u Charts - 单位缺陷数控制图
http://www.sytsma.com/tqmtools/uchart.html
Attribute control charts arise when items are compared with some standard and then are classified as to whether they meet the standard or not. The control chart is used to determine if the rate of nonconforming product is stable and detect when a deviation from stability has occurred. The argument can be made that a LCL should not exist, since rates of nonconforming product outside the LCL is in fact a good thing; we WANT low rates of nonconforming product. However, if we treat these LCL violations as simply another search for an assignable cause, we may learn for the drop in nonconformities rate and be able to permanently improve the process.
The u Chart is used when it is not possible to have an inspection unit of a fixed size (e.g., 12 defects counted in one square foot), rather the number of nonconformities is per inspection unit where the inspection unit may not be exactly one square foot...it may be an intact panel or other object, different in sizethan exactly one square foot. When it is converted into a ratio per square foot, or some other measure, it may be controlled with a u chart. Notice that the number no longer has to be integer as with the c chart.
Steps in Constructing a u Chart
- Find the number of nonconformities, c(i) and the number of inspection units, n(i), in each sample i.
- Compute u(i)=c(i)/n(i)
- Determine the centerline of the u chart:
- The u chart has individual control limits for each subgroup i.
- Plot the centerline, ubar, the individual LCL's and UCL's, and the process measurements, u(i).
- Interpret the control chart.
Example:
Besterfield Example:
data is from Besterfield (1990): Quality Control p. 185
Number Nonconformities
Day Number Non- Per
Inspected Conformities Unit
1 110 120 1.0909
2 82 94 1.1463
3 96 89 0.9271
4 115 162 1.4087
5 108 150 1.3889
6 56 82 1.4643
7 120 143 1.1917
8 98 134 1.3673
9 102 97 0.9510
10 115 145 1.2609
11 88 128 1.4545
12 71 83 1.1690
13 95 120 1.2632
14 103 116 1.1262
15 113 127 1.1239
16 85 92 1.0824
17 101 140 1.3861
18 42 60 1.4286
19 97 121 1.2474
20 92 108 1.1739
21 100 131 1.3100
22 115 119 1.0348
23 99 93 0.9394
24 57 88 1.5439
25 89 107 1.2022
26 101 105 1.0396
27 122 143 1.1721
28 105 132 1.2571
29 98 100 1.0204
30 48 60 1.2500
Calculations:
UBAR = 1.2005
UCL = ubar + 3*sqrt(ubar/n(i))
LCL = ubar - 3*sqrt(ubar/n(i))
Day CL UCL LCL Nonconformities/Unit
1 1.2005 1.513900448 0.887091405 1.09
2 1.2005 1.563485937 0.837505915 1.15
3 1.2005 1.535975424 0.865016429 0.93
4 1.2005 1.507011595 0.893980258 1.41
5 1.2005 1.51678903 0.884202823 1.39
6 1.2005 1.639741695 0.761250158 1.46
7 1.2005 1.500557911 0.900433942 1.19
8 1.2005 1.532534517 0.868457335 1.37
9 1.2005 1.525958845 0.875033008 0.95
10 1.2005 1.507011595 0.893980258 1.26
11 1.2005 1.550892833 0.850099019 1.45
12 1.2005 1.59059276 0.810399092 1.17
13 1.2005 1.537736483 0.86325537 1.26
14 1.2005 1.524375074 0.876616779 1.13
15 1.2005 1.509712226 0.891279627 1.12
16 1.2005 1.55702269 0.843969162 1.08
17 1.2005 1.527566079 0.873425774 1.39
18 1.2005 1.707693252 0.693298601 1.43
19 1.2005 1.534241668 0.866750185 1.25
20 1.2005 1.543190862 0.857800991 1.17
21 1.2005 1.529197361 0.871794491 1.31
22 1.2005 1.507011595 0.893980258 1.03
23 1.2005 1.530853298 0.870138554 0.94
24 1.2005 1.635871613 0.76512024 1.54
25 1.2005 1.548918751 0.852073102 1.20
26 1.2005 1.527566079 0.873425774 1.04
27 1.2005 1.498088223 0.90290363 1.17
28 1.2005 1.521275681 0.879716172 1.26
29 1.2005 1.532534517 0.868457335 1.02
30 1.2005 1.674935581 0.726056271 1.25
u - Chart:
c Charts - 缺陷数控制图
http://www.sytsma.com/tqmtools/cchart.html
Attribute control charts arise when items are compared with some standard and then are classified as to whether they meet the standard or not. The control chart is used to determine if the rate of nonconforming product is stable and detect when a deviation from stability has occurred. The argument can be made that a LCL should not exist, since rates of nonconforming product outside the LCL is in fact a good thing; we WANT low rates of nonconforming product. However, if we treat these LCL violations as simply another search for an assignable cause, we may learn for the drop in nonconformities rate and be able to permanently improve the process.
