30079_36.pdf

CHAPTER 36

STATISTICAL QUALITY CONTROL

Magd E. Zohdi

Department of Industrial Engineering and Manufacturing

Louisiana State University

Baton Rouge, Louisiana

36.1 MEASUREMENTS AND

QUALITY CONTROL 11 75

36.6 CONTROL CHARTS FOR

ATTRIBUTES 11 80

36.6.1 The p and np Charts 1 182

36.6.2 The c and u Charts 11 83

36.2 DIMENSION AND

TOLERANCE

11 75

36.7 ACCEPTANCE SAMPLING 118 3

36.7. 1 Double Sampling

36.3 QUALITY CONTROL 11 75

36.3.1 X, R, and <r Charts 11 75

1 1 84

36.7.2 Multiple and Sequential

Sampling

1184

36.4 INTERRELATIONSHIP OF

TOLERANCES OF

ASSEMBLED PRODUCTS 11 79

36.8 DEFENSE DEPARTMENT

ACCEPTANCE SAMPLING BY

VARIABLES

11 84

36.5 OPERATION

CHARACTERISTIC CURVE

(OC)

11 80

36.1 MEASUREMENTS AND QUALITY CONTROL

The metric and English measuring systems are the two measuring systems commonly used throughout

the world. The metric system is universally used in most scientific applications, but, for manufacturing

in the United States, has been limited to a few specialties, mostly items that are related in some way

to products manufactured abroad.

36.2 DIMENSION AND TOLERANCE

In dimensioning a drawing, the numbers placed in the dimension lines are only approximate and do

not represent any degree of accuracy unless so stated by the designer. To specify the degree of

accuracy, it is necessary to add tolerance figures to the dimension. Tolerance is the amount of variation

permitted in the part or the total variation allowed in a given dimension.

Dimensions given close tolerances mean that the part must fit properly with some other part. Both

must be given tolerances in keeping with the allowance desired, the manufacturing processes avail-

able, and the minimum cost of production and assembly that will maximize profit. Generally speaking,

the cost of a part goes up as the tolerance is decreased.

Allowance, which is sometimes confused with tolerance, has an altogether different meaning. It

is the minimum clearance space intended between mating parts and represents the condition of tightest

permissible fit.

36.3 QUALITY CONTROL

When parts must be inspected in large numbers, 100% inspection of each part is not only slow and

costly, but does not eliminate all of the defective pieces. Mass inspection tends to be careless;

operators become fatigued; and inspection gages become worn or out of adjustment more frequently.

The risk of passing defective parts is variable and of unknown magnitude, whereas, in a planned

sampling procedure, the risk can be calculated. Many products, such as bulbs, cannot be 100%

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

inspected, since any final test made on one results in the destruction of the product. Inspection is

costly and nothing is added to a product that has been produced to specifications.

Quality control enables an inspector to sample the parts being produced in a mathematical manner

and to determine whether or not the entire stream of production is acceptable, provided that the

company is willing to allow up to a certain known number of defective parts. This number of

acceptable defectives is usually taken as 3 out of 1000 parts produced. Other values might be used.

36.3.1 X, ft, and a Charts

To use quality techniques in inspection, the following steps must be taken (see Table 36.1).

1. Sample the stream of products by taking m samples, each of size n.

2. Measure the desired dimension in the sample, mainly the central tendency.

3. Calculate the deviations of the dimensions.

4. Construct a control chart.

5. Plot succeeding data on the control chart.

_ The arithmetic mean of the set of n units is the main measure of central tendency. The symbol

X is used to designate the arithmetic mean of the sample and may be expressed in algebraic terms

X,. - (Xl + X2 + X3 + . . . + XJ/n

(36.1)

where Xl9 X2, X3, etc. represent the specific dimensions in question. The most useful measure of

dispersion of a set of numbers is the standard deviation cr. It is defined as the root-mean-square

deviation of the observed numbers from their arithmetic mean. The standard deviation cr is expressed

in algebraic terms as

a= IK - xr+ <&-&+ ... + <*.-XT

(362)

1 V n

Another important measure of dispersion, used particularly in control charts, is the range R. The

range is the difference between the largest observed value and the smallest observed in a specific

sample.

