Stats Quality Control 2

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Statistical Quality Control 1

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STATISTICAL QUALITY CONTROL (SQC)

SQC consists of two major areas:

- Acceptance Sampling

- Process Control or Control Charts

Both of these statistical techniques may be applied to two kinds of data.

1. Attribute Data: when the quality characteristic being investigated is noted by either its

presence or absence and then classified as Defective or Non-Defective.

Example: Conforming or non-conforming

Pass or fail

Good or bad

2. Variable Data: The characteristics are actually measured and can take on a value along a

continuous scale.

Example: Length, Weight

Sometimes variable data can be transformed into attribute data. For example, the specifications

required for a shaft diameter (X) is 2" plus or minus 0.01".

If X falls within 1.99" and 2.01", then the shaft diameter is conforming to specifications and

hence is classified as good.

If X < 1.99" or X > 2.01", then the shaft diameter is not conforming to specifications and

hence classified as bad.

Thus, attribute data does not have information of “how much good or how much bad?” which the

variable data would have, because it would record the exact measurements of each shaft.

We will first study Acceptance Sampling.



2 Statistical Quality Control

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Acceptance Sampling:

Inspection provides a means for monitoring quality. For example, inspection may be performed on

incoming raw material, to decide whether to keep it or return it to the vendor if the quality level is not

what was agreed on. Similarly, inspection can also be done on finished goods before deciding

whether to make the shipment to the customer or not. However, performing 100% inspection isgenerally not economical or practical, therefore, sampling is used instead.

Acceptance Sampling is therefore a method used to make a decision as to whether to accept or to

reject lots based on inspection of sample(s). The objective is not to control or estimate the quality of

lots, only to pass a judgment on lots.

Using sampling rather than 100% inspection of the lots brings some risks both to the consumer and to

the producer, which are called the consumer's and the producer's risks, respectively. We encounter

making decisions on sampling in our daily affairs.

Example:

LOT (N)

SAMPLE (n)

STATISTICAL Inference is made on the quality of the lot by inspecting only the small sample drawn

from the lot.




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There are several Acceptance Sampling Plans:

- Single Sampling (Inference made on the basis of only one sample)

- Double Sampling (Inference made on the basis of one or two samples)

- Sequential Sampling (Additional samples are drawn until an inference can be made)

etc.

We will do Single Sampling plans only in this course.

Single Sampling Plans

A Single Sampling plan is characterized by n (the sample size) which is drawn from the lot and

inspected for defects. The number of defects (d) found are checked against c (the acceptance

number) and the procedure works as follows (clearly, d = 0, 1, 2, … n):

Example: Suppose n=100 and c=3, which means that if the number of defectives in the sample

(d) is equal to 0, 1, 2, or 3, then the lot will be accepted, and if d is 4 or more, then

the lot will be rejected.




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As mentioned earlier, inherent in a sampling plan are producer’s and consumer’s risk. These risks can

be depicted by the following table:

Lot is

Good Bad

Decision Accept No ErrorError

(Consumer’s

Risk)

RejectError

(Producer’s

Risk)

No Error

Formally, these risks are written as:

α : The producer's risk, is the probability that a lot with AQL will be rejected.

β : The consumer's risk, is the probability that a lot with LTPD will be accepted.

where

Acceptable Quality Level (AQL) = The quality level acceptable to the consumer

Lot Tolerance Percent

Defective (LTPD) = The level of "poor' quality that the consumer

is willing to tolerate only a small percentage

of the time.

In general, both the producer and the consumer want to minimize their risks. The choice of a well

designed sampling plan can help both the producer and the consumer maintain their respective risks at

acceptable levels to both. For example, α = 5% for AQL of 0.02 and β = 10% for LTPD of 0.08.




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Keeping c constant:

What is the effect on producer’s risk?

What is the effect on consumer’s risk?

Keeping n constant:

What is the effect on producer’s risk?

What is the effect on consumer’s risk?




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The Theory Behind Process Control

Let’s now turn our attention to the second major area of SQC, namely Process Control or Control

Charts, which directly affect the quality of a production or service process.

Every production process has a natural variation. For example, a process making shafts is adjusted sothat the shaft diameter will be 2". However, due to the natural variation in the manufacturing process,

not every shaft coming off the production line will have a diameter of exactly 2". There will be some

unexplained variation around the nominal value of 2". Therefore, some tolerance is built into the

design of the product to allow for this natural (random) variation. However, if the process goes out

of control, the variation may become more than that allowed by the design indicating the presence of

variation that can be explained(e.g., defective raw material, untrained worker, etc.). In this case some

action needs to be taken, the machine can be readjusted, replaced etc. The control charts show when

the variation in the process is within the limits of the natural variation and when it goes out of control.

Below are pictures that show various in-control and out-of-control situations for a process.




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Even when the process is in control, we need to make sure that the mean of the process is in

conformance with specifications as shown below.




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Continuous improvement in the process is possible by reducing the variation around the mean as

shown below.




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Charts Used with Variable Data:

Control charts are of two types corresponding to the type of data that is used, namely variable or

attribute data. We will study the popular control charts of both these types.

