Control Chart Concepts

STATISTICAL QUALITY IMPROVEMENT(SQQS 3063)

THEORIES & CONCEPTS OF CONTROL CHARTS

Chapter 5 2Introduction to Statistical Quality Control, 6th Edition by Douglas C. Montgomery.Copyright (c) 2009 John Wiley & Sons, Inc.

• A process is operating with only chance causes of variation present is said to be in statistical control.

• A process that is operating in the presence of assignable causes is said to be out of control.

Chance and Assignable Causes of Variation


• A control chart contains– A center line– An upper control limit– A lower control limit

• A point that plots within the control limits indicates the process is in control– No action is necessary

• A point that plots outside the control limits is evidence that the process is out of control– Investigation and corrective

action are required to find and eliminate assignable cause(s)

• There is a close connection between control charts and hypothesis testing

Statistical Basis of the Control Chart


Photolithography Example

• Important quality characteristic in hard bake is resist flow width• Process is monitored by average flow width

– Sample of 5 wafers– Process mean is 1.5 microns– Process standard deviation is 0.15 microns

• Note that all plotted points fall inside the control limits– Process is considered to be in statistical control



Shewhart Control Chart Model





More Basic Principles

• Charts may be used to estimate process parameters, which are used to determine capability

• Two general types of control charts– Variables

• Continuous scale of measurement• Quality characteristic described by central tendency and a measure

of variability– Attributes

• Conforming/nonconforming• Counts

• Control chart design encompasses selection of sample size, control limits, and sampling frequency

• Control Chart design: An important factor in control chart. This includes selection of

Sample size – how many measurements (n)Control limits – how many sigma Sampling frequency – how often (interval of ½

hour, an hour, a day, etc)


Types of Process Variability


• Stationary and uncorrelated data vary around a fixed mean in a stable or predictable manner

• Stationary and autocorrelated successive observations are dependent with tendency to move in long runs on either side of mean

• Nonstationary process drifts without any sense of a stable or fixed mean

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Reasons for Popularityof Control Charts

1. Control charts are a proven technique for improving productivity.

2. Control charts are effective in defect prevention.

3. Control charts prevent unnecessary process adjustment.

4. Control charts provide diagnostic information.

5. Control charts provide information about process capability.


CHOICE OF CONTROL LIMITS

• 3-Sigma Control Limits– Probability of type I error is 0.0027

• Probability Limits– Type I error probability is chosen directly– For example, 0.001 gives 3.09-sigma control limits

• Warning Limits– Typically selected as 2-sigma limits

• 3 Sigma control limit (99.7%):

Control limits on a control chart are commonly drawn at 3s from the center line because 3-sigma limits are a good balance point between two types of errors:

• Type I errors (e.g.1 - 0.997 = 0.0027) occur when a point falls outside the control limits even though no special cause is operating. The result is a witch-hunt for special causes and adjustment of things here and there. The tampering usually distorts a stable process as well as wasting time and energy.

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• Type II or beta errors occur when you miss a special cause because the chart isn't sensitive enough to detect it. In this case, you will go along unaware that the problem exists and thus unable to root it out.

All process control is vulnerable to these two types of

errors. The reason that 3-sigma control limits balance

the risk of error is that, for normally distributed data,

data points will fall inside 3-sigma limits 99.7% of the

time when a process is in control. This makes the witch

hunts infrequent but still makes it likely that unusual

causes of variation will be detected.17

Probability limits• Sometimes the control limit is not specified by the

multiple of , but in the form of probability limits (Type I error).

• For example: 0.001 probability limit. Then, the multiple for equals to 3.09.

• Probability limit is mentioned in a one direction, whereas it is supposed to be both direction.

US use the 3 sigma approach, but UK and western

Europe use probability limit. If the distribution of

quality characteristic is normal, then, the difference

between the two limits is very little.

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Warning limits

Some control charts have 2 control limits. • The outer limit – 3 sigma

Search for assignable causes. This is the action limit.

• The inner limit – 2 sigma

This is the warning limit. Can increase sensitivity in the control chart and thus can signal a shift more quickly (may also increase the risk of false alarm – Type I error)


Sampling and Rational Subgroups

• Although control charts are meant for monitoring a production process as a whole, taking product measurements on an assembly line can be costly, can slow down a process, or can diminish product quality

• Naturally, companies do not usually measure all products as this may be counterproductive or economically unfeasible. As a compromise, most companies select samples in the form of rational subgroups 20

Sampling

• A sample is taken at an appropriate interval to check the performance of the process

• Sampling must be made at the appropriate frequency so that:

No extra samples are taken – minimize cost Enough samples for the control chart are

taken to be most reliable for detecting out of control conditions

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Rational Subgroups• “Rational Subgroups” is an important concept

in collecting data for control charts• The subgroups or samples should be selected

so that if special (assignable ) causes are present, then:

The chance for differences between subgroups will be maximized

The chance for differences due to these assignable causes within a sample should be minimized

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• Samples should be selected in a way that explores the variation of interest to you. The variation of interest might be shift-to-shift, day-to-day, hour-to-hour, batch-to-batch, etc. One possible way of looking at control charts is that they are a statistical test to determine if the variation from subgroup to subgroup is consistent with the average variation within the subgroups. A basic principle is to choose the subgroups in a way that will give the maximum chance for the measurements in each subgroup to be alike (within) and the maximum chance for the subgroups to be different (between).


Pattern on Control Charts

A control chart may indicate an out of control

condition when one or more points fall beyond

the control limits or when

the plotted points exhibit

some nonrandom pattern

of behavior.

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Although all points are within the limit, the points

do not indicate statistical control because:• Pattern is very nonrandom in appearance;19

of 25 points plot below the center line,

while only 6 plot above• Following 4th point,

5 points in a row increase

in magnitude, a run up• There is also an unusually

long run down beginning

with 18th point

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A run is defined as as sequence of observation

of the same type. • Run up: above the center line• Run down: below the center line• Two points in a row above/below the center

line is known as run of length 2• A run of length 8 or more points has very low

probability of occurrence in a random sample of points. Therefore, if this occurs, it is often taken as a signal of an out of control condition

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• Other pattern: cyclical behavior• This indicates that a problem with the process

such as operator fatigue, raw material deliveries, heat or stress build up.

• This problem chart could be improved by eliminating or reducing the sources of variability causing this cyclic behavior

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The Cyclic Pattern



Discussion of the Sensitizing Rules

Phase I and Phase II

The construction of a standard control chart

involves two phases i.e. Phase I and Phase II.• Phase I : Primarily assist operating personnel

in bringing the process into a state of statistical control by determining if the process has been in control over the period of time the data were collected and to see if reliable control limit can be established to monitor future production

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• Phase II: Use the control chart in Phase I to monitor the process by comparing the sample statistic for each successive sample as it is drawn from the process to the control limit. –Process is assumed to be reasonably stable–Emphasis is on process monitoring, not on

bringing an unruly process into control


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Control Chart Concepts