QM Control Chart Technical Component

8/10/2019 QM Control Chart Technical Component

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D R . P H A M H U Y N H T R A M

D E PA RT M E N T O F I S EP H T R A M @ H C M I U . E D U . V N

Technical component



Contents

Review of basic probability & statistics- Probability- Types of data

- Describing dataStabilizing and improving a process withcontrol charts

- Needs of control chart- Structure of control chart- Attribute control charts- Rules of identifying out-of-control point- Possible mistakes in using control chart



Probability

Example: a bin contains 4000 screws; 2000 are goodand 2000 are defective

What is the probability of drawing a defective screw?- Classical probability- Relative frequency probability

- Difference?



Sub-group

No. ofdefective

Fraction ofdefective

Cummulativeno. ofdefective

Cummulativeno. of screw

Cummulativeof fraction

1

2

3

4

5

6

7

Subgroup size :50



Types of data

Purposes of collecting data? Attribute data

- Classification of items into categories.- E.g..: grade A, B, C

- Counts of the number of items in a given category or a proportionin a given category

- Counts of the number of occurrences per unit .- E.g..: no. of defects per batch, no. of sales per month

Variables (measurement) data- Measurement of a characteristic.

- E.g..: length of time to resolve customer complaint, weights of boxes ofdetergent

- Computation of Numerical Value from two or more measurementsof variables data.- E.g..: computation of a rectangular container, km per liter for each truck



Visually describing data

For frequency distributionTabular displaysGraphical displays

- Histogram (variable data)- Bar chart (attribute data)- Ogive

- Run chart



Histogram

The number of intervals influencesthe pattern, shape, or spread of yourHistogram.



Bar chart

The width of the bar chart has nosignificance



Run chart



Numerically describing data

Measures of central tendency:Mean, median, mode, proportion

Measures of variabilityRange, Standard diviation

Measures of shapeSkewness, kurtosis



Mode = 16

The mode is the most frequently occurring value. It is the value with the highest frequency .

Example - Mode

Given a data set:9, 10, 6, 12, 16, 14, 19, 20, 14, 15, 22, 24, 13, 16, 17, 5, 17, 18, 19, 18, 16, 22



The mean of a set of observations is theiraverage - the sum of the observed values dividedby the number of observations.

Population Mean Sample Mean

m = = x

N i

N

1 x

x

n i

n

= =

1

Arithmetic Mean or Average



Measures of Variability or Dispersion

Range Difference between maximum and minimum values

VarianceMean * squared deviation from the mean

Standard Deviation Square root of the variance

Definitions of population variance and sample variance differ slightly .



Population Variance Sample Variance

Variance and Standard Deviation

N

N

x

x

N

x

N

i N

i

N

i

2

1

1

2

1

2

2

=

=

=

=

=

2 =

1

12

1

1

2

1

2

2

=

=

=

=

=

n

n

x

x

n

x x

s

n

i n

i

n

i

2s s =





Skewed to left

Skewness

6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0

3 0

2 0

1 0

0

x

F r

e q u e n

c y

Mean < median < mode



Mean = median = mode

6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 x

3 0

2 0

1

0

0

F r e

q u e n

c y

Symmetric



Mode < median < mean

6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 x

3 0

2 0

1 0

0

F r e

q u e n

c y

Skewed to right



Kurtosis

3 . 7 2 . 9 2 . 1 1 . 3 0 . 5 - 0 . 3 - 1 . 1 - 1 . 9 - 2 . 7 - 3 . 5

7 0 0

6 0 0

5 0 0

4 0 0

3 0 0

2 0 0

1 0 0

0

X

F r

e q u e n

c y

Platykurtic - flat distribution





Leptokurtic - peaked distribution

1 0 0 - 1 0

2 0 0 0

1 0 0 0

0

Y

F r

e q u e n

c y



Interpretation of Standard Deviation

Normal distributionCalculate probabilitySkewed distribution

Unknown distribution



Normal distribution

K = 1

K=2

K=3

6826.0)11( = m m X P

9544.0)22( = m m X P

9973.0)33( = m m X P



112

114

34

75%

11

31

1

9

8

989%

114

11

161516

94%

2

2

2

= = =

= = =

= = =

Unknown distribution - Chebyshev’s Theorem

At least of the elements of any distribution

lie within k standard deviations of the mean

2

11

k

Atleast

Lie within

Standarddeviations

of the mean

2

3

4









3 Uses of Control Charts

Evaluating the past- Was the process in statistical control?

