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Statistical Process Control

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Statistical Process Control

Why do we Need SPC?

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To Help Ensure Quality

Quality means fitness for use

- quality of design

- quality of conformance

Quality is inversely proportional to variability.

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What is Quality?

• Transcendent definition: excellence– Excellence in every aspect

• PERFORMANCE

How well the output does what it is supposed to do.

• RELIABILITY

The ability of the output (and its provider) to

function as promised

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What is Quality• CONVENIENCE and ACCESSIBILITY

How easy it is for a customer to use the product or

service.

• FEATURES

The characteristics of the output that exceed the

output’s basic functions.

• EMPATHY

The demonstration of caring and individual attention to

customers.

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What is Quality• CONFORMANCE

The degree to which an output meets specifications or

requirements.

• SERVICEABILITY

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

• AESTHETICS

How a product looks, feels, tastes, etc.

• CONSISTENCY

The degree to which the performance changes over

time.

• ASSURANCE

The knowledge and courtesy of the employees and

their ability to elicit trust and confidence.

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What is Quality

• RESPONSIVENESS

Willingness and ability of employees to help

customers and provide proper services.

• PERCEIVED QUALITY

The relative quality level of the output in the eyes of

the customers.

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What is Quality

• Product-based definition: quantities of product attributes– Attributes are non-measurable types of data

• What are all the different features

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What is Quality

• User-based definition: fitness for intended use– How well does the product meet or exceed the

expected use as seen by the user

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What is Quality

• 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

• Manufacturing-based definition: conformance to specifications– 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?

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What is SPC

• SPC, Statistical Process Control, is a process that was designed in the 1930’s to characterize changes in process variation from a standard.– It can be used for both attributes and variables

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• The basic tool used in SPC is the control chart– There are various types of control charts

• 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. – 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

• In statistics, when we look at groups of numbers, they are centered in three different ways– Mode

– Median

– Mean

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Mode

• 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

• Median (X) is like the geographical center, 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

• Mean is the average of all the numbers and is designated by the symbol μ for population mean and for sample mean – 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

_X

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Xi

i = 1

n1n Σ_

X =

The sum of …

… all the numbers …

… from the first number …

… to the nth number …

… multiplied by 1 over n

The mean …

… is equal to …

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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 mean

is not geometric center but like the center of gravity

1 2 3 6 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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40 45 50 55 60

45

40

35

30

25

20

15

10

5

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40 45 50 55 60

45

40

35

30

25

20

15

10

5

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• A normal curve is considered “normal” if the following things occur– The shape is symmetrical about the mean– The mean, the mode and the median are the

same

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_X

Variation

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_X

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Workshop I

Central Tendency

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Variation• 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.

• Will there always be variation?• There are two types of variation

– Common cause variation– Special cause variation

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Common Cause

• 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

• When the operator is running the machine properly– 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– Tooling is sharp and aligned correctly

– All components are properly maintained

– Voltage is correct

– Safety interlocks are properly set

• When the material is correct– Hardness

– Size

– Thickness

– Blend

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• When the method is correct– Right tonnage machine

– Proper timing

• When the environment is correct– Ambient temperature

– Ambient humidity

– Dust and dirt

– Corrosives

• 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– 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

• Special cause is when one or more of the process specifications/conditions change– 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 – But how do we know when these problems

begin to happen?

Statistical Process Collection and Control!

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Workshop II

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– 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

Normal Process

Normal Range

Changed Process

Changed Range

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_X

Changed ProcessChanged Range

Normal Process

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– 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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_X

_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

_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– 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

Black beads and White beads

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Statistical Process Control

Gathering Meaningful Data

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Why do we need Data?

• Assessment– Assessing the effectiveness of specific

techniques or corrective actions

• Evaluation– 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– Provide information and trends that enables us

to predict when an activity will fail in the future

• Characterization– Help us understand weaknesses and strengths

of products and processes

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Deciding on the Data

• Before data is collected, a plan needs to be put in place – 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

• Base the data being collected on the hypothesis of the process to be examined – 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

• Consider the impact of the data collection process on the whole organization – 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

• Data gathering must have complete, total, 100% support from management – 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• 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– 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

• 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– 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– In many cases the two forms are the same

• 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– They are grouped based on the type of data

being collected• Attributes data

• Variables data

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Attributes

• Attributes data– Non-measurable characteristics

• Can be very subjective

– We must develop specific descriptions for each attribute

• Called operational definitions– Blush– Splay– Scratched– Color– Presence, etc.

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Attributes

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Attributes

– 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

• 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.

– 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.

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Attributes

• The description becomes more detailed as the quality becomes more critical.– It is very important that inspection is not used

to sort out chronic problems– Inspection is used to collect data

• Inspection should be as close to the process as possible and by the operators

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Attributes

• On an attributes chart, the characteristic name is used– 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

• Developed through measuring– Is very objective– Can be temperature, length, width, weight, force,

volts, amps, etc.

• Uses a measuring tool– Scale– Meter

• Scale increments are specified by design engineers, customer, etc.

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Variables

1 2 3 4 5 6

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Variables

– As with attributes, variables inspection should be done as close to process as possible, preferably by the operator.

• 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

• Important considerations for data collection– The measuring tools must be accurate and

precise• Accurate means that it is can produce similar

measurements as a standard

• Precise means repeatability

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Variables

– 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

• After the variables are being collected, they can then be visualized as a Normal Curve– 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

• 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

99.74%

.240 .242 .244 .246 .248 .250 .252 .254 .256 .258 .260

-σ σ-2σ 2σ 3σ-3σ 95.44%

68.26%

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0-2 2-3 -1 31.240 .242 .244 .246 .248 .250 .252 .254 .256 .258 .260

-σ σ-2σ 2σ 3σ-3σ

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Standard Deviation

• Whenever we talk about standard deviation, there are two types– Population standard deviation– Sample standard deviation

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Population Standard Deviation

• The population is considered the greater lot that the sample is taken from– A shipment– A days production– A shifts production

• The symbol for the population standard deviation is the Greek letter sigma (σ)

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Sample Standard Deviation

• 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

Sampling Plans

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Statistical Process Control

Pareto Diagrams

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• Vilfredo Pareto– Italy’s wealth

• 80% held by 20% of people

• Used when analyzing attributes– 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

• 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

• 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

• Draw a line graph beginning at the right-hand 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– 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

0

10

20

30

40

50

60

WrapperOpen

No Center No Wrapper Underweight

70

80

90

100

51%

20.5%14.3%

6.6%3.9%

2.3% 1.5%

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• Finally, add up each and plot as a line diagram

0

10

20

30

40

50

60

WrapperOpen

No Center No Wrapper Underweight

70

80

90

100

71.5%

51%

20.5%

14.3%

85.8%

6.6%

92.4%

3.9%

96.3%

2.3%

98.6%

1.5%

100.1%

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0

10

20

30

40

50

60

WrapperOpen

No Center No Wrapper Underweight

70

80

90

100

71.5%

51%

20.5%

14.3%

85.8%

6.6%

92.4%

3.9%

96.3%

2.3%

98.6%

1.5%

100.1%

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Workshop V

Pareto Charting

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• After the problem is corrected, continue the data collection

0

5

10

15

20

25

30

35

40

45

Wrong Shape No Center Overweight No Wrapper Short Shot Underweight

50

55

60

65

41.7%

29.1%

13.4%

7.8%4.7% 3.1%

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• Sometimes other information would be better – Use the scrap cost of each rejected part– Use rework cost of each rejected part

• This can be especially useful if the rejects are all at about the same quantity

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Here are some tips• 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.

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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 non-value added time in a process.

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• The accumulative curve can also be removed, especially if there is not distinct shift in the slope of the curve.

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Statistical Process Control

Control Charts

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X and R

• 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

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• How is it made– First, complete the header information

• This is important so that each collection can be properly understood and separated from others.

– Record the data• Not just data but significant observations.

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– Calculate the mean of each subgroup of information

– Calculate the range for each subgroup

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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

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• Calculate the control limits– The formula is

– UCLx = X + ( A2 x R )

– LCLx = X – (A2 x R )

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• Calculate the Range limits– UCLR = D4 x R

– LCLR = D3 x R

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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

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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

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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.

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Control Chart Interpretation

• 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

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• Declare whether in control or out of control

• Respond to the information

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PreControl

• PreControl is a form of X bar and R chart… without the chart.– 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

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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.

• 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– 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• 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– 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.

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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

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• We can determine how far data that is being gathered is out of specification– Let us look at some make believe data

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Using Control Charting

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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– It is unacceptable if it takes more than 14

seconds

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• Data collection has given us the following information– 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

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Estimated Standard Deviation

• The formula to find the estimated standard deviation is as follows

• This standard deviation is calculated to one more decimal point than the original data

σ = R/d2^

_

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• Now we know the grand mean, and we know the estimated standard deviation– 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

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• Now, let’s add the upper and lower specification limits to the curve– We know that the upper limit is 14 seconds– We can assume the lower limit is 0 seconds

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• Now we can see that some of the data is outside the limits that have been set, but how much?

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Z scores

• We are going to analyze the data and determine how badly we are out of specification.– Z scores will help us to determine that

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Zupper = USL - X

σ̂

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• Zupper = 14 – 10.00 / 2.00 = 2.00

• 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%

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• The Lower is done in a similar manner

Zlower = X - LSL

σ̂

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• Zlower = 10.00 - 0 / 2.00 = 5.00

• 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%

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• Total percent outside the limits is– 2.28% + 0% = 2.28%

• 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.

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• Now lets see if the process itself can provide the limited time allowed– 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

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• USL – LSL = 14

• Now we will take that specification spread and divide it by six times the estimated standard deviation– Why six times the standard deviation

• 6 x 2.00 = 12

• 14 / 12 = 1.17

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• Cp = USL - LSLσ̂6

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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– A Cp of more than one is desirable, and the

higher the better

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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

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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.

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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

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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.