Cartas de control para atributos.
Orlando Moscote Flórez
2022-05-31
- 0. Introducción.
- 1. Fracción disconforme.
- 2. Carta np.
- 3. Curva característica de operación.
- 4. Carta p revisada eliminando las observaciones 15 y 23.
- 5. Monitoreo.
- 6. Tamaño de muestra variable.
- 6. Tamaño de muestra promedio.
- 7. Carta c.
- 8. Monitoreo.
- 9. Carta u.
- 10. Curva característica de operación para la carta u.
- 11. Carta u tamaño de muestra variable.
- 12. Carta de control estandarizada pag 321.
- 13. Defectos bajos.
Más aplicaciones estadísticas en:
0. Introducción.
Los datos utilizados son tomados del libro Control estadístico de la calidad de Douglas C. Montgomery.
Datos tomados de la página 289.
nn<-rep(50,30)
dis<-c(12,15,8,10,4,7,16,9,14,10,5,6,17,12,22,8,10,5,13,
11,20,18,24,15,9,12,7,13,9,6)1. Fracción disconforme.
grap<-qcc(dis,type="p",sizes=nn,rules = shewhart.rules)
prop1<-c(dis/nn)
plot(prop1,pch=16,type="o")
data.frame(nn,dis,prop1)## nn dis prop1
## 1 50 12 0.24
## 2 50 15 0.30
## 3 50 8 0.16
## 4 50 10 0.20
## 5 50 4 0.08
## 6 50 7 0.14
## 7 50 16 0.32
## 8 50 9 0.18
## 9 50 14 0.28
## 10 50 10 0.20
## 11 50 5 0.10
## 12 50 6 0.12
## 13 50 17 0.34
## 14 50 12 0.24
## 15 50 22 0.44
## 16 50 8 0.16
## 17 50 10 0.20
## 18 50 5 0.10
## 19 50 13 0.26
## 20 50 11 0.22
## 21 50 20 0.40
## 22 50 18 0.36
## 23 50 24 0.48
## 24 50 15 0.30
## 25 50 9 0.18
## 26 50 12 0.24
## 27 50 7 0.14
## 28 50 13 0.26
## 29 50 9 0.18
## 30 50 6 0.12summary(grap)##
## Call:
## qcc(data = dis, type = "p", sizes = nn, rules = shewhart.rules)
##
## p chart for dis
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0800000 0.1600000 0.2100000 0.2313333 0.2950000 0.4800000
##
## Group sample size: 50
## Number of groups: 30
## Center of group statistics: 0.2313333
## Standard deviation: 0.421685
##
## Control limits:
## LCL UCL
## 0.05242755 0.4102391
## 0.05242755 0.4102391
## ...
## 0.05242755 0.41023912. Carta np.
granp<-qcc(dis,type="np",sizes=nn,rules = shewhart.rules)
summary(granp)##
## Call:
## qcc(data = dis, type = "np", sizes = nn, rules = shewhart.rules)
##
## np chart for dis
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.00000 8.00000 10.50000 11.56667 14.75000 24.00000
##
## Group sample size: 50
## Number of groups: 30
## Center of group statistics: 11.56667
## Standard deviation: 2.981763
##
## Control limits:
## LCL UCL
## 2.621377 20.511963. Curva característica de operación.
beta3 <- oc.curves(qcc(dis, sizes=nn, type="p", plot=TRUE))
## Warning in oc.curves.p(object, ...): Some computed values for the type II error
## have been rounded due to the discreteness of the binomial distribution. Thus,
## some ARL values might be meaningless.
4. Carta p revisada eliminando las observaciones 15 y 23.
nn2<-rep(50,28)
dis2<-c(12,15,8,10,4,7,16,9,14,10,5,6,17,12,8,10,5,13,
11,20,18,15,9,12,7,13,9,6)
grap3<-qcc(dis2,type="p",sizes=nn2,rules = shewhart.rules)
summary(grap3)##
## Call:
## qcc(data = dis2, type = "p", sizes = nn2, rules = shewhart.rules)
##
## p chart for dis2
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.080 0.155 0.200 0.215 0.265 0.400