The c Chart measures the number of nonconformities per "unit" and is denoted by c. This "unit" is commonly referred to as an inspection unit and may be "per day" or "per square foot" of some other predetermined sensible rate.
Steps in Constructing a c Chart
- Determine cbar.
There are k inspection units and c(i) is the number of nonconformities in the ith sample.
- Since the mean and variance of the underlying Poisson distribution are equal,
Thus,
and the UCL and LCL are:
- Plot the centerline cbar, the LCL and UCL, and the process measurements c(i).
- Interpret the control chart.
Example:
Farnum Example: data is from Farnum (1994): Modern Statistical Quality Control and Improvement, p. 248 Non-conforming Day Errors/1000 lines 1 6 2 7 3 7 4 6 5 8 6 6 7 5 8 8 9 1 10 6 11 2 12 5 13 5 14 4 15 3 16 3 17 2 18 0 19 0 20 1 21 2 22 5 23 1 24 7 25 7 26 1 27 5 28 5 29 8 30 8 Calculations: CBAR = 4.4667 UCL = cbar + 3*sqrt(cbar) = 10.80701366 LCL = cbar - 3*sqrt(cbar) = -1.873680327 = 0 (when LCL < 0, set LCL = 0) Day CL UCL LCL NonConforming 1 4.4667 10.80701366 0 6 2 4.4667 10.80701366 0 7 3 4.4667 10.80701366 0 7 4 4.4667 10.80701366 0 6 5 4.4667 10.80701366 0 8 6 4.4667 10.80701366 0 6 7 4.4667 10.80701366 0 5 8 4.4667 10.80701366 0 8 9 4.4667 10.80701366 0 1 10 4.4667 10.80701366 0 6 11 4.4667 10.80701366 0 2 12 4.4667 10.80701366 0 5 13 4.4667 10.80701366 0 5 14 4.4667 10.80701366 0 4 15 4.4667 10.80701366 0 3 16 4.4667 10.80701366 0 3 17 4.4667 10.80701366 0 2 18 4.4667 10.80701366 0 0 19 4.4667 10.80701366 0 0 20 4.4667 10.80701366 0 1 21 4.4667 10.80701366 0 2 22 4.4667 10.80701366 0 5 23 4.4667 10.80701366 0 1 24 4.4667 10.80701366 0 7 25 4.4667 10.80701366 0 7 26 4.4667 10.80701366 0 1 27 4.4667 10.80701366 0 5 28 4.4667 10.80701366 0 5 29 4.4667 10.80701366 0 8 30 4.4667 10.80701366 0 8
c - Chart:
p Charts - 不合格率控制图
http://www.sytsma.com/tqmtools/pchart.html
Attribute control charts arise when items are compared with some standard and then are classified as to whether they meet the standard or not. The control chart is used to determine if the rate of nonconforming product is stable and detect when a deviation from stability has occurred. The argument can be made that a LCL should not exist, since rates of nonconforming product outside the LCL is in fact a good thing; we WANT low rates of nonconforming product. However, if we treat these LCL violations as simply another search for an assignable cause, we may learn for the drop in nonconformities rate and be able to permanently improve the process.
p Charts can be used when the subgroups are not of equal size. The np chart is used in the more limited case of equal subgroups.
