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Unit V

STATISTICAL QUALITY CONTROL

B. Thilaka

Applied Mathematics

Statistical Quality Control

• Quality Control – Process control & acceptance sampling.

• Tolerance Limits

• Acceptance Sampling


• Quality- measurable / countable property of a product

• Potency of a drug, breaking strength of a yarn, outside diameter of a ball bearing, number of imperfections in a piece of cloth,….

Quality Control

The concept of quality control in manufacturing was

first advanced by Walter Shewhart

of the Bell Telephone Laboratories. He issued a

memorandum on May 16, 1924 that featured a

sketch of a modern control chart.

1931 - published a book on statistical quality

control, "Economic Control of Quality of

Manufactured Product"

set the tone for subsequent applications of

statistical methods to process control.

Quality Control

Two other Bell Labs statisticians, H.F. Dodge and H.G. Romig spearheaded efforts in applying statistical theory to sampling inspection.

The work of these three pioneers constitutes most of the theory of statistical quality and control

Quality Control - Statistical Process Control

Typical process control techniques:

1. Histograms

2. Check Sheets

3. Pareto Charts

4. Cause and Effect Diagrams

5. Defect Concentration Diagrams

6. Scatter Diagrams

7. Control Charts

Statistical Process Control

The underlying concept of statistical process control is based on a comparison of what is happening today with what happened previously.

Historical data is used to compute the initial control limits. The present data are compared against these initial limits.

Statistical Process Control

• Points that fall outside of the limits are investigated and, perhaps, some will later be discarded.

• If so, the limits would be recomputed and the process repeated.

• This is referred to as Phase I (retrospective anlaysis of data).

• Real-time process monitoring, using the limits from the end of Phase I, is Phase II (prospective monitoring of future data).


Statistical Quality Control is the process of inspecting enough product from given lots to probabilistically ensure a specified quality level.

Statistical quality control refers to the use of statistical methods in the monitoring and maintaining of the quality of products and services.


Statistical Process Control is concerned with achieving, maintaining and further improving a good process that produces desirable results reliably and consistently over time.

Applicable to any process that produces data over time according to some underlying statistical model – health care, manufacturing process.


Why and How do we achieve?

• The production processes are not perfect.

• Hence the output of these processes will not be

perfect – correct and deterministic.

• Successive runs of the same production

process will produce distinct(non-identical)

products (each fingerprint is unique -

variation).


•Alternately, seemingly similar runs of the

production process will vary, by some

degree, and impart the variation into the

some product characteristics.

•Because of these variations in the products,

we need probabilistic models and robust

statistical techniques to analyze quality of

such products.


However well a production process is

controlled, these quality measures will vary

from item to item and hence there will be a

probability distribution associated with the

population of such measurements

If all important sources of variations are under control in a production process, then the slight variations among the quality measurements will not cause any serious problem.


Such a process should produce the same

distribution of quality measurements no matter

when it is sampled, thus ensuring that this is a

“stable system.”


Variations can be broadly classified into two types.

Chance Variation & Assignable Variation

Chance(common) Variation – Variation in the quality of the product which occurs due to minor but random causes such as slight changes in temperature, pressure, metal hardness.


Assignable Variation – Variation that occurs due to non-random causes like poor quality raw material, mechanical faults.

A process is said to be in a state of statistical control, if the variability present in the process is confined to chance variation.


Statistical control is achieved by finding and eliminating all assignable variations.

Once statistical control is achieved, it is essential to maintain the same .

Control charts (Shewart charts) are principally used to detect serious deviations from a state of statistical control when or before(if possible) they occur.

Control charts

Control charts

Measurements(variables) Attributes

(Mean and Range) (p, np and c)

Control charts

Control charts for measurements:

Observations are measurements

Variables are the quality characteristics of a product that are measurable.

Control charts for attributes:

Observations are count data (the number of defectives in a sample of given data)

Attributes are the quality characteristics of a product that are identified by their presence or absence

Control charts

Control chart consists of

a central line corresponding to the average quality at which the process is to perform

lines corresponding to the upper and lower control limits – limits are chosen so that values falling between them can be attributed to chance, while values falling beyond them are interpreted as indicating a lack of control. (warning limits)

Control charts

Plotting the results obtained from samples taken periodically at frequent intervals can help determine whether the process is under control or any assignable variation has entered the process.