##
## Group sample size: 50
## Number of groups: 28
## Center of group statistics: 0.215
## Standard deviation: 0.4108223
##
## Control limits:
## LCL UCL
## 0.04070284 0.3892972
## 0.04070284 0.3892972
## ...
## 0.04070284 0.3892972granp4<-qcc(dis2,type="np",sizes=nn2,rules = shewhart.rules)
summary(granp4)##
## Call:
## qcc(data = dis2, type = "np", sizes = nn2, rules = shewhart.rules)
##
## np chart for dis2
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 4.00 7.75 10.00 10.75 13.25 20.00
##
## Group sample size: 50
## Number of groups: 28
## Center of group statistics: 10.75
## Standard deviation: 2.904953
##
## Control limits:
## LCL UCL
## 2.035142 19.464865. Monitoreo.
nn3<-rep(50,54);nn3## [1] 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
## [26] 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
## [51] 50 50 50 50dis3<-c(12,15,8,10,4,7,16,9,14,10,5,6,17,12,22,8,10,5,13,
11,20,18,24,15,9,12,7,13,9,6,9,6,12,5,6,4,6,3,7,6,2,4,3,6,5,4,8,5,6,7,5,6,3,5)
nn4<-nn3[1:30]
nn5<-nn3[31:54]
prueba<-dis3[1:30]
moni<-dis3[31:54]
q1<-qcc(prueba,sizes=nn4, type="p")
summary(q1)##
## Call:
## qcc(data = prueba, type = "p", sizes = nn4)
##
## p chart for prueba
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0800000 0.1600000 0.2100000 0.2313333 0.2950000 0.4800000
##
## Group sample size: 50
## Number of groups: 30
## Center of group statistics: 0.2313333
## Standard deviation: 0.421685
##
## Control limits:
## LCL UCL
## 0.05242755 0.4102391
## 0.05242755 0.4102391
## ...
## 0.05242755 0.4102391q2<-qcc(moni,sizes=nn5, type="p")
summary(q2)##
## Call:
## qcc(data = moni, type = "p", sizes = nn5)
##
## p chart for moni
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0400000 0.0800000 0.1100000 0.1108333 0.1200000 0.2400000
##
## Group sample size: 50
## Number of groups: 24
## Center of group statistics: 0.1108333
## Standard deviation: 0.3139256
##
## Control limits:
## LCL UCL
## 0 0.2440207
## 0 0.2440207
## ...
## 0 0.2440207qcc(prueba,sizes=nn4, type="p", newdata=moni,newsizes=nn5,ylim=c(0,0.5))
## List of 15
## $ call : language qcc(data = prueba, type = "p", sizes = nn4, newdata = moni, newsizes = nn5, ylim = c(0, 0.5))
## $ type : chr "p"
## $ data.name : chr "prueba"
## $ data : num [1:30, 1] 12 15 8 10 4 7 16 9 14 10 ...
## ..- attr(*, "dimnames")=List of 2
## $ statistics : Named num [1:30] 0.24 0.3 0.16 0.2 0.08 0.14 0.32 0.18 0.28 0.2 ...
## ..- attr(*, "names")= chr [1:30] "1" "2" "3" "4" ...
## $ sizes : num [1:30] 50 50 50 50 50 50 50 50 50 50 ...
## $ center : num 0.231
## $ std.dev : num 0.422
## $ newstats : Named num [1:24] 0.18 0.12 0.24 0.1 0.12 0.08 0.12 0.06 0.14 0.12 ...
## ..- attr(*, "names")= chr [1:24] "31" "32" "33" "34" ...
## $ newdata : num [1:24, 1] 9 6 12 5 6 4 6 3 7 6 ...
## $ newsizes : num [1:24] 50 50 50 50 50 50 50 50 50 50 ...
## $ newdata.name: chr "moni"
## $ nsigmas : num 3
## $ limits : num [1:54, 1:2] 0.0524 0.0524 0.0524 0.0524 0.0524 ...
## ..- attr(*, "dimnames")=List of 2
## $ violations :List of 2
## - attr(*, "class")= chr "qcc"6. Tamaño de muestra variable.
Datos tomados de la página 299.
muest<-c(rep(100,1),rep(80,2),rep(100,1),rep(110,2),rep(100,2),rep(90,2),rep(110,1),rep(120,3),rep(110,1),rep(80,3),rep(90,1),rep(100,4),rep(90,2) )
discon<-c(12,8,6,9,10,12,11,16,10,6,20,15,9,8,6,8,10,7,5,8,5,8,10,6,9)
q3<-qcc(discon,sizes=muest, type="p")
prop2<-c(discon/muest)
plot(prop2,pch=16,type="o")
