1. Problemas.
Histogramas, estratificación, Ishikawa, y
diagrama de Pareto.
Lizbeth Luévanos Martínez.
Universidad Tecnológica de Torreón.
Gerardo Edgar Mata.

2. La empresa “Perfumes y esencias Fraiche” elabora perfumes con diferentes
concentrados 33% y 55% de fragancia. Han recibido una demanda espacial para un
concurso de belleza; el cliente pide que se elaboren perfumes con un concentrado de
33% de fragancia con una tolerancia de ±.19, debido a la importancia de este
concurso se están trabajando con 3 maquinas para darle mayor rapidez. La persona
encargada del departamento de calidad ha decidido tomar una muestra de 300
piezas tomadas de las tres maquinas para verificar que las cosas se estén haciendo
bien. ¿Tomando en cuenta los resultados de la investigación crees que las cosas se
están haciendo bien? Los datos son los siguientes:
Primero veremos los 1500 datos tomados de las tres maquinas y posteriormente los
datos de cada una de las máquinas para poder identificar como se desempeña la
empresa:

3. 32.93 32.97 32.97 32.94 33.02 33.11 32.88 32.72 33.04 33.05 33.18 33.01 32.95 32.98 33.08 33.02 33.09 33.11 33.11 33.11
33.1 33.07 32.86 33.09 33.02 33.01 33.14 33.16 32.93 33.1 33.07 33.08 33.09 33.07 33.06 32.93 33.02 33 33.08 33.02
32.91 32.86 33.03 33.06 33 32.92 33.27 32.86 33.08 32.99 32.97 33.04 33.13 33.04 33.01 33.05 33.08 33 32.99 33
32.99 33.06 33.14 32.93 32.87 33.06 33.1 32.93 33.06 32.99 33.11 32.97 32.95 32.97 33.06 32.98 33.03 33.01 33.05 33.02
32.97 33.11 33.13 33.14 33.03 33.05 32.99 33.08 33.12 33.02 33.1 33.04 33.09 33.05 33.11 33.03 33.15 33.01 33.12 33.03
33.16 32.91 33.19 32.81 33.02 33.08 32.91 33.05 32.7 32.96 33.07 33.1 33.1 33.04 33.06 32.96 33.04 32.96 33.06 33.01
32.86 33.04 32.76 33.06 33.08 32.9 32.99 33.09 33.05 33.1 33.05 33.13 32.99 33.07 33.04 33.07 33.11 33.02 33.08 33.07
32.9 33.06 32.97 32.99 32.92 33.06 32.97 33.06 33.16 32.95 32.94 33.1 33.11 33.05 33.08 33.04 33.13 33.07 33.05 33.05
32.98 33.11 33.12 33 33.07 33.17 32.96 32.85 32.81 33.25 32.97 32.98 33.13 33.14 33.19 33.08 33.12 33.12 33.07 33.02
32.88 32.98 32.88 32.93 33.02 33.03 33.09 32.88 33 33.08 33.09 33.1 33.21 33.06 33.09 33 33.04 32.96 33.04 33.07
32.84 33.03 32.93 32.94 33.01 32.96 32.97 33.08 33.15 33.07 33.03 33.24 33.13 33.1 33.02 33.08 32.96 33.18 33.03 33.09
32.77 32.95 32.82 33.09 33.09 33.03 33.04 33.04 33.18 32.89 32.97 33.1 33.06 33.01 33.06 33.03 33.05 33.04 33.09 33.12
33.06 33.06 32.8 33.09 33.2 33.2 33.09 32.9 33.03 33.09 33.02 33.05 33.02 33.04 33.03 32.99 33.06 32.99 32.99 33.11
32.96 33.17 32.89 33.05 32.86 32.74 33.03 32.97 33.02 33.16 33.05 32.95 33.09 33.01 33.13 33 33.04 33 33 33.01
33.04 32.98 32.92 33.08 32.87 33 32.82 33.01 33.2 32.92 33.11 32.96 33.15 33.06 32.99 33.02 33.06 33.01 33.11 33.12
32.78 32.89 32.77 33.18 33.06 32.99 33 33.06 33 32.97 33.14 33.03 33.14 32.99 32.94 33.03 33.06 33.07 33.04 33.05
33.13 32.9 32.94 33.03 33.13 33.11 33.07 33.02 32.93 32.85 33.06 33.1 33.02 33.03 33.16 33.01 33.12 33.06 33.03 33.12
32.79 33.08 32.96 33.06 32.9 33.13 32.98 32.9 33.1 33.15 32.99 33.04 33 32.98 32.93 33.02 33.06 33.04 33.07 32.92
32.94 33.17 33.01 33.2 32.77 33.18 32.82 33.2 33 33.11 33.11 33.05 33.05 33.13 32.84 32.96 33.12 33.03 33.02 33.07