Evaluating the present-

Maintain an existing state of process stability (generatespecial causes signal during normal operation)- Stop management from over reacting to common

causes of variation

Predicting the near future





Constructing classification chart using PDSA

Plan- Name and flowchart the process- Purpose of the chart?- Characteristics to be charted?- Manner, size, frequency of subgroup selection?- Type of chart?- Form of recording and constructing the control chart?

Do- Record data- Calculate (average, control limits)

Study- Examine for indications of a lack of control- Review periodically, change when appropriate



Constructing classification chart using PDCA

Act- Eliminate any special causes of variation- Reduce causes of common cause variation- Review specifications in relation to the process capability

- Reconsider the purpose of the control chart



Classification charts

Data in the form of classificationsEx: conforming or non conforming

Samples of n items are periodically selected. For

these n distinct units comprising a subgroup



P chart for constant subgroup sizes

])1(

3,0max[

)1(3

n p p

p LCL

n p p pUCL

=

=

subgroupsall inunitsof number Total subgroupsall indefectiveof number Total

p =

n: subgroup sizem: number of subgroup

Centerline

Control limits:



P chart for constant subgroup sizesExample: an importer of decorative ceramic tiles concerns fraction ofcracked or broken tiles before or during transit. Each day a sample of100 tiles is drawn from the total of all tiles received from each tile vendor

Day Sample size Number ofCracked or Broken

Fraction Day Sample size Number of Crackedor Broken

Fraction

1 100 14 0.14 16 100 3 0.03

2 100 2 0.02 17 100 8 0.08

3 100 11 0.11 18 100 4 0.04

4 100 4 0.04 19 100 2 0.02

5 100 9 0.09 20 100 5 0.05

6 100 7 0.07 21 100 5 0.05

7 100 4 0.04 22 100 7 0.07

8 100 6 0.06 23 100 9 0.09

9 100 3 0.03 24 100 1 0.01

10 100 2 0.02 25 100 3 0.03

11 100 3 0.03 26 100 12 0.12

12 100 8 0.08 27 100 9 0.09

13 100 4 0.04 28 100 3 00.03

14 100 15 0.15 29 100 6 0.06

15 100 5 0.05 30 100 9 0.09



P chart for constant subgroup sizes

Centerline(p) = = 183/3000 = 0.061 p

133.0100

)061.01(061.03061.0)( == pUCL

011.0100

)061.01(061.03061.0)( == p LCL



0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Fraction LCL CL UCL



The P Chart

The fraction nonconforming, p, is usually small,say, 0.10 or less.

Because the fraction nonconforming is verysmall, the subgroup sizes must be quite large toproduce a meaningful chart.

Example: subgroup of size 20 05.0= p



P chart for variable subgroup size

)])1(

3(,0max[

)1(3

j

j

n p p

p LCL

n p p

pUCL

=

=

subgroupsall inunitsof number Total subgroupsall indefectiveof number Total

p =



Example: a highwaytoll barrier has 2 typesof toll collection: byautomatic machineand by humanoperator theautomatic lanesrequire exact change while the humanoperator lanes do not.The fraction of vehicles arriving withexact change isexamined usingcontrol chart for aseries of rush hourintervals onconsecutive day.

Day n Number with exact

change

Day n Number with exact

change

1 465 180 11 406 186

2 123 38 12 415 149

3 309 142 13 379 90

4 83 20 14 341 148

5 116 35 15 258 107

6 306 108 16 270 84

7 333 190 17 480 185

8 265 106 18 350 184

9 354 94 19 433 210

10 256 116 20 479 197



Day n FractionDefective

UCL LCL Day n FractionDefective

UCL LCL

1 465 0.387 0.468 0.332 11 406 0.458 0.473 0.327

2 123 0.309 0.533 0.267 12 415 0.359 0.472 0.328

3 309 0.460 0.484 0.316 13 379 0.237 0.475 0.325

4 83 0.241 0.561 0.239 14 341 0.434 0.480 .0320

5 116 0.302 0.536 0.264 15 258 0.415 0.491 0.309

6 306 0.353 0.484 0.316 16 270 0.311 0.489 0.311

7 333 0.571 0.481 0.319 17 480 0.385 0.467 0.333

8 265 0.400 0.490 0.310 18 350 0.526 0.479 0.3219 354 0.266 0.478 0.322 19 433 0.485 0.471 0.329

10 256 0.543 0.492 0.308 20 479 0.411 0.467 0.333



0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Fraction Defective UCL CL LCL



The np chart

The np chart is useful when it's easy to count thenumber of defective items and the sample size isalways the same .

