Apéndice A: Factores para Construir Diagramas de Control para Variables

Código

```{r}
#| label: tbl-factores
#| tbl-cap:  "Factores para construir diagramas de control para variables"

factores <- data.table::data.table(
  check.names = FALSE,
  n = c(
    2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L,
    15L, 16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L
  ),
  A = 3 / sqrt(2:25),
  A2 = 3 / (sqrt(2:25) * sapply(2:25, SixSigma::ss.cc.getd2)),
  A3 = 3 / (sqrt(2:25) * sapply(2:25, SixSigma::ss.cc.getc4)),
  c4 = sapply(2:25, SixSigma::ss.cc.getc4),
  `1/c4` = 1 / sapply(2:25, SixSigma::ss.cc.getc4),
  B3 = ifelse(
    1 - 3 * (sqrt(1 - (sapply(2:25, SixSigma::ss.cc.getc4))^2) / sapply(2:25, SixSigma::ss.cc.getc4)) < 0, 0,
    1 - 3 * (sqrt(1 - (sapply(2:25, SixSigma::ss.cc.getc4))^2) / sapply(2:25, SixSigma::ss.cc.getc4))
  ),
  B4 = 1 + 3 * (sqrt(1 - (sapply(2:25, SixSigma::ss.cc.getc4))^2) / sapply(2:25, SixSigma::ss.cc.getc4)),
  B5 = ifelse(
    sapply(2:25, SixSigma::ss.cc.getc4) - 3 * sqrt(1 - (sapply(2:25, SixSigma::ss.cc.getc4))^2) < 0, 0,
    sapply(2:25, SixSigma::ss.cc.getc4) - 3 * sqrt(1 - (sapply(2:25, SixSigma::ss.cc.getc4))^2)
  ),
  B6 = sapply(2:25, SixSigma::ss.cc.getc4) + 3 * sqrt(1 - (sapply(2:25, SixSigma::ss.cc.getc4))^2),
  d2 = sapply(2:25, SixSigma::ss.cc.getd2),
  `1/d2` = 1 / sapply(2:25, SixSigma::ss.cc.getd2),
  d3 = sapply(2:25, SixSigma::ss.cc.getd3),
  D1 = ifelse(
    sapply(2:25, SixSigma::ss.cc.getd2) - 3 * sapply(2:25, SixSigma::ss.cc.getd3) < 0, 0, sapply(2:25, SixSigma::ss.cc.getd2) - 3 * sapply(2:25, SixSigma::ss.cc.getd3)
  ),
  D2 = sapply(2:25, SixSigma::ss.cc.getd2) + 3 * sapply(2:25, SixSigma::ss.cc.getd3),
  D3 = ifelse(
    1 - 3 * sapply(2:25, SixSigma::ss.cc.getd3) / sapply(2:25, SixSigma::ss.cc.getd2) < 0, 0, 1 - 3 * sapply(2:25, SixSigma::ss.cc.getd3) / sapply(2:25, SixSigma::ss.cc.getd2)
  ),
  D4 = 1 + 3 * sapply(2:25, SixSigma::ss.cc.getd3) / sapply(2:25, SixSigma::ss.cc.getd2)
)

# knitr::kable(
#   factores,
#   # booktabs = TRUE,
#   format = "markdown",
#   # table.attr = 'data-quarto-disable-processing="true"',
#   col.names = c(
#     "$n$", "$A$", "$A_2$", "$A_3$", "$c_4$", "$1/c_4$",
#     "$B_3$", "$B_4$", "$B_5$", "$B_6$", "$d_2$", "$1/d_2$",
#     "$d_3$", "$D_1$", "$D_2$", "$D_3$", "$D_4$"
#   ),
#   digits = 4,
#   format.args = list(decimal.mark = ",", big.mark = "."),
#   escape = FALSE
# )

factores |> 
  gt::gt() |> 
  gt::cols_label(
    n = gt::html(
      "<span style='font-size: 14px;'>\\( n \\)</span>"
    ),
    A = gt::html("<span style='font-size: 14px;'>\\( A \\)</span>"),
    A2 = gt::html(
      "<span style='font-size: 14px;'>\\( A_{2} \\)</span>"
    ),
    A3 = gt::html(
      "<span style='font-size: 14px;'>\\( A_{3} \\)</span>"
    ),
    c4 = gt::html(
      "<span style='font-size: 14px;'>\\( c_{4} \\)</span>"
    ),
    `1/c4` = gt::html(
      "<span style='font-size: 14px;'>\\( 1/c_{4} \\)</span>"
    ),
    B3 = gt::html(
      "<span style='font-size: 14px;'>\\( B_{3} \\)</span>"
    ),
    B4 = gt::html(
      "<span style='font-size: 14px;'>\\( B_{4} \\)</span>"
    ),
    B5 = gt::html(
      "<span style='font-size: 14px;'>\\( B_{5} \\)</span>"
    ),
    B6 = gt::html(
      "<span style='font-size: 14px;'>\\( B_{6} \\)</span>"
    ),
    d2 = gt::html(
      "<span style='font-size: 14px;'>\\( d_{2} \\)</span>"
    ),
    `1/d2` = gt::html(
      "<span style='font-size: 14px;'>\\( 1/d_{2} \\)</span>"
    ),
    d3 = gt::html(
      "<span style='font-size: 14px;'>\\( d_{3} \\)</span>"
    ),
    D1 = gt::html(