32.89 32.93 32.96 32.96 32.9 32.8 32.83 33.05 33.11 33.02 33.06 33.17 33.02 32.98 33.13 33.06 33.22 33.05 33.09 33.03
32.99 32.91 32.96 33.11 33.02 33.09 32.91 33.01 33.05 32.76 32.99 32.93 32.98 33.1 33.05 33.07 33 33.1 33.01 33
33.01 33.11 32.96 33.02 33.05 33.04 33.03 33.09 33.16 33.05 33.04 33.08 33.03 33.06 33.05 33.07 33.11 33.09 33.16 32.98
33.19 33.02 33.15 33.12 32.89 33.08 32.93 33.09 32.99 33.06 33.16 33.11 33.15 33.12 32.96 33.03 33.1 32.97 33.17 33.03
32.95 33.05 32.99 32.98 32.91 33.05 33.22 32.93 32.9 33.13 33.06 32.98 33.09 33.01 32.97 33.05 33 33.03 32.98 33.14
32.91 33.28 32.98 32.92 33.05 32.99 32.98 32.9 32.83 32.9 33.01 33.07 33.13 33 33.02 33.1 33.1 33 33.06 33.02
32.92 32.94 32.94 33.12 33.01 33.18 33.09 33.12 32.72 33.13 33.03 32.95 33.04 32.92 33.15 33.12 33.34 32.92 33.1 33.14
33.06 32.91 33.21 32.94 32.94 33.05 33.13 32.87 33.01 32.94 32.82 33.09 33.07 32.94 33.05 33.14 33.32 33.05 33.16 33.06
33.14 33.13 33.09 33.06 33.01 33 32.95 32.83 33.03 32.92 33.18 33.09 33.15 33.23 33.09 33.21 33.11 33.05 33.09 33.1
32.84 32.97 33.26 33.06 32.9 33 32.8 33.08 33 32.81 33.06 33.17 33.06 33.08 33.11 33.14 33.02 33.09 33.11 33.16
33.11 32.85 32.93 32.97 32.9 32.99 32.9 33.07 33.02 33.15 33.09 33.11 33.13 32.99 33.05 33.08 32.99 33.1 33.17 33.12
33.17 33.12 33.18 32.95 33.11 32.82 33.01 33.24 33.03 32.9 33.13 33.14 32.91 33.04 33.09 32.94 32.99 33.12 33.11 32.92
32.93 32.86 32.83 32.85 33.18 33.05 33.03 33.16 33.16 33.17 33.15 33.06 33.16 33.11 33.22 33.03 33.06 33.03 32.94 33.36
33.01 32.96 33.06 33.22 32.93 33.02 33.1 33.14 33.01 32.87 33.1 32.99 33.15 33.13 33.02 32.97 33.21 33.07 33.13 33.11
32.91 32.84 33.1 32.94 32.98 33.07 32.77 33.01 32.88 32.96 33.02 33.09 33.06 33.16 33.03 33.32 33.16 33.23 32.99 33.15
33.08 32.88 33.21 33.01 33.01 32.98 32.93 33 32.85 32.86 33.06 33.05 33.08 33.03 33.21 32.97 33.13 33.26 33.11 33.09
33.03 33.07 32.99 32.97 33.2 33.09 33.02 33.05 33.02 33.17 33 33.05 33.12 33.14 33.03 33.21 33.18 33.13 33.26 32.94
33.01 33.01 32.94 33.31 33.06 33.14 33.02 33 33.1 33 33.14 33.01 33.23 32.97 33.08 33.02 33.18 32.94 33.11 33.16
32.89 32.81 33.07 33.14 33.18 33.21 32.99 32.89 32.84 33.04 33.18 33.13 33.01 33.13 32.87 33.18 33.09 33 33.17 33.22
32.97 33.22 32.96 33.1 33.02 32.86 32.97 32.81 32.93 33.11 32.96 33.25 33.1 33.16 33.22 33.1 33.2 33.21 33.03 33.16
33.08 33.03 33.08 33.14 33.17 33.07 33.16 33.09 33.06 33.21 33.15 33.15 33.15 33.17 33.12 32.92 33.06 33.12 33.19 33.35
33 33.15 32.84 33.02 32.93 32.98 33.09 33.05 32.98 32.99 33.14 32.97 33.04 33.15 33.16 33.07 33.09 33.09 33.16 33.33
33.11 33 32.91 33.06 32.9 32.92 32.86 33.07 32.96 32.89 33.22 33.21 33.13 33.04 33.05 33.06 32.99 33.23 33.07 33.03
32.89 32.97 32.83 32.97 33.13 33.08 32.95 33.02 32.98 32.94 33.22 33.14 32.9 32.96 33.04 33.13 33.16 32.95 33.21 32.99
33.08 33 32.96 32.87 33.1 33.25 32.97 32.89 32.95 33.09 33.12 33.04 33.11 33.2 33.13 33.19 33 33.09 33.14 33.2
33.13 32.8 33 33.14 32.81 33.15 32.9 33.13 32.91 33.09 33.3 33.18 33.16 33.07 33.11 33.1 33.13 33.07 33.08 33.13
32.87 32.93 33 32.97 33.14 33.02 32.98 32.97 33.07 33.06 33.08 33.15 33.27 33 33.12 33.11 33.17 33.11 33.02 33.12
33.04 33.15 32.96 32.86 32.98 32.95 33.02 32.91 33.01 32.84 33.03 33.12 33.09 33.1 33.18 33.02 33.02 32.98 33.04 33.12