Steps in Constructing a p Chart
- Determine the size of the subgroups needed. The size, n(i), has to be sufficiently large to have defects present in the subgroup most of the time. If we have some idea as to what the historical rate of nonconformance, p, is we can use the following formula to estimate the subgroup size:
n=3/p
- Determine the rate of nonconformities in each subgroup by using:
phat(i)=x(i)/n(i)
where:
phat(i)=the rate of nonconformities in subgroup i
x(i)=the number of nonconformities in subgroup i
n(i)= the size of subgroup i
- Find pbar; there are k subgroups.
- Estimate sigma-p if needed and determine the UCL and LCL:
- Plot the centerline, pbar, the LCL and UCL, and the process measurements, the phat's.
- Interpret the data to determine if the process is in control.
Example:
Farnum Example: data is from Farnum (1994): Modern Statistical Quality Control and Improvement, p. 242 Number Day Rejects Tested Proportion 1 14 286 0.0490 2 22 281 0.0783 3 9 310 0.0290 4 19 313 0.0607 5 21 293 0.0717 6 18 305 0.0590 7 16 322 0.0497 8 16 316 0.0506 9 21 293 0.0717 10 14 287 0.0488 11 15 307 0.0489 12 16 328 0.0488 13 21 296 0.0709 14 9 296 0.0304 15 25 317 0.0789 16 15 297 0.0505 17 14 283 0.0495 18 13 321 0.0405 19 10 317 0.0315 20 21 307 0.0684 21 19 317 0.0599 22 23 323 0.0712 23 15 304 0.0493 24 12 304 0.0395 25 19 324 0.0586 26 17 289 0.0588 27 15 299 0.0502 28 13 318 0.0409 29 19 313 0.0607 30 12 289 0.0415 Calculations: PBAR = 0.0539 UCL = pbar + 3*sqrt(pbar*(1-pbar)/n(i)) LCL = pbar - 3*sqrt(pbar*(1-pbar)/n(i)) Day CL UCL LCL Proportion 1 0.0539 0.093892049 0.013808661 0.0490 2 0.0539 0.094246721 0.013453989 0.0783 3 0.0539 0.092310827 0.015389883 0.0290 4 0.0539 0.092126068 0.015574642 0.0607 5 0.0539 0.093410843 0.014289867 0.0717 6 0.0539 0.092624795 0.015075915 0.0590 7 0.0539 0.091587368 0.016113342 0.0497 8 0.0539 0.091943946 0.015756764 0.0506 9 0.0539 0.093410843 0.014289867 0.0717 10 0.0539 0.093822229 0.013878481 0.0488 11 0.0539 0.092498288 0.015202422 0.0489 12 0.0539 0.091240619 0.016460091 0.0488 13 0.0539 0.093209857 0.014490853 0.0709 14 0.0539 0.093209857 0.014490853 0.0304 15 0.0539 0.091883814 0.015816896 0.0789 16 0.0539 0.09314354 0.01455717 0.0505 17 0.0539 0.094103724 0.013596986 0.0495 18 0.0539 0.091646103 0.016054607 0.0405 19 0.0539 0.091883814 0.015816896 0.0315 20 0.0539 0.092498288 0.015202422 0.0684 21 0.0539 0.091883814 0.015816896 0.0599 22 0.0539 0.091528906 0.016171804 0.0712 23 0.0539 0.092688517 0.015012193 0.0493 24 0.0539 0.092688517 0.015012193 0.0395 25 0.0539 0.091470715 0.016229995 0.0586 26 0.0539 0.093683678 0.014017032 0.0588 27 0.0539 0.093011904 0.014688806 0.0502 28 0.0539 0.091823966 0.015876744 0.0409 29 0.0539 0.092126068 0.015574642 0.0607 30 0.0539 0.093683678 0.014017032 0.0415p - Chart:
np Charts - 不合格数控制图
http://www.sytsma.com/tqmtools/npchart.html
Attribute control charts arise when items are compared with some standard and then are classified as to whether they meet the standard or not. The control chart is used to determine if the rate of nonconforming product is stable and detect when a deviation from stability has occurred. The argument can be made that a LCL should not exist, since rates of nonconforming product outside the LCL is in fact a good thing; we WANT low rates of nonconforming product. However, if we treat these LCL violations as simply another search for an assignable cause, we may learn for the drop in nonconformities rate and be able to permanently improve the process.
