wrapper
2
4
2.3%
1.5%
Underweight
100
92.4%
96.3%
98.6%
100.1%
90
85.8%
0011 0010 1010 1101 0001 0100
1011
80
71.5%
70
60
1
51%
50
40
30
20.5%
14.3%
20
6.6%
10
0
Wrapper
Open
No Center
3.9%
No Wrapper
2
4
2.3%
1.5%
Underweight
Workshop V
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Pareto Charting
1
2
4
• After the problem is corrected, continue the data collection
65
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60
55
50
41.7%
1
45
40
35
29.1%
30
25
20
13.4%
15
7.8%
10
0
4
4.7%
5
Wrong Shape
No Center
Overweight
No Wrapper
2
Short Shot
3.1%
Underweight
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• Sometimes other information would be
better
1
– Use the scrap cost of each rejected part
– Use rework cost of each rejected part
2
4
• This can be especially useful if the rejects
are all at about the same quantity
Here are some tips
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• Create before and after comparisons of Pareto
charts to show impact of improvement efforts.
• Construct Pareto charts using different
measurement scales, frequency, cost or time.
• Pareto charts are useful displays of data for
presentations.
• Use objective data to perform Pareto analysis
rather than team members opinions.
1
2
4
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• If there is no clear distinction between the
categories -- if all bars are roughly the same height
or half of the categories are required to account for
60 percent of the effect -- consider organizing the
data in a different manner and repeating Pareto
analysis.
• Pareto analysis is most effective when the problem
at hand is defined in terms of shrinking the
product variance to a customer target. For
example, reducing defects or elimination the nonvalue added time in a process.
1
2
4
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• The accumulative curve can also be
removed, especially if there is not distinct
shift in the slope of the curve.
1
2
4
Statistical Process Control
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Control Charts
1
2
4
X and R
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• When are they used
– When you need to assess variability
– When the data can be collected on an ongoing
basis
– When it can be collected over time
– When we are using variables
– Subgroups must be more than 1
1
2
4
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• How is it made
– First, complete the header information
1
• This is important so that each collection can be
properly understood and separated from others.
– Record the data
2
4
• Not just data but significant observations.
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– Calculate the mean of each subgroup of
information
– Calculate the range for each subgroup
1
2
4
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• When the first chart is finished, calculate
the grand average, X
– This is the average of the averages
• Calculate the average of the ranges
1
2
4
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• Calculate the control limits
– The formula is
– UCLx = X + ( A2 x R )
– LCLx = X – (A2 x R )
1
2
4
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• Calculate the Range limits
– UCLR = D4 x R
– LCLR = D3 x R
1
2
4
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Weighting Factors
Subgroup Size
A2
D3
D4
2
1.880
0
3.267
3
1.023
0
2.574
4
0.729
0
2.282
5
0.577
0
2.114
1
2
4
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• Scale the charts
– Use a scale that will ensure the numbers will all
fit in the chart and also that the new numbers
will also fit, even outside the control limits
1
2
4
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• Find the largest X value and compare to the UCL.
Use the larger
• Find the smallest X value and compare to the
LCL. Use the smaller
• Subtract the smaller from the larger and write
down the difference
• Divide the difference by 2/3 the number of lines
on the chart (30 lines on this chart) and round
upward if needed.
1
2
4
Control Chart Interpretation
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• Interpret the Data
–
–
–
–
Any point lying outside the control limits
Seven points in a row above or below the average
Seven points in a row going in one direction
Any non-random patterns
• Should look like the normal curve
– Too close to average
– Too far from average
– Cycling
1
2
4
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• Declare whether in control or out of control
• Respond to the information
1
2
4
PreControl
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• PreControl is a form of X bar and R chart…
without the chart.
1
2
– It is to be used only when a process has been
proven to be in control
– It is to be used to stop production before bad
parts are produced
4
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• A definitive variable must be selected that
will tell if the process is moving out of
control
– ie. weight, length, warp, etc.
1
2
4
• A PreControl gauge must be made to
measure the variable
– Dial indicator, weigh scale, taper gauge, etc.
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• The total specified range that is acceptable
is to be calculated for standard deviation,
and plus or minus two standard deviations
are the control limits for the PreControl
gauge
1
2
4
– The third standard deviation is red
– The second standard deviation is yellow
– The first standard deviation is green
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• Rules of PreControl
– At start up, after the process has been set to
know parameters that produces good parts
1
2
• Collect 5 samples in a row from each cavity, mold,
core, station, whatever.