data.frame(muest,discon,prop2)## muest discon prop2
## 1 100 12 0.12000000
## 2 80 8 0.10000000
## 3 80 6 0.07500000
## 4 100 9 0.09000000
## 5 110 10 0.09090909
## 6 110 12 0.10909091
## 7 100 11 0.11000000
## 8 100 16 0.16000000
## 9 90 10 0.11111111
## 10 90 6 0.06666667
## 11 110 20 0.18181818
## 12 120 15 0.12500000
## 13 120 9 0.07500000
## 14 120 8 0.06666667
## 15 110 6 0.05454545
## 16 80 8 0.10000000
## 17 80 10 0.12500000
## 18 80 7 0.08750000
## 19 90 5 0.05555556
## 20 100 8 0.08000000
## 21 100 5 0.05000000
## 22 100 8 0.08000000
## 23 100 10 0.10000000
## 24 90 6 0.06666667
## 25 90 9 0.100000006. Tamaño de muestra promedio.
n<-mean(muest)
muest2<-rep(n,25)
q4<-qcc(discon,sizes=muest2, type="p")
n<-mean(muest)
muest2<-rep(n,25)
q4<-qcc(discon,sizes=muest2, type="p")
beta4 <- oc.curves(qcc(dis, sizes=n, type="p", plot=TRUE))
## Warning in oc.curves.p(object, ...): Some computed values for the type II error
## have been rounded due to the discreteness of the binomial distribution. Thus,
## some ARL values might be meaningless.
beta4## 0 0.01 0.02 0.03 0.04
## 0.000000e+00 6.265357e-01 8.619122e-01 9.494606e-01 9.816945e-01
## 0.05 0.06 0.07 0.08 0.09
## 9.934399e-01 9.976744e-01 9.991838e-01 9.997093e-01 9.998536e-01
## 0.1 0.11 0.12 0.13 0.14
## 9.997369e-01 9.991363e-01 9.973791e-01 9.931400e-01 9.842936e-01
## 0.15 0.16 0.17 0.18 0.19
## 9.679861e-01 9.410310e-01 9.006000e-01 8.450274e-01 7.744591e-01
## 0.2 0.21 0.22 0.23 0.24
## 6.911060e-01 5.989983e-01 5.033102e-01 4.094627e-01 3.222558e-01
## 0.25 0.26 0.27 0.28 0.29
## 2.452391e-01 1.804225e-01 1.283200e-01 8.823582e-02 5.867043e-02
## 0.3 0.31 0.32 0.33 0.34
## 3.773214e-02 2.347580e-02 1.413323e-02 8.234920e-03 4.644566e-03
## 0.35 0.36 0.37 0.38 0.39
## 2.536011e-03 1.340632e-03 6.861673e-04 3.400162e-04 1.631100e-04
## 0.4 0.41 0.42 0.43 0.44
## 7.573731e-05 3.403291e-05 1.479561e-05 6.221134e-06 2.528941e-06
## 0.45 0.46 0.47 0.48 0.49
## 9.934413e-07 3.769223e-07 1.380413e-07 4.876679e-08 1.660632e-08
## 0.5 0.51 0.52 0.53 0.54
## 5.446273e-09 1.718730e-09 5.213931e-10 1.518795e-10 4.243185e-11
## 0.55 0.56 0.57 0.58 0.59
## 1.135483e-11 2.906375e-12 7.104580e-13 1.655830e-13 3.672791e-14
## 0.6 0.61 0.62 0.63 0.64
## 7.737903e-15 1.545134e-15 2.917503e-16 5.195843e-17 8.703570e-18
## 0.65 0.66 0.67 0.68 0.69
## 1.367170e-18 2.007224e-19 2.744387e-20 3.480535e-21 4.076622e-22
## 0.7 0.71 0.72 0.73 0.74
## 4.388515e-23 4.319026e-24 3.863164e-25 3.119902e-26 2.258401e-27
## 0.75 0.76 0.77 0.78 0.79
## 1.453329e-28 8.238108e-30 4.070726e-31 1.732875e-32 6.269901e-34
## 0.8 0.81 0.82 0.83 0.84
## 1.898611e-35 4.726459e-37 9.474085e-39 1.492320e-40 1.794733e-42
## 0.85 0.86 0.87 0.88 0.89
## 1.592242e-44 9.996426e-47 4.221225e-49 1.125661e-51 1.750662e-54
## 0.9 0.91 0.92 0.93 0.94
## 1.433398e-57 5.399486e-61 7.795705e-65 3.339486e-69 2.917158e-74
## 0.95 0.96 0.97 0.98 0.99
## 2.905275e-80 1.245431e-87 3.700676e-97 1.263449e-110 1.031882e-133
## 1
## 0.000000e+007. Carta c.
Datos tomados de la página 311.