When a sample point falls beyond the control limits or even when the points fall within the control limits if a trend or some other pattern is visible, preventive action is needed.

Control charts

The control limits are used to assess the stability of the process and to identify

unusual events (outliers)

Control charts are thus quite useful both for monitoring if processes get worse and

for testing and verifying improvement ideas

Control charts

A control chart will tell whether a process is out of control, unstable, or unpredictable.

A stable process produces predictable results consistently.

Processes that are “out of control” need to be stabilized before they can be improved.

Control charts

Process is in control:

No points outside the control limits

The number of points above and below the

central line are about the same.

Points are seemingly random above and below

the central line.

Most points are near central line, and only a

few are close to control limits.

Basic Assumption: Central Limit Theorem (Unit II)

Control charts

Process monitoring with control charts is an important component within an overall process evaluation and improvement framework in healthcare.

Control charts

Control chart methods are used extensively in health care

WHERE? HOW?

To verify improvements in the time between insulin injections.

Control charts In cases of epidemiological concerns

such as surgical- site infections, bacteremia, Clostridium difficile toxin-positive stool assays, medical intensive-care-unit (ICU) nosocomial infections, and needlestick injuries.

Can you relate to something else?

EBOLA

Control charts

In many industries, processes will be divided into stages and these principles of SQC are applied to the individual stages separately.

Unless a product is found to meet the specified norms, it will not be permitted to move to the next stage.

This stage-wise SQC scheme ensures that the desired quality level is maintained at all levels.

Control charts

Examples???

Food Processing

Wine-making

The right type of grapes,

Crushing and fermentation,

Processing and aging

Control charts for measurements

The chart :

Determination of the three lines:

If the process mean µ and standard deviation σ are known and if it is reasonable to assume that the measurements are samples from a normal population (ideal case), then we assert with probability (1-α) that the mean of a random sample of size n will fall between

n

zandn

z 2/2/

x


The chart :

Central line is given by -------------

LCL is given by ------------------

UCL is given by ------------------

x


The chart :

It is common industrial practice to use three-sigma limits obtained by substituting 3 for zα/2. (α=0.0027 – Recall properties of Normal distribution)

Hence, Central line : µ

Control Limits : µ±Aσ, A=3/√n

x


If a control chart is being constructed for a

new process, the µ and σ (the population

parameters) will not be known and hence must

be estimated from the sample data.

For establishing control limits, the general rule

of thumb is that at least 30 time points be

sampled before the control limits are

calculated.

For each of the k samples, we compute the

sample mean, the sample variance, and the

range .


The chart :

To form an unbiased estimator of µ from sample means we calculate

Further is a better estimator than

The average of the above values is not an unbiased estimator of σ2.

However, computation of the mean standard deviation is avoided by using the range data instead of calculating the adjusted standard deviation.

x

jX .1

1

k

j

jXk

X

1

` 22

jj Sn

nS

.2

jS


The chart:

We compute

The control chart values for the chart are

Central line:

Control Limits :

.1

1

k

j

jRk

R

x

x

X

RAX 2


The R- chart :

While the chart monitors the central value of the process, the R-chart monitors the variation of the process.

As in the case of the chart, the three values of the R- chart in the various cases are as follows:

x

x


The R- chart :

If the underlying population is normal, with σ known, then

Central line =

UCL =

LCL =

2d

2D

1D


The R- chart :

If σ is unknown, then it may be estimated from past data as in the case of the chart. Further

Central line =

UCL =

LCL =

x

R

RD4

RD3

Control charts for measurements Thus, there is a close connection between the use

of control charts and the testing of hypothesis. A point on the chart that is out of control corresponds to a sample for which the null hypothesis is rejected.

Specifically, control-chart techniques provide sequential, temporally ordered sets of tests. The position of the individual points, possible trends, patterns exhibited by the points representing successive samples provide an insight into the state and progress of the process.

Remark: Analyze R-chart before the chart. (unit 3)

x

x


1. Given the values of the sample mean

and sample range R for 10 samples, each of size 5. Draw the appropriate mean and range charts and comment on the state of control of the process.

