process improvement dr. ron tibben-lembke. statistics
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Process Improvement
Dr. Ron Tibben-Lembke
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Statistics
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Measures of Variability Range: difference between largest and smallest
values in a sample Very simple measure of dispersion R = max - min
Variance: Average squared distance from the mean Population (the entire universe of values) variance:
divide by N Sample (a sample of the universe) var.: divide by N-1
Standard deviation: square root of variance
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Skewness Lack of symmetry Pearson’s coefficient
of skewness:0246810121416
0246810121416
0246810121416
Skewness = 0 Negative Skew < 0
Positive Skew > 0
s
Medianx )(3
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Kurtosis Amount of peakedness
or flatness
Kurtosis < 0 Kurtosis > 0
Kurtosis = 04
4)(
ns
xx
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4 -2 0 2 4 6
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Subgroup Size All data plotted on a control chart represents the
information about a small number of data points, called a subgroup.
Variability occurs within each group Only plot average, range, etc. of subgroup Usually do not plot individual data points Larger group: more variability Smaller group: less variability Control limits adjusted to compensate Larger groups mean more data collection costs
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Number of data points Ideally have at least 2 defective points per
sample for p, c charts Need to have enough from each shift, etc.,
to get a clear picture of that environment At least 25 separate subgroups for p or np
charts
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Control Chart Usage Only data from one process on each chart Putting multiple processes on one chart
only causes confusion 10 identical machines: all on same chart or
not?
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Attribute Control Charts Tell us whether points in tolerance or not
p chart: percentage with given characteristic (usually whether defective or not)
np chart: number of units with characteristic c chart: count # of occurrences in a fixed area
of opportunity (defects per car) u chart: # of events in a changeable area of
opportunity (sq. yards of paper drawn from a machine)
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p Chart Control Limits
# Defective Items in Sample i
Sample iSize
UCL p zp
n
p
X
n
p
ii
k
ii
k
(1 - p)
1
1
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p Chart Control Limits
# Defective Items in Sample i
Sample iSize
UCL p zp p)
n
p
X
n
p
ii
k
ii
k
(1
1
1
z = 2 for 95.5% limits; z = 3 for 99.7% limits
# Samples
n
n
k
ii
k
1
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p Chart Control Limits
# Defective Items in Sample i
# Samples
Sample iSize
z = 2 for 95.5% limits; z = 3 for 99.7% limits
UCL p z
LCL p z
n
n
kp
X
n
p
p
ii
k
ii
k
ii
k
1 1
1
and
n
p p) (1
p p)
n
(1
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p Chart ExampleYou’re manager of a 500-room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control (use z = 3)?
© 1995 Corel Corp.
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p Chart Hotel Data
No. No. NotDay Rooms Ready Proportion
1 200 16 16/200 = .0802 200 7 .0353 200 21 .1054 200 17 .0855 200 25 .1256 200 19 .0957 200 16 .080
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p Chart Control Limits
n
n
k
ii
k
1 14007
200
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p Chart Control Limits
16 + 7 +...+ 16
p
X
n
ii
k
ii
k
1
1
1211400
0864.n
n
k
ii
k
1 14007
200
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p Chart Control Limits Solution
pp 3 0864 3.n
p p) (1
200
.0864 * (1-.0864)
p
X
n
ii
k
ii
k
1
1
1211400
0864.n
n
k
ii
k
1 14007
200
16 + 7 +...+ 16
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p Chart Control Limits Solution
0864 0596 1460. . . or & .0268
pp 3 0864 3.n
p p) (1
200
.0864 * (1-.0864)
p
X
n
ii
k
ii
k
1
1
1211400
0864.n
n
k
ii
k
1 14007
200
16 + 7 +...+ 16
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0.00
0.05
0.10
0.15
1 2 3 4 5 6 7
P
Day
p Chart Control Chart Solution
UCL
LCL
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Table 7.1 p.193 Enter the data, compute the average, calculate
standard deviation, plot lines
P Chart of Number Cracked
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Sample
Pro
port
ion
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Dealing with out of control Two points were out of control. Were
there any “assignable causes?” Can we blame these two on anything
special? Different guy drove the truck just those 2 days. Remove 1 and 14 from calculations. p-bar down to 5.5% from 6.1%, st dev, UCL,
LCL, new graph
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Figure 7.4, p. 196P Chart of Number Cracked
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Sample
Pro
port
ion
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Different Sample sizes Standard error varies inversely with sample size Only difference is re-compute for each data
point, using its sample size, n. Why do this? The bigger the sample is, the more
variability we expect to see in the sample. So, larger samples should have wider control limits.
If we use the same limits for all points, there could be small-sample-size points that are really out of control, but don’t look that way, or huge sample-size point that are not out of control, but look like they are.
Judging high school players by Olympic/NBA/NFL standards.
n
pp )1(
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Fig. 7.6P Chart of Exact Change, p.202
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Sample
Pro
portio
n
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How not to do it If we calculate n-bar, average sample size,
and use that to calculate a standard deviation value which we use for every period, we get: One point that really is out of control, does not
appear to be OOC 4 points appear to be OOC, and really are not.
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5 false readingsFig. 7.6 DONE WRONG!!
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Sample
Pro
port
ion
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C-Chart Control Limits # defects per item needs a new chart How many possible paint defects could
you have on a car? C = average number defects / unit Each unit has to have same number of
“chances” or “opportunities” for failure
UCL c zC c
LCL zC cc
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Figure 7.9C Chart Blemishes, p.211
0
2
4
6
8
10
12
14
16
1 3 5 7 9 11 13 15 17 19 21 23 25
Sample
Sam
ple
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Small Average Counts For small averages, data likely not
symmetrical. Use Table 7.8 to avoid calculating UCL,
LCL for averages < 20 defects per sample Aside:
Everyone has to have same definitions of “defect” and “defective”
Operational Definitions: we all have to agree on what terms mean, exactly.
