statistical quality control introduction · def: statistical quality control (sqc) is defined as...

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1 Statistical Quality Control Introduction Statistical Quality Control (SQC) is a technique, which is used for applying statistical theory to industrial mass production for ensuring uniformity in the items as also for facilitating in meeting the specification requirements. In any manufacturing process, some variation in quality, as measured by a parameter (s), of items is expected as a result of chance. Statistical quality control has its origin nearly 1920. Mr. Dodge H.E. of the Bell telephone laboratories in USA, first used normal distributions in statistical methods, to develop control charts for checking the quality of the products by tests on random samples of products manufactured by a factory. The main concept of any production process is to maintain good quality of the product manufacturing or produced. Now a day the manufacturing process is interested to see that the product confirms to contain specifications or standards which are not rejected by the customers. It is observed that in all production processes, no two items produced even by a machine are exactly alike. When an accurate measurement is taken, variations in quality of the items (threading of screws, thickness of wires, etc.) are inevitable. Specifications recognize this fact by giving tolerance range with in which measurement must fall for the items to be acceptable. A direct method to ensure that all of the items or parts fall within the specification limits is to inspect each and every item is costly and the time consuming and the defective items are traced only after production. Also 100% inspection cannot ensure that only satisfactory products will come out of the factory. However, the use of statistical methods like control charts is much more effective in controlling the manufacturing process ensuring more or less information quality of product at lower cost of production. The main objective of SQC is to find out i) the extent to which the products satisfies the specifications ii) the limits of variations and iii) the causes of variations. Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous flow of products manufactured by an industry. This can also be defined as “an economic and effective system of maintaining and improving the quality of output throughout the whole operating process of specification”, predictors and inspection based on continuous testing of random samples. Advantages of SQC Statistical Quality Control has several advantages over 100% inspection of a system. 1. Only sample of output inspected, inspection costs are tremendously reduced. 2. It is easy to apply. 3. Inspection Works is considerably reduced, efficiency is increased.

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Page 1: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

1

Statistical Quality Control

Introduction

Statistical Quality Control (SQC) is a technique, which is used for applying statistical

theory to industrial mass production for ensuring uniformity in the items as also for facilitating in

meeting the specification requirements. In any manufacturing process, some variation in quality,

as measured by a parameter (s), of items is expected as a result of chance.

Statistical quality control has its origin nearly 1920. Mr. Dodge H.E. of the Bell telephone

laboratories in USA, first used normal distributions in statistical methods, to develop control

charts for checking the quality of the products by tests on random samples of products

manufactured by a factory.

The main concept of any production process is to maintain good quality of the product

manufacturing or produced. Now a day the manufacturing process is interested to see that the

product confirms to contain specifications or standards which are not rejected by the customers.

It is observed that in all production processes, no two items produced even by a machine

are exactly alike. When an accurate measurement is taken, variations in quality of the items

(threading of screws, thickness of wires, etc.) are inevitable. Specifications recognize this fact

by giving tolerance range with in which measurement must fall for the items to be acceptable. A

direct method to ensure that all of the items or parts fall within the specification limits is to

inspect each and every item is costly and the time consuming and the defective items are traced

only after production.

Also 100% inspection cannot ensure that only satisfactory products will come out of the

factory. However, the use of statistical methods like control charts is much more effective in

controlling the manufacturing process ensuring more or less information quality of product at

lower cost of production.

The main objective of SQC is to find out i) the extent to which the products satisfies the

specifications ii) the limits of variations and iii) the causes of variations.

Def: Statistical quality control (SQC) is defined as the statistical methods applied for the

maintenances of uniform quality in a continuous flow of products manufactured by an industry.

This can also be defined as “an economic and effective system of maintaining and

improving the quality of output throughout the whole operating process of specification”,

predictors and inspection based on continuous testing of random samples.

Advantages of SQC

Statistical Quality Control has several advantages over 100% inspection of a system.