Sample No. 1 2 3 4 5 6

Mean 43 49 37 44 45 37

Range 5 6 5 7 7 4

Sample No. 7 8 9 10

Mean 51 46 43 47

Range 8 6 4 6

X


Solution:

Consider

= 44.2

Also,

= 5.8

iXN

X1

47...37494310

1

iRN

R1

6...56510

1


From the table of control chart constants, for sample size n=5, we have

A2 = 0.577, D3 = 0 and D4 = 2.115.

Hence Control Limits for chart are:

Central Line : = 44.2

LCL = - A2 = 44.2- 0.577x5.8 =40.85

UCL = + A2 = 44.2+0.577x5.8 =47.55

X

X

X R

X R


Further, the Control Limits for R chart are:

CL= =5.8

LCL = D3 =0

UCL= D4 =2.115x5.8=12.27.

We now draw the required two control charts

RR

R


State of Control:

Since all the points in the range chart lie

within the control lines, the process is under control as far as the variability of the sample

values is concerned.

However in the mean chart two points lie above

the upper control line and two points lie below

the lower control line. The process is thus not under control as far as the average of the

sample values is concerned.

On the whole, the process is out of control.


2. A machine fills boxes with dry cereal. 15

samples of 4 boxes are drawn randomly. The

weights of the sample boxes are shown as follows. Draw the control charts for the sample

mean and sample range and determine whether

the process is in a state of control.

Sample Number : 1 2 3 4 5

Weights of boxes(X): 10.0 10.3 11.5 11.0 11.3

10.2 10.9 10.7 11.3 11.6

11.3 10.7 11.4 10.7 11.0

12.4 11.7 12.4 11.4 12.1




11.4 11.4 12.1 13.1 12.1

10.7 11.1 12.7 13.1 10.7

11.0 10.3 10.7 12.4 11.5



11.9 12.1 11.1 12.1 11.9

11.8 11.6 12.1 13.1 11.7

11.3 11.4 11.7 12.0 12.1


Solution:

Since the given data is raw data, we first

compute the means and ranges of the given samples.

Sample 1 2 3 4 5 6 7

No. (i)

43.9 43.6 46.0 44.2 46.9 43.8 44.1

11.0 10.9 11.5 11.1 11.7 11.0 11.0

2.9 1.4 1.7 0.7 0.8 0.7 1.1

X

iX

iR


(i) 8 9 10 11 12 13 14 15

47.8 49.6 45.6 47.5 47.0 47.0 49.1 46.3

12.0 12.4 11.4 11.9 11.8 11.8 12.3 11.6

2.0 2.1 1.4 1.2 0.7 1.0 1.2 1.5

Now,

= 11.56

X

iX

iR

iXN

X1

6.11...5.119.100.1115

1

15

4.173


Further,

= 1.33

From the table of control chart constants, for

sample size n=4, we have

A2 = 0.729, D3 = 0 and D4 = 2.282.

iRN

R1

5.1...7.14.14.215

1

15

9.19


Hence Control Limits for chart are:


LCL = - A2 = 11.56- 0.729x1.33=10.59

UCL = + A2 = 11.56+0.729x1.33=12.53

Further, the Control Limits for R chart are:

CL= =1.33

LCL = D3 =0

UCL= D4 =2.282x1.33=3.04

X

X

X

R

R

X

R

R

R


We now draw the required two control charts as given below:


State of Control:

Since all the points lie within the upper and lower control lines in both the mean chart and the R-chart, the process is under control.


3. The values of sample mean and sample

standard deviation for 15 samples each of size 4, drawn from a production process are given below. Draw the appropriate control charts for the process average and process variability. Comment on the state of control.

Sample No. 1 2 3 4 5 6 Mean 15.0 10.0 12.5 13.0 12.5 13.0 S. D. 3.1 2.4 3.6 2.3 5.2 5.4


7 8 9 10 11 12 13 14 15

13.5 11.5 13.5 13.0 14.5 9.5 12.0 10.5 11.5

6.2 4.3 3.4 4.1 3.9 5.1 4.7 3.3 3.3

Solution:

HW


4. The following data gives the coded measurements

of 10 samples each of size 5, drawn from a

production process at intervals of 1 hour. Draw the appropriate control charts and comment on the state

of control of the process.

