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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
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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
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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?
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Sub-group
No. ofdefective
Fraction ofdefective
Cummulativeno. ofdefective
Cummulativeno. of screw
Cummulativeof fraction
1
2
3
4
5
6
7
Subgroup size :50
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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
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Visually describing data
For frequency distributionTabular displaysGraphical displays
- Histogram (variable data)- Bar chart (attribute data)- Ogive
- Run chart
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Histogram
The number of intervals influencesthe pattern, shape, or spread of yourHistogram.
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Bar chart
The width of the bar chart has nosignificance
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Run chart
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Numerically describing data
Measures of central tendency:Mean, median, mode, proportion
Measures of variabilityRange, Standard diviation
Measures of shapeSkewness, kurtosis
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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
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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
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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 .
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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 =
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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
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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
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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
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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
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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
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Interpretation of Standard Deviation
Normal distributionCalculate probabilitySkewed distribution
Unknown distribution
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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
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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
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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
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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
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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
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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
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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:
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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
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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
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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
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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
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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 =
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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
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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
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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
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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.
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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
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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.
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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:
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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
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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
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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
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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
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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