32.81 33.02 33.24 33.08 32.88 32.97 32.9 32.91 33.14 32.85 32.9 33.08 33.09 33.03 33.1 33.09 33.13 33.12 33.2 33
33.01 32.95 32.81 33.09 32.8 32.88 33.09 32.95 33.03 32.98 33.09 32.95 33.13 33 33.01 32.89 33.05 33.22 33.25 33.22
33.11 32.93 32.92 33.05 32.91 33.01 32.98 32.95 33.01 32.93 33.09 33.04 33.12 33.04 33.25 33.09 33.17 33.27 33.06 33.05
33.05 33.08 33.16 33.15 33 33.03 33.02 32.89 33.02 33.08 33.03 33.14 33 33.06 32.97 33.02 32.93 33.14 33.23 32.98
33.05 33.01 33.02 33.05 33 33.06 32.98 33.07 33.15 33.08 33.06 33.17 32.96 33.17 33.15 33.12 33.01 33 33.29 33.26
32.99 33.06 32.99 33.13 33.02 33.06 33.09 33.02 32.99 33.11 33.12 33.07 33.22 33.05 33.09 33.13 32.95 33.14 33.17 33.15
33.13 33.09 33.13 33.02 33.02 33.13 33.04 33.11 33.06 33.1 33.19 33.24 33.19 33.17 33.1 33.14 33.17 32.93 33 32.96
33.05 33.08 32.94 32.96 32.97 33.1 33.06 33.11 33.06 33.1 33.21 33.18 33.07 33.27 33.03 33.15 33 33.21 33.03 33.08
33.11 32.99 33.08 32.92 32.97 32.99 33.02 33.08 33.07 33.18 33.15 33.22 33.01 33.12 33.01 33.04 33.13 33.11 33.18 33.06
33.02 33.12 33.04 33.06 33.02 33.02 33.04 33.06 33.04 33.06 32.97 33.2 33.07 33.15 33.1 33.13 33.08 33.04 33.04 33.09
33.11 33.07 33.01 33.08 33.05 33 33.08 33.08 33.15 33.09 33 33.16 33.1 33.18 33.13 33.24 33.32 33.13 33.19 33.13
33.14 33.04 32.98 33.04 33.09 33.05 33.03 33.05 33.05 33.04 33.02 33.21 33.17 33.06 33.04 33.27 33.08 33.18 33.11 33.03
33.16 33.15 33.02 33 33.11 33.1 32.96 32.99 33.13 33.09 33.24 32.89 33.08 33.14 33.06 33.1 33.14 33.24 32.88 33.18
33.02 33.07 33.07 33.01 33.14 33.11 33.08 33.07 33.14 33.01 33.13 33 33.03 33.12 33.11 33.14 33.15 32.96 33.1 33.07
33.1 33.18 33.07 33.01 33.02 32.94 33.01 33.05 32.98 33.11 32.97 32.94 33.11 33.08 33.06 33.12 33.12 33.11 33.14 33.01
33.05 32.99 33.05 33.01 33.1 33.03 33.05 33.05 33 33.06 33.25 33.15 33.2 33.05 33.07 33.21 32.99 33 33.21 33.03
33.06 33.07 33.11 33.1 33.07 33.03 32.98 33.05 33.15 33 32.93 33.09 32.89 33.11 33.16 32.99 33.18 33.1 33.01 33.17
33.01 33.04 33.08 33.01 32.94 33.15 32.99 33.15 33.07 33.03 33.19 33.2 33.11 33.16 33.13 33.16 33.02 33.12 33.02 33.14
33.02 33.04 33.03 33.08 33.11 32.94 33.02 33.02 33.15 33.01 32.98 33.11 33.12 33.05 33.29 33.2 33.09 33.17 33.13 33.01
33.13 33.06 32.96 33.04 33.05 33.1 33.06 33.07 33.08 32.94 33.07 33.03 33.16 33.05 33.12 33.02 33.16 33.01 32.95 33.12
33.02 33.09 33.08 33.11 32.97 33.09 33.21 33.05 33.09 33.02 33.17 32.94 33.2 33.07 33.06 33.09 33.33 33.16 33.22 33.05
33.07 33.03 33.11 33.07 33.2 33.1 33.13 33.04 33.19 32.99 33.24 33.02 33.11 33.15 33.14 33.19 33.03 33.11 33.06 33.07
33.02 33 33.03 33.18 32.98 33.08 33.06 33.07 33.01 33.11 33.05 32.93 33.22 33.23 33.18 32.92 33.15 33.06 33.29 33.14
32.95 33.1 33.07 33.09 33.04 33.09 33.05 33.08 33 32.95 33.15 33.27 33.08 33.31 33.14 33.19 33.27 32.83 33.3 33.07
33.08 33.18 33.13 33.12 33.08 33.02 33.04 33.01 33.07 33.06 33.16 33.02 33.07 32.96 33.01 33.07 33.12 33.14 33.07 33.06
33.06 33.04 33.07 33.04 32.96 32.96 33.08 32.98 33.08 33.01 33.22 33.02 33.13 33.2 33.19 33.08 33.18 33.05 33.21 33
33.05 33.01 33.04 33.05 32.95 33.07 32.98 33.16 33.04 33.05 32.98 33.06 33.2 33.15 33.03 33.08 33.04 33.16 33.11 32.99
33.14 33.14 33.08 32.98 33.04 33.01 33.03 33.11 33.03 33 33.15 33.25 33.05 33.19 33.27 33.07 33.05 33.12 33.19 33.11

4. En esta tabla el mayor valore es de: 33.36 y el mínimo es de: 32.7, como es evidente
se observa que en estos dos valores están por fuera de la tolerancia pedida por el
cliente.
A continuación se realizara la tabla de frecuencias agrupadas en 10 intervalos:
"Reales" 'Frecuencias'' Medidas de Tendencia Central
Limite Inferior Limite Superior Marca de Clase Fi Fa Fr Fra fi * xi xl-x*fi xl-x´2*fi
32.695 32.765 32.730 3 3 0.20% 0.20% 98.19 1.0599 0.374490935
32.765 32.835 32.8 14 17 0.93% 1.13% 459.2 3.9664 1.123730228
32.835 32.905 32.870 49 66 3.27% 4.40% 1610.63 10.4524 2.229626331
32.905 32.975 32.94 125 191 8.33% 12.73% 4117.5 17.9142 2.567338939
32.975 33.045 33.01 285 476 19.00% 31.73% 9407.85 20.8943 1.531830781
33.045 33.115 33.08 465 941 31.00% 62.73% 15382.2 1.5407 0.005104853
33.115 33.185 33.15 355 1296 23.67% 86.40% 11768.25 23.6738 1.578724586
33.185 33.255 33.220 155 1451 10.33% 96.73% 5149.1 21.1864 2.895902951
33.255 33.325 33.29 37 1488 2.47% 99.20% 1231.73 7.6474 1.580616993
33.325 33.395 33.36 12 1500 0.80% 100.00% 400.32 3.3202 0.918666138
49624.97 111.6557 14.80603273
media= 33.08331333
des. Media= 0.074437129
Varianza= 0.009870688
des. Estándar= 0.099351339
Observaciones:
1.-Los valores más grandes se encuentran un poco cargados hacia el centro
mientras que los mínimos en los extremos, esto nos indica que la distribución no es
normal.
2.- La media aritmética del proceso varia un poco a el valor deseado esperado por el
cliente, esto nos indica que posiblemente las cosas no anden tan bien.
3.- como indicábamos al inicio los valores máximos y mínimos están por fuera de los
valores deseados, (máximo: 33.6, USL: 33.19 y el valor mínimo: 32.7, LSL: 32.81)
Ahora revisaremos el histograma:

5. 200
180
LSL TV USL
media.
160
140
120
100
80
60
40
20
0
32.7 32.8 32.9 33 33.1 33.2 33.3 33.4
Observaciones:
1.- De forma más clara notamos que algunos valores están fuera del límite de
especificación.
2. Observamos que hay una pequeña variación entre el TV y la media.
3.- La desviación estándar no es tan grande, pero si hay valores que sobresalen los
limites.
4.- A pesar de que la mayoría de los valores se encuentran centrados hay varios que
sobresalen pero, solo el USL, esto indica que los valores sobrepasan la tolerancia
mayor.

6. Ahora hay que identificar en que maquina se presente el problema;
Maquina 1:
32.9 33 33 32.9 33 33.1 32.9 32.7 33 33.1
33.1 33.1 32.9 33.1 33 33 33.1 33.2 32.9 33.1
32.9 32.9 33 33.1 33 32.9 33.3 32.9 33.1 33
33 33.1 33.1 32.9 32.9 33.1 33.1 32.9 33.1 33
33 33.1 33.1 33.1 33 33.1 33 33.1 33.1 33
33.2 32.9 33.2 32.8 33 33.1 32.9 33.1 32.7 33
32.9 33 32.8 33.1 33.1 32.9 33 33.1 33.1 33.1
32.9 33.1 33 33 32.9 33.1 33 33.1 33.2 33
33 33.1 33.1 33 33.1 33.2 33 32.9 32.8 33.3
32.9 33 32.9 32.9 33 33 33.1 32.9 33 33.1
32.8 33 32.9 32.9 33 33 33 33.1 33.2 33.1
32.8 33 32.8 33.1 33.1 33 33 33 33.2 32.9
33.1 33.1 32.8 33.1 33.2 33.2 33.1 32.9 33 33.1
33 33.2 32.9 33.1 32.9 32.7 33 33 33 33.2
33 33 32.9 33.1 32.9 33 32.8 33 33.2 32.9
32.8 32.9 32.8 33.2 33.1 33 33 33.1 33 33
33.1 32.9 32.9 33 33.1 33.1 33.1 33 32.9 32.9
32.8 33.1 33 33.1 32.9 33.1 33 32.9 33.1 33.2
32.9 33.2 33 33.2 32.8 33.2 32.8 33.2 33 33.1
32.9 32.9 33 33 32.9 32.8 32.8 33.1 33.1 33
33 32.9 33 33.1 33 33.1 32.9 33 33.1 32.8
33 33.1 33 33 33.1 33 33 33.1 33.2 33.1
33.2 33 33.2 33.1 32.9 33.1 32.9 33.1 33 33.1
33 33.1 33 33 32.9 33.1 33.2 32.9 32.9 33.1
32.9 33.3 33 32.9 33.1 33 33 32.9 32.8 32.9
32.9 32.9 32.9 33.1 33 33.2 33.1 33.1 32.7 33.1
33.1 32.9 33.2 32.9 32.9 33.1 33.1 32.9 33 32.9
33.1 33.1 33.1 33.1 33 33 33 32.8 33 32.9
32.8 33 33.3 33.1 32.9 33 32.8 33.1 33 32.8
33.1 32.9 32.9 33 32.9 33 32.9 33.1 33 33.2
33.2 33.1 33.2 33 33.1 32.8 33 33.2 33 32.9
32.9 32.9 32.8 32.9 33.2 33.1 33 33.2 33.2 33.2
33 33 33.1 33.2 32.9 33 33.1 33.1 33 32.9
32.9 32.8 33.1 32.9 33 33.1 32.8 33 32.9 33
33.1 32.9 33.2 33 33 33 32.9 33 32.9 32.9
33 33.1 33 33 33.2 33.1 33 33.1 33 33.2
33 33 32.9 33.3 33.1 33.1 33 33 33.1 33
32.9 32.8 33.1 33.1 33.2 33.2 33 32.9 32.8 33
33 33.2 33 33.1 33 32.9 33 32.8 32.9 33.1
33.1 33 33.1 33.1 33.2 33.1 33.2 33.1 33.1 33.2
33 33.2 32.8 33 32.9 33 33.1 33.1 33 33
33.1 33 32.9 33.1 32.9 32.9 32.9 33.1 33 32.9
32.9 33 32.8 33 33.1 33.1 33 33 33 32.9
33.1 33 33 32.9 33.1 33.3 33 32.9 33 33.1
33.1 32.8 33 33.1 32.8 33.2 32.9 33.1 32.9 33.1
32.9 32.9 33 33 33.1 33 33 33 33.1 33.1
33 33.2 33 32.9 33 33 33 32.9 33 32.8
32.8 33 33.2 33.1 32.9 33 32.9 32.9 33.1 32.9
33 33 32.8 33.1 32.8 32.9 33.1 33 33 33
33.1 32.9 32.9 33.1 32.9 33 33 33 33 32.9

7. Los datos se agruparan en 10 intervalos.
"Reales" 'Frecuencias'' Medidas de Tendencia Central
Limite Inferior Limite Superior Marca de Clase Fi Fa Fr Fra fi * xi xl-x*fi xl-x´2*fi
32.695 32.760 32.728 3 3 0.006 0.0060 98.1825 0.9313 0.289118981
32.760 32.825 32.7925 9 12 0.018 0.0240 295.1325 2.2090 0.542167142
32.825 32.890 32.858 34 46 0.068 0.0920 1117.155 6.1350 1.106992182
32.890 32.955 32.9225 74 120 0.148 0.2400 2436.265 8.5426 0.986153126
32.955 33.020 32.9875 92 212 0.184 0.4240 3034.85 4.6405 0.234065811
33.020 33.085 33.0525 116 328 0.232 0.6560 3834.09 1.6890 0.024591258
33.085 33.150 33.1175 92 420 0.184 0.8400 3046.81 7.3195 0.582341011
33.150 33.215 33.183 57 477 0.114 0.9540 1891.4025 8.2399 1.191162835
33.215 33.280 33.2475 17 494 0.034 0.9880 565.2075 3.5625 0.746561691
33.280 33.345 33.3125 6 500 0.012 1.0000 199.875 1.6474 0.452299162
33.345 33.41 33.3775 0 500 1.0000 Totales.
Totales: 16518.97 44.9166 6.1554532
media= 33.03794
des. Media= 0.08983312
Varianza= 0.012310906
des. Estándar= 0.110954524
Observaciones:
1.- Los valores mayores están centrados mientras que los menores se encuantrane n
los extremos.
2.- La desviación estándar es de S=0.110954524 por el momento no la podemos
comparar pero cuando obtengamos el valor de las otras dos maquinas lo
compararemos.
3.- El LSL=32.81 mientras que el mínimo de estos datos es de 33.31 y el USL= 33.19
mientras el máximo es de 33.31 los cual indica que estos valores no son los que
esperábamos y refirma la hipótesis de que algo anda mal.

8. 200
LSL TV USL
180
media.
160
140
120
100
80
60
40
20
0
32.60 32.70 32.80 32.90 33.00 33.10 33.20 33.30 33.40 33.50
Observaciones:
1.- Entre la media y el TV hay una pequeña variación, la mayoría de los valores están
un poco alejados del TV.
2.-Nuevamente se nota que un gran número de valores sobrepasa más el USL,
también el LSL pero hay más valores fuera del USL. Lo que nos indica que el
concentrado del perfume sale con más fragancia que la que se pide.

9. Maquina 2:
33.05 33.08 33.16 33.15 33 33.03 33.02 32.89 33.02 33.08
33.05 33.01 33.02 33.05 33 33.06 32.98 33.07 33.15 33.08
32.99 33.06 32.99 33.13 33.02 33.06 33.09 33.02 32.99 33.11
33.13 33.09 33.13 33.02 33.02 33.13 33.04 33.11 33.06 33.1
33.05 33.08 32.94 32.96 32.97 33.1 33.06 33.11 33.06 33.1
33.11 32.99 33.08 32.92 32.97 32.99 33.02 33.08 33.07 33.18
33.02 33.12 33.04 33.06 33.02 33.02 33.04 33.06 33.04 33.06
33.11 33.07 33.01 33.08 33.05 33 33.08 33.08 33.15 33.09
33.14 33.04 32.98 33.04 33.09 33.05 33.03 33.05 33.05 33.04
33.16 33.15 33.02 33 33.11 33.1 32.96 32.99 33.13 33.09
33.02 33.07 33.07 33.01 33.14 33.11 33.08 33.07 33.14 33.01
33.1 33.18 33.07 33.01 33.02 32.94 33.01 33.05 32.98 33.11
33.05 32.99 33.05 33.01 33.1 33.03 33.05 33.05 33 33.06
33.06 33.07 33.11 33.1 33.07 33.03 32.98 33.05 33.15 33
33.01 33.04 33.08 33.01 32.94 33.15 32.99 33.15 33.07 33.03
33.02 33.04 33.03 33.08 33.11 32.94 33.02 33.02 33.15 33.01
33.13 33.06 32.96 33.04 33.05 33.1 33.06 33.07 33.08 32.94
33.02 33.09 33.08 33.11 32.97 33.09 33.21 33.05 33.09 33.02
33.07 33.03 33.11 33.07 33.2 33.1 33.13 33.04 33.19 32.99
33.02 33 33.03 33.18 32.98 33.08 33.06 33.07 33.01 33.11
32.95 33.1 33.07 33.09 33.04 33.09 33.05 33.08 33 32.95
33.08 33.18 33.13 33.12 33.08 33.02 33.04 33.01 33.07 33.06
33.06 33.04 33.07 33.04 32.96 32.96 33.08 32.98 33.08 33.01
33.05 33.01 33.04 33.05 32.95 33.07 32.98 33.16 33.04 33.05
33.14 33.14 33.08 32.98 33.04 33.01 33.03 33.11 33.03 33
33.18 33.01 32.95 32.98 33.08 33.02 33.09 33.11 33.11 33.11
33.07 33.08 33.09 33.07 33.06 32.93 33.02 33 33.08 33.02
32.97 33.04 33.13 33.04 33.01 33.05 33.08 33 32.99 33
33.11 32.97 32.95 32.97 33.06 32.98 33.03 33.01 33.05 33.02
33.1 33.04 33.09 33.05 33.11 33.03 33.15 33.01 33.12 33.03
33.07 33.1 33.1 33.04 33.06 32.96 33.04 32.96 33.06 33.01
33.05 33.13 32.99 33.07 33.04 33.07 33.11 33.02 33.08 33.07
32.94 33.1 33.11 33.05 33.08 33.04 33.13 33.07 33.05 33.05
32.97 32.98 33.13 33.14 33.19 33.08 33.12 33.12 33.07 33.02
33.09 33.1 33.21 33.06 33.09 33 33.04 32.96 33.04 33.07
33.03 33.24 33.13 33.1 33.02 33.08 32.96 33.18 33.03 33.09
32.97 33.1 33.06 33.01 33.06 33.03 33.05 33.04 33.09 33.12
33.02 33.05 33.02 33.04 33.03 32.99 33.06 32.99 32.99 33.11
33.05 32.95 33.09 33.01 33.13 33 33.04 33 33 33.01
33.11 32.96 33.15 33.06 32.99 33.02 33.06 33.01 33.11 33.12
33.14 33.03 33.14 32.99 32.94 33.03 33.06 33.07 33.04 33.05
33.06 33.1 33.02 33.03 33.16 33.01 33.12 33.06 33.03 33.12
32.99 33.04 33 32.98 32.93 33.02 33.06 33.04 33.07 32.92
33.11 33.05 33.05 33.13 32.84 32.96 33.12 33.03 33.02 33.07
33.06 33.17 33.02 32.98 33.13 33.06 33.22 33.05 33.09 33.03
32.99 32.93 32.98 33.1 33.05 33.07 33 33.1 33.01 33
33.04 33.08 33.03 33.06 33.05 33.07 33.11 33.09 33.16 32.98
33.16 33.11 33.15 33.12 32.96 33.03 33.1 32.97 33.17 33.03
33.06 32.98 33.09 33.01 32.97 33.05 33 33.03 32.98 33.14
33.01 33.07 33.13 33 33.02 33.1 33.1 33 33.06 33.02