Sample 1 2 3 4 5 6

Number(i)

9 10 10 8 7 12

Coded 15 11 13 13 9 15

Measurements 14 13 8 11 10 7

(X) 9 6 12 10 4 16

13 10 7 13 5 10


(i) 7 8 9 10

9 15 10 16

9 15 13 14

(X) 9 10 14 12

13 13 7 14

5 17 11 14

Solution: HW

Control charts for attributes

The control charts seen earlier plots quality characteristics that can be measured and expressed numerically. We measure weight, height, position, thickness, etc.

If a particular quality characteristic cannot be represented numerically, or if it is impractical to do so, then using a quality characteristic to sort or classify an item that is inspected into one of two "buckets“ is a good alternate option.


An example of a common quality characteristic classification would be designating units as "conforming units" or "nonconforming units".

Another quality characteristic criteria would be sorting units into "non defective" and "defective" categories.

Quality characteristics of these types are called attributes.


We observe that there is a difference between

"nonconforming to an engineering specification"

and "defective" -- a nonconforming unit may still function fine and be, in fact, not defective at all,

while a part can be defective and not function as

desired.

Examples of quality characteristics that are

attributes

the number of failures in a production run

the proportion of malfunctioning wafers in a lot


Two fundamental control charts for attributes:

Fraction-defective chart called a “p chart”

Number-of –defects chart called a “c chart”


p-chart is a type of control chart used to monitor the proportion of nonconforming units in a sample, where the nonconforming sample proportion is defined as the ratio of the number of nonconforming units to the sample size, n


c-chart is a type of control chart used to monitor "count"-type data, typically total number of nonconformities per unit. It is also occasionally used to monitor the total number of events occurring in a given unit of time.


The c-chart differs from the p-chart in that it accounts for the possibility of more than one nonconformity per inspection unit, and that (unlike the p-chart and u-chart) it requires a fixed sample size.

The p-chart models "pass"/"fail"-type inspection

only, while the c-chart (and u-chart) give the ability to distinguish between (for example) 2 items which fail inspection because of one fault each and the same two items failing inspection with 5 faults each; in the former case, the p-chart will show two non-conformant items, while the c-chart will show 10 faults.


Nonconformities may also be tracked by type or location which can prove helpful in tracking down assignable causes.

Control charts for attributes(p)

Control Limits for fraction-defective chart :

(Link this to the sampling theory of proportions – unit 3)

(To what distribution will we associate this???)

BINOMIAL

(Ultimately, Normal curve approximation)

Control charts for attributes (p)

Hence, if a standard is given, that is, if the fraction defective should take some pre-assigned value p , the central line is p and the three-sigma limits are used for the control limits.

In other words,

Central line : p

Control limits:

n

ppp

)1(3


If no standard is given(practical situations), p will have to be estimated from past data.

If k samples are available, di is the number of defectives in the ith sample, and ni is the number of observations in the ith sample, p is estimated as the proportion of defectives in the combined sample, as

k

k

nnn

dddp

...

...

21

21


Hence the three control-chart values are

Central Line :

UCL :

LCL :

p

n

ppp

)1(3

n

ppp

)1(3


Note :

1. If p is small, as may often be the case, then,

the LCL may be a negative quantity. In this case, the LCL is assumed to be zero and effectively, only the UCL is used.

2. If the binomial distribution cannot be approximated to a normal curve(when??? – unit I), then and only then the UCL may be obtained directly from the Binomial tables or the Poisson approximation to the Binomial may be used.

Control charts for attributes (np)

The np-chart is a type of control chart used to

monitor the number of nonconforming units in a

sample.

It is an adaptation of the p-chart and used in

situations where it is easier to interpret process

performance in terms of concrete numbers of

units rather than the somewhat more abstract proportion.

Control charts for attributes (np)

• The np-chart differs from the p-chart in only the

three following aspects:

• The control limits are , where n is the sample size and is the estimate of the

long-term process mean established during

control-chart setup.

• The number nonconforming (np), rather than the

fraction nonconforming (p), is plotted against the

control limits.