The np Chart can be used for the special case when the subgroups are of equal size. Then it is not necessary to convert nonconforming counts into the proportions phat(i). Rather, one can directly plot the counts x(i) versus the subgroup number i.
Steps in Constructing an np Chart
- Determine the size of the subgroups needed. The size, n, has to be sufficiently large to have defects present in the subgroup most of the time. If we have some idea as to what the historical rate of nonconformance, p, is we can use the following formula to estimate the subgroup size:
n=3/p
- Find find pbar.
- Find the UCL and LCL where:
- Plot the centerline pbar, the LCL and UCL, and the process nonconforming counts, the x(i)'s.
- Interpret the control chart. Only if a point is outside the +/- 3 sigma range is the process considered to be out of control.
Example:
Farnum Example: data is from Farnum (1994): Modern Statistical Quality Control and Improvement, p. 245 Sample Day Non-conforming Size 1 10 100 2 12 100 3 10 100 4 11 100 5 6 100 6 7 100 7 12 100 8 10 100 9 6 100 10 11 100 11 9 100 12 14 100 13 16 100 14 21 100 15 20 100 16 12 100 17 11 100 18 6 100 19 10 100 20 10 100 21 11 100 22 11 100 23 11 100 24 6 100 25 9 100 Calculations: PBAR = 0.1088 CL = 10.8800 UCL = n*pbar + 3*sqrt(n*pbar*(1-pbar)) LCL = n*pbar + 3*sqrt(n*pbar*(1-pbar)) Day CL UCL LCL NonConforming 1 10.8800 20.22164354 1.538356462 10.0000 2 10.8800 20.22164354 1.538356462 12.0000 3 10.8800 20.22164354 1.538356462 10.0000 4 10.8800 20.22164354 1.538356462 11.0000 5 10.8800 20.22164354 1.538356462 6.0000 6 10.8800 20.22164354 1.538356462 7.0000 7 10.8800 20.22164354 1.538356462 12.0000 8 10.8800 20.22164354 1.538356462 10.0000 9 10.8800 20.22164354 1.538356462 6.0000 10 10.8800 20.22164354 1.538356462 11.0000 11 10.8800 20.22164354 1.538356462 9.0000 12 10.8800 20.22164354 1.538356462 14.0000 13 10.8800 20.22164354 1.538356462 16.0000 14 10.8800 20.22164354 1.538356462 21.0000 15 10.8800 20.22164354 1.538356462 20.0000 16 10.8800 20.22164354 1.538356462 12.0000 17 10.8800 20.22164354 1.538356462 11.0000 18 10.8800 20.22164354 1.538356462 6.0000 19 10.8800 20.22164354 1.538356462 10.0000 20 10.8800 20.22164354 1.538356462 10.0000 21 10.8800 20.22164354 1.538356462 11.0000 22 10.8800 20.22164354 1.538356462 11.0000 23 10.8800 20.22164354 1.538356462 11.0000 24 10.8800 20.22164354 1.538356462 6.0000 25 10.8800 20.22164354 1.538356462 9.0000
np - Chart:
