Statistics Quality: Control - Statistical Process Control and Using Control Charts

Processes

Processing an application for admission to a university and deciding

whether or not to admit the student.

Reviewing an employee’s expense report for a business trip and

issuing a reimbursement check.

Hot forging to shape a billet of titanium that will become part of a

medical implant for hip, knee, or shoulder replacement.

How processes are like populations

Think of a population containing all the outputs that would

be produced by the process if it ran forever in its present

state.

The outputs produced today or this week are a sample

from this population.

Systematic approach to process improvement A systematic approach to process improvement is captured in

the Plan-Do-Check-Act (PDCA). Plan the intended work. Then Do the implementation of the solution or change. Check to see if improvement efforts have been successful. Act by implementing the changes.

Process improvement toolkit

Statistical tools: Histogram – visualize the process in terms of location,

variability, distribution.

Pareto Chart – rank data about the frequency of process problems or causes.

Scatterplot – investigate whether two variables are related, identify root cause of problems.

Control Charts – monitor the process, alert us when the process has changed.

Process improvement toolkit Non statistical tools:

Flowchart

Cause and effect diagram

Process improvement toolkit : Flowchart

Describing processes graphically: A flowchart is a picture of the stages of a process

Process improvement toolkit: cause-and-effect diagram

Describes processes graphically

Organizes the logical relationships between the inputs

and stages of a process and an output.

Statistical process control

Goal: make a process stable over time and then keep it stable unless planned changes are made.

All processes have variation.

Statistical stability means the pattern of variation remains stable, not there there is no variation in the variable measured.

Statistical process control

Statistical Control: A variable that continues to be described by the same distribution when observed over time is said to be in control.

Control charts: Statistical tools that monitor a process and alert us when the process has been disturbed so that it is now out of control. This is a signal to find and correct the cause of the disturbance.

The idea of statistical process control A process that is in control has only common cause variation.

Common cause variation is the inherent variability of the system, due to many small causes that are always present.

When the normal functioning of the process is disturbed by some unpredictable event, special cause variation is added to the common cause variation.

We hope to be able to discover what lies behind special cause variation and eliminate that cause to restore the stable functioning of the process.

Control charts

Control charts distinguish between the common cause variation and the special cause variations.

A control chart sounds an alarm when it sees too much variation.

The point X indicates a data point for sample number 13 that is “out of control.”

charts for process monitoring Two-step procedure for applying control charts to a process:

1. Chart setup stage

a) Collect data from the process and assess stability.

b) Establish control by uncovering and removing special causes.

c) Set up control charts to maintain control.

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charts for process monitoring 2. Process monitoring

a) Observe the process operating in control for some time.

b) Understand usual process behavior.

c) Have a long run of data from the process.

d) Keep control charts to monitor the process because a special

cause could occur at any time.

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charts for process monitoring

Process monitoring conditions: o Measure a quantitative variable x that has a Normal

distribution. o Examples: diameter of a part, time to respond to a

customer call, etc. o The process has been operating in control for a long

period, so that we know the process mean µ and the process standard deviation σ that describe the distribution of x as long as the process remains in control.

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Law of Large Numbers

If we keep taking larger and larger samples, the statistic is guaranteed to get closer and closer to the parameter µ

Law of Large Numbers:

Draw observations at random from any population with finite mean µ. As the number of observations drawn increases, the mean of the observed values gets closer and closer to the mean µ of the population

It is reasonable to use to estimate µ

X

X

X

Central Limit Theorem

If a population has ( an approximate) N(µ, σ) distribution, then the sample mean of n independent observations has the Ν (µ, σ/√n) distribution

X

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68-95-99.7 Rule

68% of the data will be between µ − σ and µ + σ

95% of the data will be between µ − 2σ and µ + 2σ

99.7% of the data will be between µ − 3σ and µ + 3σ

charts for process monitoring 1. Take samples of size n from the process at regular intervals.

2. Plot the means of these samples against the order in which the

samples were taken.

3. We know that the sampling distribution of under the process-

monitoring conditions is Normal with a mean µ and a standard

deviation .

4. Draw a solid center line on the chart at height µ.

x

x

x

/ nσ

- Continued - The 99.7 part of the 68-95-99.7 rule for Normal

distributions says that as long as the process remains in control, 99.7% of the values of will fall within three standard deviations of the mean.

Draw dashed control limits on the chart at these heights.

The control limits mark off the range of variation in sample means that we expect when the process remains in control.

x

charts for process monitoring x

xµ

3n

σµ +

3n

σµ −

Sample mean that is out of control.