• The sample size, n, is constant.

)1(3 ppnpn p

Control charts for attributes (c)

In many situations, it is necessary to control the number of defects in a unit of product, rather than the fraction defective or the number of defectives.

Example:

number of defects per hundred yards in carpeting industry

number of defects per roll in newsprint industry.


Control Limits for no-of-defects chart :

(To what distribution will we associate this???)

POISSON - WHY???


Hence, the central line for a number-of-defects chart is the parameter λ of the Poisson distribution, and the three sigma control limits is based on the fact that the standard deviation of this distribution is √λ. (what will the limits be ?)

If λ is unknown, that is, if no standard value is given, then its value is estimated from at least 20 values of c observed from past data.


If k is the number of units of product available for

estimating λ, and if ci is the number of defectives

in the ith unit, then λ is estimated by

The control chart values are

Central line :

UCL :

LCL :

k

i

ick

c1

1

c

cc

cc


Note:

1. np-chart (sample sizes are the same) and p-

chart (sample sizes may be different – in this

case ,for all i)are used

when ≥0.05 or ≥4

2. c-chart is used when ≥4 or when

is small compared with the maximum number

of defects in the given data.

p pn

c c

nnn i 25.175.0


5. 15 samples of 200 items each were drawn from the output of a process. The number of defective items in the samples are given below. Prepare a control chart for the fraction defective and comment on the state of control.

Sample No. (i) 1 2 3 4

No. of defective (np) 12 15 10 8


(i) 5 6 7 8 9 10

(np) 19 15 17 11 13 20

`(i) 11 12 13 14 15

(np) 10 8 9 5 8


Solution:

Consider ∑np=12+15+10+..+5+8=180

= 12

Also,

(since each sample contains 200 items).

The control limits are thus given by

npN

pn1

15

180

200

12p


Central line :

LCL :

= 0.01

06.0p

n

ppp

)1(3

200

094.006.0306.0

x


UCL :

=0.11

The fraction defectives (values of p) for the given samples are:

0.06, 0.075, 0.05, 0.04, 0.095, 0.075, 0.085, 0.055, 0.065, 0.1, 0.05, 0.04, 0.045, 0.025, 0.04.

n

ppp

)1(3

200

094.006.0306.0

x


Since all the sample points lie between the LCL and UCL lines, the process is under control.


6. Construct a control chart for defectives for the

following data and comment on the state of control

Sample No. : 1 2 3 4

No. inspected : 90 65 85 70

No. of defectives : 9 7 3 2

5 6 7 8 9 10

80 60 70 95 90 75

9 5 3 9 6 7


Solution:

Since the size of the sample varies from sample to sample, we do not construct the np-chart.

We use the p-chart only if

for all i.

Now, = (90+65+…+90+75)/10

= 800/10 = 80

nnn i 25.175.0

inN

n1


The values of ni are all between 60 and 100 .

Hence, we now draw the p-chart.

Now, = (Total no. of defectives)/

(Total no. of items inspected)

= 60/800 = 0.075

Hence the control limits are given by

p


CL :

LCL :

= -0.013

Since LCL cannot be negative, it is assumed to be zero.

UCL :

075.0p

80

025.0075.03075.0

)1(3

x

n

ppp

163.080

025.0075.03075.0

)1(3

x

n

ppp


The values of pi for the various samples are 0.100, 0.108, 0.035, 0.029, 0.113, 0.063, 0.043, 0.095, 0.067, 0.093

Since all the sample points lie within the control limits, the process is under control.


7. In a factory producing spark plugs, the number of defectives found in the inspection of 15 lots of 100 each is given below. Draw the control chart for the number of defectives and comment on the state of control.

Sample No. (i) 1 2 3 4

No. of defectives (np) 9 10 12 8


(i) 5 6 7 8 9 10

(np) 6 4 6 3 4 5

(i) 11 12 13 14 15

(np) 4 7 9 3 4

Solution:

Consider ∑np = 5+10+12+….+3+4=90


= 90/15 = 6

and = 6/100 = 0.06 ( since each sample contains 100 items)

The control limits for the np-chart are

CL : = 6

npN

pn1

p

pn


LCL : = 6-3√(6x0.94)=-1.12

Since the LCL cannot be negative, LCL=0.