10. Agruparemos los datos en 10 intervalos:
"Reales" 'Frecuencias'' Medidas de Tendencia Central
Limite Inferior Limite Superior Marca de Clase Fi Fa Fr Fra fi * xi xl-x*fi xl-x´2*fi
32.835 32.885 32.860 1 1 0.002 0.0020 32.86 0.2237 0.05004169
32.885 32.935 32.91 1 2 0.002 0.0040 32.91 0.1737 0.03017169
32.935 32.985 32.960 18 20 0.036 0.0400 593.28 2.2266 0.27543042
32.985 33.035 33.01 79 99 0.158 0.1980 2607.79 5.8223 0.42910351
33.035 33.085 33.06 163 262 0.326 0.5240 5388.78 3.8631 0.09155547
33.085 33.135 33.11 142 404 0.284 0.8080 4701.62 3.7346 0.09821998
33.135 33.185 33.16 75 479 0.15 0.9580 2487 5.7225 0.43662675
33.185 33.235 33.210 17 496 0.034 0.9920 564.57 2.1471 0.27117873
33.235 33.285 33.26 4 500 0.008 1.0000 133.04 0.7052 0.12432676
33.285 33.335 33.31 0 500 0 1.0000 0 0.0000 0
16541.85 24.6188 1.806655
media= 33.0837
des. Media= 0.0492376
Varianza= 0.00361331
des. Estándar= 0.060110814
Observaciones.
1.-La mayoría de los valores están centrados y solo los valores mas mínimos están a
los extremos esto indica que la distribución no es normal.
2.- La desviación estándar de esta máquina es de S=0.060110814 mientras que la
de la otra máquina fue de S= 0.110954524 hasta el momento la maquina 2 tiene más
controlado su proceso.
3.- El valor mínimo es de 32.84 y el LSL= 33.19; mientras el máximo es de 33.24 y el
USL = 33.19 lo cual es evidente concluir que los valores siguen estando fuera de las
tolerancias pedidas por el cliente.

11. 200
LSL TV USL
180
media .
160
140
120
100
80
60
40
20
0
32.7 32.8 32.9 33 33.1 33.2 33.3 33.4
Observaciones.
1.- Es muy claro que la desviación estándar en más pequeña, y si la media hubiera
estado centrad, hubieran quedado las tres desviaciones estándar dentro de los
límites.
2.- Nuevamente se nota que los valores del USL sobrepasan a esta misma, mientras
en el LSL los datos no lo sobrepasan. Se siguen observando muchos errores.

12. Maquina 3.
33.03 32.95 33.04 32.92 33.15 33.12 33.34 32.92 33.1 33.14
32.82 33.09 33.07 32.94 33.05 33.14 33.32 33.05 33.16 33.06
33.18 33.09 33.15 33.23 33.09 33.21 33.11 33.05 33.09 33.1
33.06 33.17 33.06 33.08 33.11 33.14 33.02 33.09 33.11 33.16
33.09 33.11 33.13 32.99 33.05 33.08 32.99 33.1 33.17 33.12
33.13 33.14 32.91 33.04 33.09 32.94 32.99 33.12 33.11 32.92
33.15 33.06 33.16 33.11 33.22 33.03 33.06 33.03 32.94 33.36
33.1 32.99 33.15 33.13 33.02 32.97 33.21 33.07 33.13 33.11
33.02 33.09 33.06 33.16 33.03 33.32 33.16 33.23 32.99 33.15
33.06 33.05 33.08 33.03 33.21 32.97 33.13 33.26 33.11 33.09
33 33.05 33.12 33.14 33.03 33.21 33.18 33.13 33.26 32.94
33.14 33.01 33.23 32.97 33.08 33.02 33.18 32.94 33.11 33.16
33.18 33.13 33.01 33.13 32.87 33.18 33.09 33 33.17 33.22
32.96 33.25 33.1 33.16 33.22 33.1 33.2 33.21 33.03 33.16
33.15 33.15 33.15 33.17 33.12 32.92 33.06 33.12 33.19 33.35
33.14 32.97 33.04 33.15 33.16 33.07 33.09 33.09 33.16 33.33
33.22 33.21 33.13 33.04 33.05 33.06 32.99 33.23 33.07 33.03
33.22 33.14 32.9 32.96 33.04 33.13 33.16 32.95 33.21 32.99
33.12 33.04 33.11 33.2 33.13 33.19 33 33.09 33.14 33.2
33.3 33.18 33.16 33.07 33.11 33.1 33.13 33.07 33.08 33.13
33.08 33.15 33.27 33 33.12 33.11 33.17 33.11 33.02 33.12
33.03 33.12 33.09 33.1 33.18 33.02 33.02 32.98 33.04 33.12
32.9 33.08 33.09 33.03 33.1 33.09 33.13 33.12 33.2 33
33.09 32.95 33.13 33 33.01 32.89 33.05 33.22 33.25 33.22
33.09 33.04 33.12 33.04 33.25 33.09 33.17 33.27 33.06 33.05
33.03 33.14 33 33.06 32.97 33.02 32.93 33.14 33.23 32.98
33.06 33.17 32.96 33.17 33.15 33.12 33.01 33 33.29 33.26
33.12 33.07 33.22 33.05 33.09 33.13 32.95 33.14 33.17 33.15
33.19 33.24 33.19 33.17 33.1 33.14 33.17 32.93 33 32.96
33.21 33.18 33.07 33.27 33.03 33.15 33 33.21 33.03 33.08
33.15 33.22 33.01 33.12 33.01 33.04 33.13 33.11 33.18 33.06
32.97 33.2 33.07 33.15 33.1 33.13 33.08 33.04 33.04 33.09
33 33.16 33.1 33.18 33.13 33.24 33.32 33.13 33.19 33.13
33.02 33.21 33.17 33.06 33.04 33.27 33.08 33.18 33.11 33.03
33.24 32.89 33.08 33.14 33.06 33.1 33.14 33.24 32.88 33.18
33.13 33 33.03 33.12 33.11 33.14 33.15 32.96 33.1 33.07
32.97 32.94 33.11 33.08 33.06 33.12 33.12 33.11 33.14 33.01
33.25 33.15 33.2 33.05 33.07 33.21 32.99 33 33.21 33.03
32.93 33.09 32.89 33.11 33.16 32.99 33.18 33.1 33.01 33.17
33.19 33.2 33.11 33.16 33.13 33.16 33.02 33.12 33.02 33.14
32.98 33.11 33.12 33.05 33.29 33.2 33.09 33.17 33.13 33.01
33.07 33.03 33.16 33.05 33.12 33.02 33.16 33.01 32.95 33.12
33.17 32.94 33.2 33.07 33.06 33.09 33.33 33.16 33.22 33.05
33.24 33.02 33.11 33.15 33.14 33.19 33.03 33.11 33.06 33.07
33.05 32.93 33.22 33.23 33.18 32.92 33.15 33.06 33.29 33.14
33.15 33.27 33.08 33.31 33.14 33.19 33.27 32.83 33.3 33.07
33.16 33.02 33.07 32.96 33.01 33.07 33.12 33.14 33.07 33.06
33.22 33.02 33.13 33.2 33.19 33.08 33.18 33.05 33.21 33
32.98 33.06 33.2 33.15 33.03 33.08 33.04 33.16 33.11 32.99
33.15 33.25 33.05 33.19 33.27 33.07 33.05 33.12 33.19 33.11