1. Only sample of output inspected, inspection costs are tremendously reduced.

2. It is easy to apply.

3. Inspection Works is considerably reduced, efficiency is increased.

Page 2: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

2

4. SQC ensures an early detection of faults

5. It ensures that a minimum waste of reject production.

6. This enables a process to be brought into and held in a state of statistical control.

7. It is preferred as the only course where 100% inspection cannot be carried out without

destroying all the products inspected.

8. It is used to detect whether or not a change in the production process results in a

significant change in quality.

[Types of Quality Control

The control of quality of a product manufactured by a factory can be classified under two

heads. Viz., Process Control and Product Control

1. Process Control

2. Product Control (or Lot control)

Control Charts

Control chart is a most important tool of statistical Quality control. It is a simple and

most powerful graphical device developed by Dr. W. A. Shewhart in 1924.

The main feature of the Control Chart method is the drawing of conclusions about the production

process on the basis of samples drawn from the production process at regular intervals.

This graph consists of a central horizontal line corresponding to the average value of the

characteristic, together with upper and lower control limits between which the values of the

sample statistic should fall.

These charts are used to detect lack of control in any manufacturing process and to

indicate any future trouble that any creep in the process, when the control charts indicate that any

assignable cause is present, steps should be taken to locate the trouble and correct it before more

defective products are produced. If these charts indicate that only random variations are present

in the products, then no further inspection will be necessary.

We know that the Process Control is achieved through Control Charts where as Product

Control is accomplished through Sampling Inspection Plans. There are two principal types of

control charts namely, Control Charts for Variables, and Control Charts for Attributes. Variables

are quality characteristics that can be measured and expressed in numbers, namely length of a

screw, diameter of a wire, etc.

Attributes are usually refer to the classification of a quality characteristic into any one of

two classes, either conforming or not conforming to specifications, as in acceptance or rejection

by sampling inspection plans. Attributes may be judged either by the proportion of defectives or

by the number of defects per unit.

Causes of Variations in Quality Characteristics

The causes of variations in quality characteristics can be classified into two ways.

1. Assignable causes of variation (Assignable variation)

2. Chance causes of variation (chances variation)

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3

Control Charts for Variables

There are three commonly used control charts for variables:

i) x -Chart or Mean Chart

ii) R- Chart or Range Chart

iii) - Chart or Standard Deviation Chart

General procedure to draw Control Chart for variables

The construction of control chart is based on a sequence of suitable sample statistics

(namely, x , R, ) calculated from samples drawn at random from the products of the

manufacturing process.

3- Control Limits

Let be the quality-characteristic of an item manufactured.

Let nxxxx . . . ,3,2,1 be a random sample of size ‘n’ drawn from the given process and let ‘t’ be a

statistic computed from the sample such that

itE )( and 2)( itV

If the process is under control, will remain more or less the same from sample to

sample and hence the variations in the values of t from sample to sample is due to chance causes.

If the statistic t is normally distributed with mean i and variance 2

i , then by the area

distribution of normal curve we have

33

i

itP

=0.9974

iiii tP 33 =0.9974

iitP 3 = 0.9974

iitP 3 = 0.0026

This shows that if the process is under control, then the observed values of the statistic t

lie between the limits ii 3 and ii 3 in 99.74% of the cases and they lie outside these

limits in 0.26% cases that is very small.

In this case, ii 3 is called the Upper Control Limits (UCL) and ii 3 , the Lower

Control Limit (LCL).

Similarly, we can find 2- control limits when observed values of ‘t’ lie between ii 2 and

ii 2 in 95.44% of the cases.

Thus if for any random sample, the computed value of ‘t’ lies within the 3- limits. (i.e.

within ii 3 ), then we say that the process is in state of statistical control and in this case the

process is operating under the influence of chance causes of variation only. But if the computed

value of ‘t’ lies outside the 3- limits, ii 3 , then it is an indication the lack of statistical

Page 4: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

4 control. In this case, some assignable causes are present in the process and they should be

traced, identified and eliminated.

A control chart consists of one vertical line and three horizontal lines.

Quality scale is marked according to the quality characteristic of each sample along the

vertical line. The central line representing the average quality of the samples is plotted as a bold

horizontal line CC . Upper Control Limit (UCL) and Lower Control Limit (LCL) are plotted as

dotted horizontal lines AA and BB respectively, above and below the central line CC , each of

them being at a distance of i3 from CC

Sample Numbers are shown at equal distances on a horizontal line at the bottom of

the control chart.

In control chart, if all the plotted points corresponding to the given samples lie within the

control limits i.e. in the region between the two dotted lines AA and BB , then the process is said

to be in a state of control. Otherwise the process is not in a state of control. In such a case,

production process should be stopped assignable causes present in the process must be traced,

identified and eliminated before the production starts again.

Before drawing a control chart, it is important to group the manufactured items into

suitable samples in such a manner that the variation in quality among the items within the same

sample is small, but between the items of any two samples is as large as possible. Such a sample

is called a rational sub-group. The variation in quality between any two samples is due to

chance causes, but appreciable variation within the same sample, if any, is due to assignable

causes which are to be identified and removed.

Control Chart for Mean or x -Chart

A control chart for Mean or x -Chart is the control chart on which the means of the

samples drawn from a production process at equal intervals of time are plotted as points. It

indicates the fluctuations of the means of the samples about the mean of the process. x -Chart is

used to determine whether these fluctuations are due to chance causes or assignable causes. The

construction of x -Chart involve the following steps.

i) Mean ix and Range iR of samples drawn

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Suppose we draw k independent random samples each of size n (usually k lies between

20 and 30 and n = 4 or 5) from the lot of items to be examined. Let 1x , 2x ,. . . , kx be the

sample means and 1R ,

2R , . . . , kR the sample ranges of the k samples. Then

n

iix

th sample in the nsobservatio theof sum

And iii

SLR = Largest observation – Smallest observation, in the ith sample (Since Range

iR is the difference between the largest and smallest observations)

ii) Mean of the sample means x and mean of the sample ranges R :

The mean of all the k sample means is denoted by x and the mean of the all the k sample

ranges is denoted by R as follows:

x k

xxx k

21

= k

x i and R =

k

RRRk

21 =

k

Ri

iii) Computation of Control Limits UCL x and LCL x

We assume that the quality characteristic X follows normal distribution with mean and

standard deviation . Here the process mean is and the process standard deviation is

The sample mean x approximately follows normal distribution with

)(xE and the Standard Error of x , i.e., n

xES

)(.

The 3 control limits for x chart are

)(.3 xES = n

3 = A where A =

n

3, is a constant whose values for

different values of n from 2 to 25 are given in table. The value of A are usually given in a

problem.

Case I: Standards are given: If and are given

i.e., If = and = are known then the central line is CL =

And UCL x = Upper Control limit of x = +A

LCL x = Lower Control limit of x = - A where A =n

3

Case II: Standards are not given: If and are not given

If and are not known then x is taken as the estimate of . We can estimate

either with the help of the mean of the sample ranges R or on the basis of the means of the

sample standard deviations s and quality control factors as shown below.

i) With the help of R ,

= the estimate of = 2d

R, where

2d is a quality control factor

depending on the size of the sample and it can be obtained from the Statistical Table.

In this case , the 3 control limits for x chart are

Page 6: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

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nd

Rx

2

3 = RAx 2 where

2A =nd2

3 is the constant (

2A is quality control factor

depending on the sample size n)

Now the central line is CL = x (If and are not given)

The Upper Control Limit UCL x = RAx 2

The Lower Control Limit LCL x = RAx2

The values of 2A are n = 2 to n = 25 are given in table.

ii) With the help of mean of the sample standard deviations s :

The estimate of denoted by

and is taken as

=

2c

s,

(where 2

c is the quality control factor depending on sample size n.)

In this case , the 3 control limits for x chart are:

nc

sx

2

3 = sAx 1

where 1A =

nc2

3 is the constant (it is also called quality control factor depending on the

sample size n)

Now the Central Line is CL x = x

The Upper Control Limit UCL x = sAx 1

The Lower Control Limit LCL x = sAx 1

Construction of x chart:

To draw the x chart on a graph paper, we represent the sample numbers on a horizontal

scale at the bottom of the control chart and statistic x along the vertical scale. Sample means 1x ,

2x ,. . . kx are then plotted as points against the k sample number 1, 2, . . . k. The central line

CL= or x , UCL x and LCL x are drawn as dark dotted lines at the computed values.

Comments on x chart:

If all the plotted points corresponding to the given samples fall within the control limits

i.e., in the region between the two dotted lines BB and AA drawn at the UCL x and LCL x ,

then we say that the process is in a state of control.

But if one or more points fall outside the Upper and Lower lines i.e., above AA or BB

below, then the process is not in a state of statistical control. In such a case the production

process is stopped, assignable causes present in the process is traced, identified and eliminated

before the production starts again

Control Chart for Range or R-Chart:

Page 7: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

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R chart is used to control the variability in the quality of products manufactured by a

given process. The method of constructions of R-chart is similar to that of x - chart.

The 3- control limits for R- chart are:

The Upper Control Limit UCLR = R +3R

The Lower Control Limit LCLR = R - 3R

Where R is the standard deviation of all the k sample ranges R1, R2 .... .Rk, each of size ‘n’.

Remark: Control limits can also be determined for Range or R-Chart by using quality control

factors D1, D2, D3 and D4 whose values depends on the sample size ‘n’ as

i) If the process standard deviation is known then the control limits are:

The Upper Control Limit UCLR = D2 The Lower Control Limit LCLR = D1

And the Central Line is CLR = d2

ii) If the process standard deviation is not known, then the control limits are:

The Upper Control Limit UCLR = D4 R

The Lower Control Limit LCLR =D3 R

And the Central Line is CLR= R

The sample numbers 1, 2, . . . k are represent along a horizontal line and the range statistic R

represent along the vertical line on a graph paper. Sample range R1, R2, . . . . Rk are then plotted

as points against the respective sample numbers 1, 2,…, k.

The Central Line CL at R , UCLR and LCLR at the computed values are drawn as dark dotted lines

CC , AA & BB respectively as

Note: If all the points fall between the two control lines BB and AA then the variation in

quality of the manufactured products is in a state of control. Otherwise the process is not in a

state of statistical control.

Observation (Choice of x and R Chart) :

In order to examine whether a production process is in s state of control or not, it is

necessary to study both x and R charts together. x chart indicate the variations between the

Page 8: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

8 samples, where as R chart indicates variations within the samples. So a production process is

said to be under control if both the charts indicates that the process is in a state of control.

It is better to construct R chart first. If R chart shows that dispersion (or variation) of the

quality in the process is not in a state of control, it is better to construct x chart until the quality

dispersion is brought under control. If a point or a set of points fall outside the control limits in

x -chart, but R chart indicates a state of control, then we say that the process average has shifted

considerably.

Note:

In any control chart, the calculated Value of Lower Control Limit (LCL) becomes

negative, then we take it as zero, because neither x nor R can be negative.

Applications

Example 1: Construct X chart for the following data.

Sample

Numbers

Observations

I II III

1 32 36 42

2 28 32 40

3 39 52 28

4 50 42 31

5 42 45 34

6 50 29 21

7 44 52 35

8 22 35 44

Also determine whether the process is under control or not and prepare a control chart

Solution:-

From the given data, n=3 and k=8

Sample

Numbers

Observations Total

x

Sample

mean

x

Sample

Range-R I II III

1 32 37 42 111 37.00 42-32 = 10

2 28 32 40 100 33.33 40-28 = 12

3 39 52 28 119 39.67 52-28 = 24

4 50 42 31 123 41.00 50-31 = 19

5 42 45 34 121 40.33 42-34 = 11

6 50 29 21 100 33.33 50-21 = 29

7 44 52 35 131 43.67 52-35 = 17

8 22 35 44 101 33.67 44-22 = 22

Total 906 302.00 144R

From the above table 00.302 x 144R

The mean of the means,

Page 9: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

9

ix

kx

1 =

8

00.302

= 75.37

The mean of the ranges,

iRk

R1

=8

144

= 18

(From the table of control chart, for the sample size n = 3, we have 023.12 A )

The Central Line x

CL x = 75.37

The control limits for �̅� Chart are RAx 2

i.e., The Control Limits for mean chart are RAx 2 = 18023.175.37

Therefore the Upper Control Limit x

UCL = RAx 2

= 18023.175.37 164.56

The Lower Control Limit x

LCL = RAx 2

= 18023.175.37 336.19

Now plot these points on the following graph with xLCL

, xCL

and x

UCL

Sample Number 1 2 3 4 5 6 7 8

Sample Mean x 37.00 33.33 39.67 41.00 40.33 33.33 43.67 33.67

x or Mean Chart

Since all the sample means lie between the control limits, the process is in under control

Example 2: A plastic manufacturer extrudes blanks for use in the manufactures of eye-glass

temples. Specifications requires that the thickness of these blanks have x =0.150 and =0.002

inch. Calculate 2-signa and 3-sigma upper and lower control limits for means of samples 5 and

prepare a control chart

Solution:

From the given data,

5n , x = 0.150 and = 0.002

For 2 Control Limits are:

Page 10: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

10

n

x

25

002.02150.0

0017888.0150.0

The Upper Control Limit, x

UCLn

x

2

0017888.0150.0 = 0.1517 inch

Therefore the Upper Control Limit (UCL) =0.1517 inch

The Lower Control Limit, x

LCL = n

x

2

0017888.0150.0 = 0.1517 inch

Therefore the Lower Control Limit (LCL) = 0.1482 inch

The Central Line x

CL x = 150.0 inch

For 3 Control Limits are:

n

x

35

002.03150.0

00268.0150.0

The Upper Control Limit, x

UCLn

x

3

00268.0150.0 =0.1526

Therefore the Upper Control Limit (UCL) =0.1526 inch

The Lower Control Limit, x

LCL = n

x

3

00268.0150.0 = 0.14732

Therefore the Lower Control Limit (LCL) = 0.14732 inch

The Central Line x

CL x = 150.0 inch

Page 11: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

11 Example 3: A machine is set to deliver packets of a given weight. Ten samples of size 5 each

were recorded. Below are given relevant data

Sample Number 1 2 3 4 5 6 7 8 9 10

Range 7 7 4 9 8 7 12 4 11 5

Draw the R-chart and comment on its state of control.

Solution:

From the given data, ten samples of size 5 each were recorded, and their range were

Sample Number 1 2 3 4 5 6 7 8 9 10 Total

Range 7 7 4 9 8 7 12 4 11 5 74

The Control Limits for the Range- chart,

Since 4.710

74R

Then the Central Line is CL= R = 7.4;

And the control limits are D4 R and D3 R when 𝜎 is unknown ( ;0D3 115.2D4 )

The Upper Control Limit is RUCL = RD4

115.2 7.4 = 15.651

The Lower Control Limit is RLCL = RD3 = 0 7.4 = 0

Plot the following points on the graph along with RLCL , R , RLCL

Sample Number 1 2 3 4 5 6 7 8 9 10 Total

Range 7 7 4 9 8 7 12 4 11 5 74

From the control chart for Range, we see that the plotted points corresponding to all the sample

numbers fall within the control limits and, therefore, the process is in a state of control,

regarding the weights of the deliver packets.

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12

Control charts for Attributes

There are three types of control charts for Attributes:

(i) p- chart or control chart for fraction defective (or proportion of defectives);

(ii) np-chart or control chart for the number of defectives;

(iii) c- chart or control chart for the number of defects per unit.

P- Chart or control chart for fraction defective (or proportion of defectives)

The Control Chart p, i.e., p-chart is used to see the variation in sample units of

manufacturing process is there are not (variation in yes or no type), i.e., there are only two

possible outcomes: either the item is defective or it is not defective.

A product or service is defective if it fails to conform to specifications or a standard in some

respect. For example, consider the case of a customer calling the company to place an order. The

customer would probably not like to have the phone ring 10 to 15 times before it is answered.

Suppose you have determined that the operational definition for answering the phone in a timely

fashion is "to answer the phone on three or fewer rings." Using this definition, you could monitor

the fraction of phone calls answered or not answered in a timely fashion. If a phone call is

answered on or before the third ring, the item (answering the phone call) is not defective. If the

phone call is not answered on or before the third ring, the item is defective.

You use a p control chart when you have yes/no type data. Let P be the fraction defective in a

production process. Suppose a random sample of n units is drawn from the process and let d be

the number of defective units in the sample. Then the fraction defective p in the sample is given

by

n

dp or npd

Then for the sampling distribution of the statistic p, we have

E(p) = P and S.E. of p =n

PQ

n

PP )1( PQ 1

The 3-𝜎 control limits for p-chart are = ).(.3 pESp n

PPp

)1(3

Case I :- Standards are given

If P be the given value of P, then the control limits for p-chart are

n

PPP

)1(3

,

Now Therefore the Upper Control Limit for p is

i.e pUCLn

PPP

)1(3

And the Lower Control Limit p

LCL n

PPP

)1(3

The Central Line is pCL = P

Case II :- Standards are not given

If P is not given, then p is an unbiased estimate of P andk

pppp k

21 where ip is the

number of defectives in the ith sample.

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The control limits for p-chart are n

)p(pp

13 ,

Now therefore the Upper Lower Control Limit for p are

i.e., p

UCL n

)p(pp

13 )p(pAp 1

And p

LCL = n

)p(pp

13 )p(pAp 1

Where nA /3 is called quality control factor which is constant.

The Central Line is pCL = p

Construction of p- chart is exactly similar to that of x chart and R-chart. In this case we

represent the fraction defectives p1, p2, … .pk of the k samples along the vertical scale and they

are plotted as points against the corresponding sample numbers 1,2, … k

np- chart or control Chart for the number of defectives

If the sample size n is constant, i.e., the number of units in all the samples (or Sub–

groups) is the same, then d is the number of defective in a sample is equal to np, where p is the

faction defective in the sample.

For the sampling distribution of the statistic d,

nPnpEdE )()( and

nPQnpESdES )(.)(.

where P is the process (population) fraction defective and PQ 1

The 3 control limits for np-chart are

)(3 npSEnp = nPQnp 3 where PQ 1

Case 1:- Standards are given

If P be the given value of P, then the control limits for np-chart are

QPnPn 3 ,

Now the Upper Control Limit for np is

i.e np

UCL QPnPn 3

And the Lower Control Limit np

LCL QPnPn 3

The Central Line is pCL = Pn

Case-II Standards not given

If P is not given or not known, then p is an unbiased estimate of P and

n

dp =

k

d

where ip is the number of defectives in the ith sample.

The 3 control limits for np-chart are )p(pnpn 13 ,

Now the Upper and Lower Control Limit for np is

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14

i.e., np

UCL = )p(pnpn 13

And np

LCL = )p(pnpn 13

The Central Line is pCL = pn

Note: If the sample size n is constant for all the samples i.e. nnnnk

21(say ), then

np-chart (or d-chart) should be used. But if the sample size varies from sample to sample, the p-

chart is more suitable to examine the state of control.

C-chart or control chart for the number of defects per unit It is constructed to control the number of defects per unit of the products. Every defective

unit may contain one or more defects. If a variable c represents the number of defects observed

per unit (i.e per item), then c follows Poisson distribution with parameter, say m. Thus the 3-𝜎

control limits for c -chart are based on the Poisson distribution whose standard deviation is m

As the statistic c is distributed as a Poisson variable with parameter m , we have mcE )( and

standard error of c is standard deviation of c is equal to m

Case I: Standards are given

If m be the given value of m, then the 3 control limits for c-chart are

mm 3 ,

Now the Upper Lower Control Limits for c are

i.e c

UCL mm 3 And c

LCL mm 3

The Central Line is pCL = m

Case –II:- Standard are not given

If m is not given, then c is an unbiased estimate of m and

k

ccccm k

21

where i

c is the number of defects observed in the ith item of the product.

The 3 control limits for c-chart are cc 3 ,

Now the Upper and Lower Control Limit for c are

i.e., c

UCL cc 3 And c

LCL = cc 3

The Central Line is pCL = c

Note: If in any case lower control limit comes out to be negative, we take it as zero, since c

LCL

is always greater than or equal to zero.

Page 15: Statistical Quality Control Introduction · Def: Statistical quality control (SQC) is defined as the statistical methods applied for the maintenances of uniform quality in a continuous

15

Applications

Example 1: 15 samples of 200 items each wave drawn from the output of a process. The

number of defective items in the samples is given below. Prepare a control charts for the fraction

defective and comment are the state of control.

Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of

Defective (np) 12 15 10 8 19 15 17 11 13 20 10 8 9 5 8

Sol:

From the given data, n = 200 Since each sample contains 200 items; and k = 15

Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total

Number of

Defective (np) 12 15 10 8 19 15 17 11 13 20 10 8 9 5 8 180

and from the above table 180np

k

nppn

12

15

180

121

np 06.012

200

1

For the P-chart

The Central Line CLP = p =0.06

The control limits are )1(3 PPP

200

)06.01(06.0306.0

200

09406.0306.0

01679.0306.0200

0564.0306.0

05038.006.0

The upper control limits for p-chart is

UCLP = n

PPP

)1(3

= 0.06 + 0.05.38 = 0.11038

The lower control limits for p-chart is

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16

LCLP n

PPP

)1(3

= 0.06 - 0.05038 = 0.0096

The fraction defectives (values of p) for the given samples are : 0.06, 0.07, 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.

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

Remark: If the sub-group size n is not constant, then we find two sets of control limits for p-

chart, one set for the largest value of n and the other set for the smallest value of n. if all the

plotted points fall within the inner control limits corresponding to the smallest value of n, then

the process in statistically under control. But if one or more sample points fall outside the

control limits, then the process is out of control and action must be taken to locate the assignable

causes and to remove them. If one or more sample points fall between the two sets of limits,

then the actual control limits should be computed.

Example 2: 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:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of

Defective (np) 5 10 12 8 6 4 6 3 4 5 4 7 9 3 4

Sol:

From the given data, n = 100; each sample contains 100 items, and k = 15,

Sample:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total

Number of

Defective (np) 5 10 12 8 6 4 6 3 4 5 4 7 9 3 4 90

From the table, np 90

Now pn = 69015

11np

k

and P = 06.06100

16

1

n

for the np. Chart:

Central line np

CL = 6Pn

The control limits for np chart are: n )1(3 PPnP

= )06.01(636 = 64.53694.0636

= 3749.236 = 1247.76

The upper control limit for np chart is np

UCL n )1(3 PPnP

= 1247.76 = 13.1247.

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17

The lower control limit for np chart in np

UCL )1(3 PPnPn

= 6 - 7.1247 = -1.1247

Since LCLnp cannot be negative; 0np

UCL

Since al the sample points lie between the upper and lower control lines, the process in under

control.

Example 3: Ten pieces of cloth out of different rolls of equal length contained the following

number of defects: 3, 0, 2, 8, 4, 2, 1, 3, 7, 1. Prepare a c-chart and state whether the production

process is in a state of statistical control.

Sol.

If c be the number of defects in an item, then

c = 3, 0, 2, 8, 4, 2, 1, 3, 7, 1.

k

cc

10

1731248203

10

31 1.3

The control limits for the c-chart are

ccUCLc

3

1.331.3 282.51.3 382.8

And c

LCL = cc 3 282.51.3 182.2

Here we take c

LCL as zero

The central line is c

CL = c 3.1

From the c-chart shown in Fig. we see that all the plotted-point fall between the upper and

lower control limits.

Hence the production process is in a state of statistical control.