UCL : = 6+3√(6x0.94) = 13.12

)1(3 ppnpn

)1(3 ppnpn


Since all the sample points lie between the upper and lower control limits, the process is under control.


8. 10 samples each of size 50 were inspected and the number of defectives in the inspection were 2,1,1,2,3,5,5,1,2,3. Draw the appropriate control chart for the defectives and comment on the state of control.

Solution : HW

(Hint : The sample sizes are the same. Either chart, p or np will do)


9. 15 tape recorders were examined for quality control test. The number of defects in each tape recorder is recorded below. Draw the appropriate control chart and comment on the state of control.

Unit No. (i) 1 2 3 4 5 6

No. of defects (c) 2 4 3 1 1 2

7 8 9 10 11 12 13 14 15

5 3 6 7 3 1 4 2 1


Solution:

Since the number of defects per sample containing only one item is given, we use the c-chart.

Now,

= 45/15=3

icN

c1

12...34215

1


Aside: Observe that even though <4, we use only the c-chart at this point of time.

We now compute the three control limits :

CL : = 3

LCL : = 3-3√3 = -2.20

Since LCL cannot be negative, LCL=0

UCL : = 3+3√3 = 8.20

c

ccc 3

cc 3


Since all the points lie within the control limits, the process is under control.

10. A plant produces paper for newsprint and rolls of paper are inspected for defects. The results of inspection of 20 rolls of papers are given below. Draw the c chart and comment on the state of control.


Roll No (i) 1 2 3 4 5

No. of defects (c) 19 10 8 12 15

(i) 6 7 8 9 10 11 12 13

(c) 22 7 13 18 13 16 14 8

14 15 16 17 18 19 20

7 6 4 5 6 8 9

Solution : HW

Tolerance Limits

In quality control, the limiting values between

which measurements must lie if a product is to be accepted is loosely defined as tolerance

limits.

(Contrast this with confidence limits of unit 3)

(Confidence limits are the lower and upper boundaries / values of a confidence interval, that

is, the values which define the range of a

confidence interval).

Tolerance Limits

Recall ”A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.”

If reliable information is available about the distribution underlying the measurement in question, the natural tolerance limits are clearly easily obtained.

Tolerance Limits

In most practical situations, the true values of µ and σ are not known and therefore, the tolerance limits are based on the mean and standard deviation of a random sample.

While in a normal population, µ±1.96σ are the tolerance limits that include 95% of the population, the same cannot be said of for the limits based on sample mean and sample standard deviation.

Tolerance Limits

However, it is possible to determine a constant K so that “one can assert with

(1-α) 100% confidence that the proportion of the population contained between

and is at least P”, the given proportion of the population.

Can you now differentiate between confidence limits(of unit 3) and tolerance limits?

Ksx Ksx

Tolerance Limits

Confidence limits are used to estimate a

parameter of a population.

Tolerance limits are used to indicate between

what limits one can find a certain proportion of the population

Further observe that when n becomes large, the length of a confidence interval approaches zero

while the tolerance limits approach the

corresponding values of the population.

Acceptance Sampling

Acceptance sampling is an important field of statistical quality control that was popularized by Dodge and Romig and originally applied by the U.S. military to the testing of bullets during World War II. If every bullet was tested in advance, no bullets would be left to ship. If, on the other hand, none were tested, malfunctions might occur in the field of battle, with potentially disastrous results.

Acceptance Sampling

Dodge reasoned that a sample should be picked at random from the lot, and on the basis of information that was yielded by the sample, a decision should be made regarding the disposition of the lot. In general, the decision is either to accept or reject the lot. This process is called Lot

Acceptance Sampling or just Acceptance Sampling.

Acceptance Sampling

Acceptance sampling is "the middle of the road"

approach between no inspection and 100% inspection. There are two major classifications of

acceptance plans: by attributes ("go, no-go") and

by variables. The attribute case is the most

common for acceptance sampling.

The main purpose of acceptance sampling is to

decide whether or not the lot is likely to be

acceptable, not to estimate the quality of the lot.