Stopping Rules One attempt to increase the power of control charts is by

adding supplementary stopping rules based on runs.

2 of 3 consecutive points fall outside warning (2-sigma) limits, but within control (3-sigma) limits.

4 of 5 consecutive points fall beyond 1-sigma limits, but within control limits.

8 consecutive points fall on one side of the centerline.

Assessing improvement efforts

a) shows a case where the control chart demonstrates a successful attempt to decrease the time needed to obtain lab results.

b) The control chart indicates no impact from the attempted process improvement.

Types of control charts There are two main types of control charts:

Variable Control Charts: , range, standard deviation, and cusum charts.

Attribute Control Charts: percent defective, number defective, and nonconformities charts.

x

General procedure for control charts Three-sigma (3σ) control charts for any statistic Q: 1. Take samples from the process at regular intervals and plot the

values of the statistic Q against the order in which the samples were

taken.

2. Draw a center line on the chart at height µQ, the mean of the statistic

when the process is in control.

3. Draw upper and lower control limits on the chart 3 standard

deviations of Q (σQ) above and below the mean.

4. The chart produces an out-of-control signal when a plotted point

lies outside the control limits.

s charts for process monitoring charts are easy to interpret if the process standard

deviation remains fixed.

Even the simplest description of a distribution should give both a measure on center and a measure of spread.

We must monitor both the process center, using an chart, and the process spread, using an s chart

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s charts for process monitoring – continued - Note that the standard deviation s does not have a

Normal distribution. The sampling distribution of s is skewed to the right.

Nonetheless, control charts for any statistics are based on the “plus or minus three standard deviations” idea, motivated by the 68-95-99.7% rule for Normal distributions.

Control charts are intended to be practical tools that are easy to use.

s charts for process monitoring – continued -

1. The mean of s is a constant times the process standard deviation σ.

This is the center line of an s chart.

2. The standard deviation of s is also a constant times the process

standard deviation.

3. The values of the c4 and c5 depend upon the sample size. For large

samples, c4 is close to 1.

4s cµ σ=

5s cσ σ=

s charts for process monitoring The control limits for an s chart are

The control chart constants c4, B5, and B6 depend on the sample size n.

6

5

UCLLCL

BB

σσ

=

=

Control chart constants

s charts for process monitoring

4s cµ σ=

5LCL B σ=

6UCL B σ=

In control Out of control

Comparing to s control charts

Do both types of control charts show the same information?

Here are two control charts for mesh tension

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Comparing to s control charts

Lack of control on an s chart is due to special causes that affect the observations within a sample differently.

examples: non-uniform material, new and poorly trained operator, mixing results from several machines or several operators.

Look at the s chart first.

Lack of control on an chart responds to s-type causes as well as to longer-range changes in the process, so it is important to eliminate the s-type causes first.

examples of longer-range change: new raw material that differs from that used in the past or a gradual drift in the process level caused by wear in a cutting tool.

x

x

Using control charts and R charts: an R chart is based on the sample range for spread instead of the sample standard deviation. Range = largest observation – smallest observation. Less informative than s charts.

Range is easier to calculate than standard deviation

The control limits for this chart type are: D3R(lower control limit) and D4R(upper control limit) Where R is the average of the ranges, and D3 and D4 (anti-biasing constants) are given in Table E.

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Process control record sheet

Setting up control charts

Comments on statistical control

Focus on the process rather than on the products. Inspection of finished products can ensure good quality, but is expensive and comes too late in the in the process.

Choose key points in the process using flowcharts and cause-and-effect diagrams.

Monitor the process at key points

Comments on statistical control Rational subgroups.

We want the variation within a sample to reflect only the item-to-item chance variation that (when in control) results from many small common causes.

Should we choose a random sample?

Samples of consecutive items are rational subgroups when we are monitoring the output of a single activity that does the same thing over and over again.

Don’t confuse control with capability!

There is no guarantee that a process in control produces products of satisfactory quality.

“Satisfactory quality” is measured by comparing the product to some standard outside the process, set by technical specifications, customer expectations, or the goals of the organization.

Statistical quality control only pays attention to the internal state of the process.

Don’t confuse control with capability!

Capability refers to the ability of a process to meet or exceed the requirements placed on it.

If a process is not in control, it is hard to tell if it is capable or not.

If a process is in control but does not have adequate capability, fundamental changes in the process are needed. Better training for workers, new equipment, better raw materials, etc.

Hope you enjoyed and retain! Good bye!