• Each are checked to the definitive PreControl gauge
• All five of each must be in the green
4
– If not, the process must be brought under control
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• Once all five are in the green the process is
started up
• At selected intervals (typically one an hour
or once every half hour) two samples in a
row are selected from each cavity, etc.
1
2
4
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– If the both are green, process is ok
– If the first is green, but the second is yellow, ok
but be alert
– If both are yellow, stop the process and go back
to start up
– If either is red, stop the process and go back to
start up
1
2
4
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• We can determine how far data that is being
gathered is out of specification
1
– Let us look at some make believe data
2
4
Using Control Charting
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1
2
4
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• We are investigating the time it will take for
a process to pick up a part, move it to
another area and place it on a conveyor
1
– It is unacceptable if it takes more than 14
seconds
2
4
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• Data collection has given us the following
information
1
2
– The grand mean is calculated to be 10.00
seconds
– Our sample size has been 5 observations each
hour
– Our calculated average range is 4.653
4
Estimated Standard Deviation
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• The formula to find the estimated standard
deviation is as follows
_
^ = R/d
σ
2
1
2
4
• This standard deviation is calculated to one
more decimal point than the original data
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• Now we know the grand mean, and we
know the estimated standard deviation
1
2
– From this we can calculate the location of the
“tails” of the distribution curve
– Add three of the estimated standard deviations
to the grand mean for the upper tail
– Subtract three of the estimated standard
deviations to the grand mean for the lower tail
4
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1
2
4
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• Now, let’s add the upper and lower
specification limits to the curve
1
– We know that the upper limit is 14 seconds
– We can assume the lower limit is 0 seconds
2
4
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1
2
4
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• Now we can see that some of the data is
outside the limits that have been set, but
how much?
1
2
4
Z scores
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• We are going to analyze the data and
determine how badly we are out of
specification.
1
– Z scores will help us to determine that
2
4
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Zupper = USL - X
σ^
1
2
4
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• Zupper = 14 – 10.00 / 2.00 = 2.00
1
2
• Now we will look at the following table to
see what percentage that is
• We see that the Z score is .0228, which
when changed to a percentage is 2.28%
4
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1
2
4
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• The Lower is done in a similar manner
Zlower = X - LSL
σ^
1
2
4
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• Zlower = 10.00 - 0 / 2.00 = 5.00
1
2
• Now we will look at the following table to
see what percentage that is
• We see that the Z score is so small that it is
insignifican, or it is 0%
4
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• Total percent outside the limits is
– 2.28% + 0% = 2.28%
1
2
• From this information, we can be pretty
sure that if we look at this process
continually, we will probably find that
2.28% of the time, we will be over the 14
seconds we are limiting the process to.
4
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• Now lets see if the process itself can
provide the limited time allowed
1
2
– We are going to look at the width of the
calculated spread vs. the specified spread
– The allowed spread is the difference between
the lower specification limit and the upper
specification limit
4
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• USL – LSL = 14
• Now we will take that specification spread
and divide it by six times the estimated
standard deviation
2
4
– Why six times the standard deviation
• 6 x 2.00 = 12
• 14 / 12 = 1.17
1
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• Cp = USL - LSL
6 σ^
1
2
4
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1
2
4
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• From the previous slide, it is obvious that
we want to make sure that the actual spread
is less than the specified spread
1
– A Cp of more than one is desirable, and the
higher the better
2
4
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• The only problem with this is we can have a
Cp of 2, but still be outside the limits of the
specification
• So there is another calculation that will
measure this. It is called Cpk
1
2
4
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• Cpk is simple. It is the smallest of the Z
scores, or Zmin, divided by 3
• In our case, that would be 2.00 / 3 = .67
• As in the Cp, we would want the result to be
greater than 1.
1
2
4
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• So in our example we had a Z score of 2.00,
which told us that 2.28% of the time we
were more than 14 seconds
• We also know that the process spread was
1.17, which means we can maintain the
proper time
• But we also found that the Cpk was .67,
which tells us we are outside of the limits
1
2
4
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• Our conclusion?
– We have a good process, but we need to find a
way to correct it so that we are well within the
limits as specified.
1
2
4