Acceptance Sampling

Acceptance sampling is employed when one or

several of the following hold:

Testing is destructive

The cost of 100% inspection is very high

100% inspection takes too long

This procedure is equivalent to a test of the null

hypothesis that the proportion defective p in the

lot equals some specified value p0 against the

alternative that it equals p1, where p1>p0.

(One tailed test- WHY?? -up for discussion)

Acceptance Sampling

11. The specifications for a certain quality characteristic are 15.0±6.0 ( in coded values). 15 samples of 4 readings each gave the following values for and R.

Sample No. (i) : 1 2 3

: 16.1 15.2 14.2

R : 3.0 2.1 5.6

X

X

Acceptance Sampling

(i): 4 5 6 7

: 13.9 15.4 15.7 15.2

R : 2.4 4.1 2.7 2.3

(i): 8 9 10 11

: 15.0 16.5 14.9 15.3

R : 3.8 5.0 2.9 13.8

(i): 12 13 14 15

: 17.8 15.9 14.6 15.2

R : 14.2 4.8 5.0 2.2

X

X

X

Acceptance Sampling

Compute the control limits for and R-charts using the above data for all the samples. Hence examine if the process is in control. If not, remove the doubtful samples and recompute the values of and . After testing the state of control, estimate the tolerance limits and find if the process will meet the required specifications.

X

X

R

Acceptance Sampling

Solution:

We first compute the following values for the given data of 15 samples.

= 15.39

XN

X1

2.15....2.151.1615

1

15

9.230

Acceptance Sampling

= 4.93

We now compute the control limits for the

- chart.


RN

R1

2.2...1.20.315

1

9.7315

1x

X

X

Acceptance Sampling

LCL : - A2 = 15.39-0.729x4.93 =11.80

UCL : +A2 = 15.39+0.729x4.93 =18.98

The control limits of the R-chart are

CL : = 4.93

LCL : D3 = 0

UCL : D4 = 11.25

X R

X R

RR

R

Acceptance Sampling

From the values calculated above, we see that though the process is under control with respect to the average ( - chart), it is not under control with respect to the variability (R- chart), since the R values of sample numbers 11 and 12 (13.8 and 14.2 respectively) exceed the UCL = 11.25

Hence the process is not under control.

X

Acceptance Sampling

We now remove the samples numbered11 and 12 from the given data and recompute the values for the remaining 13 samples only.

= 15.22

= 3.53

2.156.149.159.14....2.151.1613

1X

13

8.197

2.20.48.59.2...1.20.313

1R

9.4513

1x

Acceptance Sampling

We now compute the revised control limits for both the charts.

The control limits for the - chart are:

CL : = 15.22

LCL : - A2 = 15.22- 0.729x3.53=12.65

UCL : - A2 =15.22+0.729x3.53=17.79

The control limits for the R-chart are:

X

X

X R

RX

Acceptance Sampling

CL : = 3.53

LCL : D3 = 0

UCL : D4 = 11.25

We now observe that the process is under control with respect to the 13 samples considered.

We now compute the tolerance limits using the revised values of and .

RR

RX

Acceptance Sampling

The tolerance limits are given by :

Thus the required tolerance limits are (10.08, 20.36).

Since these tolerance limits lie within the specification limits (9.0,21.0), the process meets the required specifications.

2

3

d

RX

14.522.15059.2

53.3322.15

x

Acceptance Sampling

12. The specifications for a certain quality characteristic are (60±24) in coded values. The table below gives the measurements obtained in 10 samples. Find the tolerance limits for the process and test if the process meets the specifications.

Acceptance Sampling

Sample No. (i) : 1 2 3 4 5

Measurements (X) : 75 48 57 61 55

66 79 55 71 68

50 53 53 66 58

62 61 61 69 62

52 49 72 77 75

70 56 53 63 64

Acceptance Sampling

(i) 6 7 8 9 10

49 74 67 66 62

(X) 98 63 70 65 68

65 62 68 58 66

64 57 56 52 68

66 62 61 58 73

64 64 66 50 68

Acceptance Sampling

Solution : HW

(Hint: In control, does not meet the specifications – under discussion)

statistical quality control a… · sqc are applied to the individual stages separately. unless a...

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