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U charts: areas of opportunity vary Like C chart, counts
number of defects per sample
No. opportunities per sample may differ
Calculate defects / opportunity, plot this.
Number of opportunities is different for every data point
Table 7.13
ia
uu
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U Chart of Defects, p. 222
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
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
Sample
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Variable Control Charts Focus on unit-to-unit variability
x chart: subgroup average R chart: subgroup range I chart: average, subgroup size of one MR moving range chart: one data point per
subgroup s chart: standard deviation with more than 10
samples per subgroup
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R Chart Type of variables control chart
Interval or ratio scaled numerical data
Shows sample ranges over time Difference between smallest & largest values in
inspection sample
Monitors variability in process Calculate the range of each data sample:
Maximum – Minimum Calculate average range:
k
RR
k
ii
1
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R Chart – Control Limits How much variability
should there be in the R values?
Depends on process variability,
We don’t know that, only the R values.
We could get it from here:
233 / dRddR
RDLCL
RDUCL
R
R
3
4
But this seems a lot easier:
Look up values in Table B-1, p. 786
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Control Chart Limits
n A2 D3 D4
2 1.880 0 3.267
3 1.023 0 2.574
4 0.729 0 2.282
5 0.577 0 2.114
6 0.483 0 2.004
7 0.419 0.076 1.924
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You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?
Hotel Example
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Hotel Data
Day Delivery Time
1 7.30 4.20 6.10 3.45 5.552 4.60 8.70 7.60 4.43 7.623 5.98 2.92 6.20 4.20 5.104 7.20 5.10 5.19 6.80 4.215 4.00 4.50 5.50 1.89 4.466 10.10 8.10 6.50 5.06 6.947 6.77 5.08 5.90 6.90 9.30
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R &X Chart Hotel Data
SampleDay Delivery Time Mean Range
1 7.30 4.20 6.10 3.45 5.55 5.32
7.30 + 4.20 + 6.10 + 3.45 + 5.55 5
Sample Mean =
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R &X Chart Hotel Data
SampleDay Delivery Time Mean Range
1 7.30 4.20 6.10 3.45 5.55 5.32 3.85
7.30 - 3.45Sample Range =
Largest Smallest
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R &X Chart Hotel Data
SampleDay Delivery Time Mean Range
1 7.30 4.20 6.10 3.45 5.55 5.32 3.852 4.60 8.70 7.60 4.43 7.62 6.59 4.273 5.98 2.92 6.20 4.20 5.10 4.88 3.284 7.20 5.10 5.19 6.80 4.21 5.70 2.995 4.00 4.50 5.50 1.89 4.46 4.07 3.616 10.10 8.10 6.50 5.06 6.94 7.34 5.047 6.77 5.08 5.90 6.90 9.30 6.79 4.22
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R
R Chart Control Limits
R
k
ii
k
1 3 85 4 27 4 227
3 894. . .
.
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R Chart Control Limits Solution
From B-1 (n = 5)
R
R
k
UCL D R
LCL D R
ii
k
R
R
1
4
3
3 85 4 27 4 227
3 894
(2.114) (3.894) 8 232
(0)(3.894) 0
. . ..
.
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02468
1 2 3 4 5 6 7
R, Minutes
Day
R Chart Control Chart Solution
UCL
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X Chart Control Limits
k
RR
k
XX
RAXUCL
k
ii
k
ii
X
11
2
Sample Range at Time i
# Samples
Sample Mean at Time i
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X Chart Control Limits
UCL X A R
LCL X A R
X
X
kR
R
k
X
X
ii
k
ii
k
2
2
1 1
From Table B-1
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R &X Chart Hotel Data
SampleDay Delivery Time Mean Range
1 7.30 4.20 6.10 3.45 5.55 5.32 3.852 4.60 8.70 7.60 4.43 7.62 6.59 4.273 5.98 2.92 6.20 4.20 5.10 4.88 3.284 7.20 5.10 5.19 6.80 4.21 5.70 2.995 4.00 4.50 5.50 1.89 4.46 4.07 3.616 10.10 8.10 6.50 5.06 6.94 7.34 5.047 6.77 5.08 5.90 6.90 9.30 6.79 4.22
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X Chart Control Limits
X
X
k
R
R
k
ii
k
ii
k
1
1
5 32 6 59 6 797
5 813
3 85 4 27 4 227
3 894
. . ..
. . ..
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X Chart Control Limits
From B-1 (n = 5)
X
X
k
R
R
k
UCL X A R
ii
k
ii
k
X
1
1
2
5 32 6 59 6 797
5 813
3 85 4 27 4 227
3 894
5 813 0 58 * 3 894 8 060
. . ..
. . ..
. . . .
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X Chart Control Limits Solution
From Table B-1 (n = 5)
X
X
k
R
R
k
UCL X A R
LCL X A R
ii
k
ii
k
X
X
1
1
2
2
5 32 6 59 6 797
5 813
3 85 4 27 4 227
3 894
5 813 (0 58)
5 813 (0 58)(3.894) = 3.566
. . ..
. . ..
. .
. .
(3.894) = 8.060
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X ChartControl Chart Solution*
02468
1 2 3 4 5 6 7
X, Minutes
Day
UCL
LCL
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General Considerations, X-bar, R Operational definitions of measuring
techniques & equipment important, as is calibration of equipment
X-bar and R used with subgroups of 4-9 most frequently 2-3 is sampling is very expensive 6-14 ideal
Sample sizes >= 10 use s chart instead of R chart.