13. Datos agrupados en 10 intervalos.
"Reales" 'Frecuencias'' Medidas de Tendencia Central
Limite Inferior Limite Superior Marca de Clase Fi Fa Fr Fra fi * xi xl-x*fi xl-x´2*fi
32.815 32.873 32.844 2 2 0.004 0.0040 65.688 0.5733 0.164320393
32.873 32.931 32.902 7 9 0.014 0.0180 230.314 1.6005 0.365920943
32.931 32.989 32.960 28 37 0.056 0.0740 922.88 4.7778 0.815266046
32.989 33.047 33.018 46 83 0.092 0.1660 1518.828 5.1813 0.583595951
33.047 33.105 33.076 106 189 0.212 0.3780 3506.056 5.7914 0.316419805
33.105 33.163 33.134 129 318 0.258 0.6360 4274.286 0.4340 0.001459828
33.163 33.221 33.192 106 424 0.212 0.8480 3518.352 6.5046 0.399147293
33.221 33.279 33.250 52 476 0.104 0.9520 1729 6.2069 0.740883754
33.279 33.337 33.308 15 491 0.03 0.9820 499.62 2.6605 0.471869827
33.337 33.395 33.366 9 500 0.018 1.0000 300.294 2.1183 0.498565912
16565.318 35.8484 4.357449752
media= 33.130636
des. Media= 0.071696816
Varianza= 0.0087149
des. Estándar= 0.093353626
Observaciones:
1.- Recopilando las desviaciones estándar de las maquinas concluimos que; la
maquina uno tiene una S=0.110954524, maquina 2 S=0.060110814 mientras que la
maquina 3 la desviación estándar es igual a S=0.093353626 lo que nos indica que la
maquina 2 tiene más controlado su proceso.
2.-Nuevamente los valores sobrepasan los límites; máximo: 33.36, USL= 33,19
mientras el mínimo es de 32,82 y el LSL= 32.81. Como lo mencionamos
anteriormente laos valores no sobrepasan el LSL solo el USL, esto no dice que los
perfumes están siendo llenados con más concentrado que el que se pide.

14. 200
LSL TV USL
180
160 media.
140
120
100
80
60
40
20
0
32.7 32.8 32.9 33 33.1 33.2 33.3 33.4
Observaciones:
1.- La media y el TV están muy separados, la distribución no es normal.
2.- A pesar de que los errores en los extremos no son tantos siguen sobresaliendo de
los límites de ULS,
3.- Como mencionamos en el inicio los valores de USL fueron sobrepasados por los
datos, esto indica que las cosas no se están haciendo bien y que hay factores que
están afectando estos resultados recordemos que este evento es muy importante y
que se está dejando entre dicho la importancia de la empresa.
Con estos resultados la persona de calidad puede alentar a los de producción sobre
este problema y atacarlo a tiempo. Así que se procedió a buscar el problema:

15. 50 1.200
causas Fi Fa Fr Fr a 45
M.O 45 39 0.390 0.390 1.000
40
M. ambiente 20 62 0.230 0.620
35
Maquinaria 15 80 0.180 0.800 0.800
30
M. prima 10 90 0.100 0.900
25 0.600
Mètodo 6 98 0.080 0.980
20
Medidas 4 100 0.020 1.000 0.400
15
10
0.200
5
0 0.000
0 0 0 0 0 0
Según la investigación realizada los operadores debido a que se trabaja a marchas
forzadas no estaban dando el 100 en sus actividades, además de que no se les daba

16. descanso; esto lo resintieron los operadores y el efecto se vio en la calidad del
producto que elaboran.