Examples: the number of defective circuit boards,meals in a restaurant, teller interactions in a bank,invoices, or bills.





0

2

4

6

8

10

12

14

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Cracked or Broken LCL CL UCL



Count chart

Defective item ≠ Defects The count chart will help evaluate process stability whenthere can be more than one defect per unit.Examples :

the number of defective elements on a circuit board,the number of defects in a dining experience--order wrong, food toocold, wrong checkthe number of defects in bank statement, invoice, or bill.

This chart is especially useful when you want to knowhow many defects there are not just how manydefective items there are.



C chart

yopportunit of areasof Number

observed eventsof number Total c =

The c chart is useful when it's easy to count the number of defectsand the sample size is always the same.

Center line:

Control limits:



Consider the output of apaper mill: the productappears at the end of a web and is rolled onto aspool called a reel. Every

reel is examined for blemishes, which are theimperfection.

Reel Number of blemishes

Reel Number of blemishes

1 4 14 92 5 15 13 5 16 1

4 10 17 6

5 6 18 106 4 19 3

7 5 20 7

8 6 21 4

9 3 22 8

10 6 23 7

11 6 24 9

12 7 25 7

13 11 Total 150

00.625

150 c(c )Centerline ===

35.13636c3c(c)UCL ===

35.1636c3c(c)LCL ===

C chart – Example





U charts

Control chart for the count of the number of eventsover a given area of opportunity. However, the areaof opportunity varies from observation toobservation

U charts considers the number of events as afraction of the total size of the area of opportunity in which these events were possible

Consider the case of the manufacture of a certain grade of plastic. The



g pplastic is produced in rolls, with samples taken 5 times daily. Becauseof the nature of the process, the square footage of each sample variesfrom inspection lot to inspection lot

Inspection lot

Squarefeet ofplastic

Area ofopportun

ity (in100

squarefeet)

No ofdefects in

lot

Defectsper 100square

feet

Inspection lot

Squarefeet ofplastic

Area ofopportun

ity (in100

squarefeet)

No ofdefects in

lot

Defectsper 100square

feet

1 200 2 5 2.5 16 180 1.80 4 2.22

2 250 2.5 7 2.8 17 80 0.80 1 1.25

3 100 1 3 3 18 100 1.00 2 2.00

4 90 0.9 2 2.22 19 140 1.40 3 2.14

5 120 1.2 4 3.33 20 120 1.20 4 3.33

6 80 0.8 1 1.25 21 250 2.50 2 0.80

7 200 2 10 5 22 130 1.30 3 2.31

8 220 2.20 5 2.27 23 220 2.20 1 0.45

9 140 1.40 4 2.86 24 200 2.00 5 2.50

10 80 0.8 2 2.50 25 100 1.00 2 2.0011 170 1.70 1 0.59 26 160 1.60 4 2.50

12 90 0.90 2 2.22 27 250 2.50 12 4.80

13 200 2.00 5 2.50 28 80 0.80 1 1.25

14 250 2.50 12 4.80 29 150 1.50 5 3.33

15 230 2.30 4 1.74 30 210 2.10 4 1.90

Total 2420 24.2 67 2370 23.7 53



51.290.47

120ft.sq.100

defectsof numberAverageCL(u) ===

Inspectionlot

Number ofInspection

Unit a i

LCL UCL Inspectionlot

Number ofInspection

Unit a i

LCL UCL

1 2 0 5.9 16 1.80 0 6.1

2 2.5 0 5.5 17 0.80 0 7.8

3 1 0 7.3 18 1.00 0 7.3

4 0.9 0 7.5 19 1.40 0 6.55 1.2 0 6.8 20 1.20 0 6.8

6 0.8 0 7.8 21 2.50 0 5.5

7 2 0 5.9 22 1.30 0 6.7

8 2.20 0 5.7 23 2.20 0 5.7

9 1.40 0 6.5 24 2.00 0 5.9

10 0.8 0 7.8 25 1.00 0 7.3

11 1.70 0 6.2 26 1.60 0 6.3

12 0.90 0 7.5 27 2.50 0 5.5

13 2.00 0 5.9 28 0.80 0 7.8

14 2.50 0 5.5 29 1.50 0 6.4

15 2.30 0 5.6 30 2.10 0 5.8



0

1

2

3

4

5

6

7

8

9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Number of Inspection Unit ai LCL CL UCL

Documents

QM Control Chart Technical Component