circ<-c(21,24,16,12,15,5,28,20,31,25,20,24,16,19,10,17,13,22,18,39,30,24,16,19,17,15)
q5<-qcc(circ, type="c")
summary(q5)##
## Call:
## qcc(data = circ, type = "c")
##
## c chart for circ
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 5.00000 16.00000 19.00000 19.84615 24.00000 39.00000
##
## Group sample size: 1
## Number of groups: 26
## Center of group statistics: 19.84615
## Standard deviation: 4.454902
##
## Control limits:
## LCL UCL
## 6.481447 33.210868. Monitoreo.
circ2<-c(16,18,12,15,24,21,28,20,25,19,18,21,16,22,19,12,14,9,16,21)
q6<-qcc(circ2, type="c")
summary(q6)##
## Call:
## qcc(data = circ2, type = "c")
##
## c chart for circ2
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 9.00 15.75 18.50 18.30 21.00 28.00
##
## Group sample size: 1
## Number of groups: 20
## Center of group statistics: 18.3
## Standard deviation: 4.27785
##
## Control limits:
## LCL UCL
## 5.46645 31.133559. Carta u.
Datos tomados de la página 317.
tama<-rep(5,20)
total<-c(10,12,8,14,10,16,11,7,10,15,9,5,7,11,12,6,8,10,7,5)
q10<-qcc(total,sizes=tama,type="u")
summary(q10)##
## Call:
## qcc(data = total, type = "u", sizes = tama)
##
## u chart for total
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 1.00 1.40 2.00 1.93 2.25 3.20
##
## Group sample size: 5
## Number of groups: 20
## Center of group statistics: 1.93
## Standard deviation: 1.389244
##
## Control limits:
## LCL UCL
## 0.06613305 3.79386710. Curva característica de operación para la carta u.
beta5 <- oc.curves(qcc(total, sizes=tama, type="u", plot=TRUE))
## Warning in oc.curves.c(object, ...): Some computed values for the type II error
## have been rounded due to the discreteness of the Poisson distribution. Thus,
## some ARL values might be meaningless.
11. Carta u tamaño de muestra variable.
Datos tomados de la página 320.
defec<-c(14,12,20,11,7,10,21,16,19,23)
mue<-c(10,8,13,10,9.5,10,12,10.5,12,12.5)
q11<-qcc(defec,sizes=mue,type="u")
summary(q11)##
## Call:
## qcc(data = defec, type = "u", sizes = mue)
##
## u chart for defec
##
## Summary of group statistics:
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.7368421 1.1750000 1.5119048 1.3972447 1.5721154 1.8400000
##
## Summary of group sample sizes:
## sizes 8 9.5 10 10.5 12 12.5 13
## counts 1 1.0 3 1.0 2 1.0 1
##
## Number of groups: 10
## Center of group statistics: 1.423256
## Standard deviation: 1.193003
##
## Control limits:
## LCL UCL
## 0.2914739 2.555038
## 0.1578852 2.688626
## ...
## 0.4109593 2.43555212. Carta de control estandarizada pag 321.
upro<-c(defec/mue);upro## [1] 1.4000000 1.5000000 1.5384615 1.1000000 0.7368421 1.0000000 1.7500000
## [8] 1.5238095 1.5833333 1.8400000ubarra<-c(sum(defec)/sum(mue));ubarra## [1] 1.423256zeta1<-(upro-ubarra)/(sqrt(ubarra/mue));zeta1## [1] -0.06164389 0.18194872 0.34818035 -0.85685012 -1.77339822 -1.12191886
## [7] 0.94876140 0.27311859 0.46481431 1.23504583plot(zeta1,type="o",ylim=c(-3,3))
abline(h=0,lty=2)
abline(h=-3,lty=2)
abline(h=3,lty=2)
13. Defectos bajos.
Datos tomados de la página 326.
tiempo<-c(286,948,536,124,816,729,4,143,431,8,2837,596,81,227,603,492,1199,1214,2831,96)
y<-tiempo^0.2777;y## [1] 4.809865 6.709029 5.726497 3.813671 6.435412 6.237053 1.469576 3.967682
## [9] 5.390069 1.781509 9.096180 5.897744 3.388335 4.510954 5.916898 5.591891
## [17] 7.161238 7.186005 9.090833 3.552031qcc(y, type="xbar.one")
## List of 11
## $ call : language qcc(data = y, type = "xbar.one")
## $ type : chr "xbar.one"
## $ data.name : chr "y"
## $ data : num [1:20, 1] 4.81 6.71 5.73 3.81 6.44 ...
## ..- attr(*, "dimnames")=List of 2
## $ statistics: Named num [1:20] 4.81 6.71 5.73 3.81 6.44 ...
## ..- attr(*, "names")= chr [1:20] "1" "2" "3" "4" ...
## $ sizes : int [1:20] 1 1 1 1 1 1 1 1 1 1 ...
## $ center : num 5.39
## $ std.dev : num 2.09
## $ nsigmas : num 3
## $ limits : num [1, 1:2] -0.888 11.661
## ..- attr(*, "dimnames")=List of 2
## $ violations:List of 2
## - attr(*, "class")= chr "qcc"|-|-|-|
O.M.F.
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