R = X.(max) - X,.(min)

(36.3)

Even_ though the distribution of the X values in the universe can be of any shape, the distribution of

the X values tends to be close to the normal distribution. The larger the sample size and the more

nearly normal the universe, the closer will the frequency distribution of the average X's approach

the normal curve, as in Fig. 36.1.

_ According to the statistical theory, (the Central Limit Theory) in the long run, the average of the

X values will be the same as /u,, the average of the universe. And in the long run, the standard

deviation of the frequency distribution X values, cr-, will be given by

cr-x = -7=

(36.4)

where cr is the standard deviation of the universe. To construct the control limits, the following steps

are taken:

Table 36.1 Computational Format for Determining X, R, and <r

Standard

Sample

Mean

Range

Deviation

Number

Sample Values

cr1

Xn, X12, . . . , Xln

cr[

X21, X22, • • . , X2n

cr2

Xml, Xm2, . . . , Xmn

cr'm

Fig. 36.1 Normal distribution and percentage of parts that will fall within a limits.

1. Calculate the average of the average X as follows:

X = 2) Xt/m i = 1, 2, . . . , m

(36.5)

2. Calculate the average deviation, <r where

0-=^ o-'t/m i = 1, 2, . . . , m

(36.6)

Statistical theory predicts the relationship between cr and cr-. The relationship for the 3 a- limits or

the 99.73% limits is

A,a = 3cr-

(36.7)

This means that control limits are set so that only 0.27% of the produced units will fall outside the

limits. The value of 3cr- is an arbitrary limit that has found acceptance in industry.

The value of Al calculated by probability theory is dependent on the sample size and is given in

Table 36.2. The formula for 3<r control limits using this factor is

CL(X} = X± A,a

(36.8)

Once the control chart (Fig. 36.2) has been established, data (X/s) that result from samples of

the same size n are recorded on it. It becomes a record of the variation of the inspected dimensions

over a period of time. The data plotted should fall in random fashion between the control limits

99.73% of the time if a stable pattern of variation exists.

So long as the points fall between the control lines, no adjustments or changes in the process are

necessary. If five to seven consecutive points fall on one side of the mean, the process should be

checked. When points fall outside of the control lines, the cause must be located and corrected

immediately. _

Statistical theory also gives the expected relationship between R (2/^/m) and cr-. The relationship

for the 3cr- limits is

A2R = 3cr-

(36.9)

The values for A2 calculated by probability theory, for different sample sizes, are given in Table 36.2.

The formula for 3cr control limits using this factor is

CL(X) = X±A2R

(36.10)

In control chart work, the ease of calculating R is usually much more important than any slight

Table 36.2 Factors for X, R, <r, and X Control Charts

Factors for X

Chart

From

RA2

1.880

1.023

0.729

0.577

0.483

0.419

0.373

0.337

0.308

0.285

0.266

0.249

0.235

0.223

0.212

0.203

0.194

0.187

0.180

0.173

0.167

0.162

0.157

0.153

Factors for R

Chart

Lower

0.076

0.136

0.184

0.223

0.256

0.284

0.308

0.329

0.348

0.364

0.380

0.393

0.404

0.414

0.425

0.434

0.443

0.451

0.459

Factors for a'