X and R-Charts (mean and range charts) are commonly used in dealing with variable data to monitorthe quality of a manufacturing process. The reason that both the charts have to be used together is

that both the mean and the variation (spread) have to be under control. Recall that the variable data

consists of actual measurements (e.g., shaft lengths, weight of bags in lbs, etc.). Let us take an

example of variable data that is pertinent for the acid content in a certain chemical product. The

operator measured and recorded the acid content of a sample of 4 units at a time at regular intervals

for at least 25 times. This variable data and the calculations performed with it are shown on the

following table. Also, given are the variable control charts (X and R charts) for the data.




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The Control Limits (UCL = Upper Control Limit and LCL = Lower Control Limit with the mean of

the data as the central line) for X and R Charts are established as follows:

X -Charts

X

X

g

i

i

g

=∑

UCL X x x= + 3σ LCL X x x= − 3σ

where: X = average of subgroup averages (the central line in the chart)

X i = average of the ith subgroup

g = number of subgroups

σ x is further estimated using the range information (i.e., 3σ x = A R2 ); as such the control limit

calculations are much simplified. The simplified control limits are as follows:

UCL X A R x= +

2 LCL X A R x= −

2

where A2 is a factor available in tables for different sample sizes (see table below).

R-Charts:

R

R

g

i

i

g

=∑

UCL R R R= + 3σ LCL R R R= − 3σ

where R = average of subgroup ranges (the central line in the chart) Ri = range if the ith subgroup

g = number of subgroups

Similarly, control limit calculations are much simplified and are:

UCL D R R = 4 LCL D R R = 3

where D3 and D4 are factors available in tables for different sample sizes (see table below).

Factors for Control Limitsn A2 D4 D3

2 1.880 3.268 0.0

3 1.023 2.574 0.0

4 0.729 2.282 0.0

5 0.577 2.114 0.0

6 0.483 2.004 0.0




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Let us now calculate the control limits for the given data, starting first with the Range (R) chart. This

is done first because the X chart requires R in determining its control limits. Therefore, naturally

we need to first check if the R chart is under control and use that R in the control limits of theX

chart.

n = __________ R = __________ X = __________

A2 = __________ D3 = __________ D4 = __________

R-Chart:

UCL = D4 R LCL = D3 R

= =

Does the R chart show that the process is under control? Yes or No and why?

X -Chart:

UCL = X + A2 R LCL = X - A2 R

= =

Does the X chart show that the process is under control? Yes or No and why?




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Another Example:

The St. Patrick's Hospital is starting a quality improvement project on the time to admit a patient

using X and R Charts. Determine the limits for the X and R charts and check to see if there are any

out-of-control points.

OBSERVATION OBSERVATION

Subgroup

NumberX1 X2 X3 X R Subgroup

NumberX1 X2 X3 X R

1 6.0 5.8 6.1 13 6.1 6.9 7.4

2 5.2 6.4 6.9 14 6.2 5.2 6.8

3 5.5 5.8 5.2 15 4.9 6.6 6.6

4 5.0 5.7 6.5 16 7.0 6.4 6.1

5 6.7 6.5 5.5 17 5.4 6.5 6.7

6 5.8 5.2 5.0 18 6.6 7.0 6.8

7 5.6 5.1 5.2 19 4.7 6.2 7.1

8 6.0 5.8 6.0 20 6.7 5.4 6.7

9 5.5 4.9 5.7 21 6.8 6.5 5.2

10 4.3 6.4 6.3 22 5.9 6.4 6.0

11 6.2 6.9 5.0 23 6.7 6.3 4.6

12 6.7 7.1 6.2 24 7.4 6.8 6.3

n = __________ R = __________ X = __________

A2 = __________ D3 = __________ D4 = __________

R-Chart:

UCL = D4 R LCL = D3 R

= =

Does the R chart show that the process is under control? Yes or No and why?

X -Chart:

UCL = X + A2 R LCL = X - A2 R

= =

Does the X chart show that the process is under control? Yes or No and why?




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Charts Used with Attribute Data

P-Chart, also known as the fraction or percent defective chart, is commonly used in dealing with

attribute data to monitor the quality of a manufacturing process. The mean percent defective ( p ) is

the central line. The upper and lower control limits are constructed as follows:

The mean proportion defective ( p ): The standard deviation of p:

p =Total Number of Defectives

Total Number Inspectedσ p

p p

n=

−( )1

where n = sample size.

Control Limits are:

UCL p Z p= + ∗σ LCL p Z p= − ∗σ

or

UCL p Z p p

n= + ∗

−( )1 LCL p Z

p p

n= − ∗

−( )1

Usually the Z value is equal to 3 (as was used in the X and R charts), since the variations within

three standard deviations are considered as natural variations. However, the choice of the value of Zdepends on the environment in which the chart is being used, and on managerial judgment.




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Example:

A computer manufacturer collects data from the final test of its product starting from the end of

January and all through February. Each day a sample of 2000 items are inspected and the number of

items in the sample that do not conform to specifications is recorded. The data is shown below:

Subgroup Number Number Percent Subgroup Number Number Percent

Number Inspected Defective Defective Number Inspected Defective Defective

(day) (day)

1 2000 55 13 2000 47

2 2000 18 14 2000 31

3 2000 50 15 2000 38

4 2000 42 16 2000 28

5 2000 39 17 2000 30

6 2000 52 18 2000 113

7 2000 47 19 2000 58

8 2000 34 20 2000 34

9 2000 29 21 2000 19

10 2000 53 22 2000 30

11 2000 45 23 2000 17

12 2000 26 24 2000 46

n = __________ p = __________ σp = __________

UCL = __________ LCL = __________

Stats Quality Control 2

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