Chart

Lower

0.030

0.118

0.185

0.239

0.284

0.321

0.354

0.382

0.406

0.428

0.448

0.466

0.482

0.497

0.510

0.523

0.534

0.545

0.555

0.565

Factors for X

Chart

From

RE2

2.660

1.772

1.457

1.290

1.184

1.109

1.054

1.011

0.975

0.946

0.921

0.899

0.881

0.864

0.848

0.830

0.820

0.810

0.805

0.792

0.783

0.776

0.769

0.765

a = R/d2

1.128

1.693

2.059

2.326

2.539

2.704

2.847

2.970

3.078

3.173

3.258

3.336

3.407

3.472

3.532

3.588

3.640

3.687

3.735

3.778

3.819

3.858

3.895

3.931

Sample

Sizen

From

a>A1

3.759

2.394

1.880

1.596

.410

.277

.175

.094

.028

0.973

0.925

0.884

0.848

0.817

0.788

0.762

0.738

0.717

0.698

0.680

0.662

0.647

0.632

0.619

Upper

C>4

3.268

2.574

2.282

2.114

2.004

1.924

1.864

1.816

1.171

1.744

1.717

1.692

1.671

1.652

1.636

1.621

1.608

1.597

1.586

1.575

1.566

1.557

1.548

1.540

Upper

3.267

2.568

2.266

2.089

1.970

1.882

1.815

1.761

1.716

1.679

1.646

1.618

1.594

1.572

1.552

1.534

1.518

1.503

1.490

1.477

1.466

1.455

1.445

1.435

From

aE,

5.318

4.146

3.760

3.568

3.454

3.378

3.323

3.283

3.251

3.226

3.205

3.188

3.174

3.161

3.152

3.145

3.137

3.130

3.122

3.114

3.105

3.099

3.096

3.095

Fig. 36.2 Control chart X.

theoretical advantage that might come from the use of a. However, in some cases where the mea-

surements are costly and it is necessary that the inferences from a limited number of tests be as

reliable as possible, the extra cost of calculating cr is justified. It should be noted that, because Fig.

36.2 shows the averages rather than individual values, it would have been misleading to indicate the

tolerance limits on this chart. It is the individual article that has to meet the tolerances, not the

average of a sample. Tolerance limits should be compared to the machine capability limits. Capability

limits are the limits on a single unit and can be calculated by

capability limits = X ± 3cr

(36.11)

a = R/d2

Since cr' = Vrc cr-, the capability limits can be given by

capability limits (X) = X ± 3Vn cr-

(36.12)

= X ± E^

(36.13)

- X ± E2R

(36.14)

The values for d2, E^ and E2 calculated by probability theory, for different sample sizes, are given

in Table 36.2.

Figure 36.3 shows the relationships among the control limits, the capability limits, and assumed

tolerance limits for a machine that is capable of producing the product with this specified tolerance.

Capability limits indicate that the production facility can produce 99.73% of its products within these

limits. If the specified tolerance limits are greater than the capability limits, the production facility

is capable of meeting the production requirement. If the specified tolerance limits are tighter than

the capability limits, a certain percentage of the production will not be usable and 100% inspection

will be required to detect the products outside the tolerance limits.

_ To detect changes in the dispersion of the process, the R and cr charts are often employed with

X and X charts.

The upper and lower control limits for the R chart are specified as

UCL(R) = D4R

(36.15)

LCL(R) = D3R

(36.16)

Figure 36.4 shows the R chart for samples of size 5.

The upper and lower control for the T chart are specified as

UCL(cr) = B4a

(36.17)

LCL(cr) = B3o-

(36.18)

The values for D3, Z)4, B3, and B4 calculated by probability theory, for different sample sizes, are

given in Table 36.2.

36.4 INTERRELATIONSHIP OF TOLERANCES OF ASSEMBLED PRODUCTS

Mathematical statistics states that the dimension on an assembled product may be the sum of the

dimensions of the several parts that make up the product. It states also that the standard deviation of

the sum of any number of independent variables is the square root of the sum of the squares of the

standard deviations of the independent variables. So if

X = X1 ±X2± ...XH

(36.19)

X = Xl±X2± ...Xn

(36.20)

<r(X) = V(cr1)2 + (cr2)2 + . . . + (crn)2

(36.21)

Whenever it is reasonable to assume that the tolerance ranges of the parts are proportional to their

respective cr' values, such tolerance ranges may be combined by taking the square root of the sum

of the squares:

T = V71? + T\ + T\ + . . . + T2n

(36.22)

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