      "<span style='font-size: 14px;'>\\( D_{1} \\)</span>"
    ),
    D2 = gt::html(
      "<span style='font-size: 14px;'>\\( D_{2} \\)</span>"
    ),
    D3 = gt::html(
      "<span style='font-size: 14px;'>\\( D_{3} \\)</span>"
    ),
    D4 = gt::html(
      "<span style='font-size: 14px;'>\\( D_{4} \\)</span>"
    ),
  ) |> 
  gt::tab_spanner(
    label = gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Observaciones} \\)</span>"
    ),
    id = "observaciones", # Identificador único
    columns = c(n)
  ) |> 
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Factores  para los límites de control} \\)</span>"
    ),
    id = "factores_limites-control_1", # Identificador único
    columns = c(A, A2, A3)
  ) |> 
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Factores  para la línea central} \\)</span>"
    ),
    id = "factores_linea_centasl_1", # Identificador
    columns = c(c4, `1/c4`)
  ) |> 
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Factores  para los límites de control} \\)</span>"
    ),
    id = "factores_limites-control_2", # Identificador único
    columns = c(B3, B4, B5, B6)
  ) |> 
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Factores  para la línea central} \\)</span>"
    ),
    id = "factores_linea_centasl_2", # Identificador
    columns = c(d2, `1/d2`)
  ) |>
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Factores  para los límites de control} \\)</span>"
    ),
    id = "factores_limites-control_3", # Identificador único
    columns = c(d3, D1, D2, D3, D4)
  ) |> 
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Diagrama de control } \\bar{x} \\)</span>"
    ),
    id = "factores_limites_control_xbarra", # Identificador único
    columns = c(A, A2, A3, c4, `1/c4`)
  ) |> 
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Diagrama de control } S \\)</span>"
    ),
    id = "factores_limites_control_S", # Identificador único
    columns = c(B3, B4, B5, B6)
  ) |>
  gt::tab_spanner(
    label =  gt::html(
      "<span style='font-size: 14px;'>\\( \\text{Diagrama de control } R \\)</span>"
    ),
    id = "factores_limites_control_R", # Identificador único
    columns = c(d2, `1/d2`, d3, D1, D2, D3, D4)
  ) |>
  gt::fmt_number(
    decimals = 4, sep_mark = ".", dec_mark = ",",
    drop_trailing_zeros = TRUE
  ) |> 
  gt::opt_align_table_header(align = "left") |>
  gt::cols_align(
    align = "center",
    columns = everything()
  ) |> 
  gt::tab_style(
    style = list(
      gt::cell_text(weight = "bold")
    ),
    locations = gt::cells_body(
      columns = n
    )
  ) # |>
  # gt::opt_interactive(
  #   use_filters = TRUE,
  #   use_compact_mode = TRUE,
  #   use_text_wrapping = FALSE
  # ) |> 
  # gt::opt_row_striping(row_striping = TRUE)
```

Tabla A.1: Factores para construir diagramas de control para variables

	\( \text{Diagrama de control } \bar{x} \)					\( \text{Diagrama de control } S \)				\( \text{Diagrama de control } R \)
\( \text{Observaciones} \)	\( \text{Factores para los límites de control} \)			\( \text{Factores para la línea central} \)		\( \text{Factores para los límites de control} \)				\( \text{Factores para la línea central} \)		\( \text{Factores para los límites de control} \)
\( n \)	\( A \)	\( A_{2} \)	\( A_{3} \)	\( c_{4} \)	\( 1/c_{4} \)	\( B_{3} \)	\( B_{4} \)	\( B_{5} \)	\( B_{6} \)	\( d_{2} \)	\( 1/d_{2} \)	\( d_{3} \)	\( D_{1} \)	\( D_{2} \)	\( D_{3} \)	\( D_{4} \)
2	2,1213	1,88	2,6587	0,7979	1,2533	0	3,2665	0	2,6063	1,1284	0,8862	0,8525	0	3,6859	0	3,2665
3	1,7321	1,0233	1,9544	0,8862	1,1284	0	2,5682	0	2,276	1,6926	0,5908	0,8884	0	4,3577	0	2,5746
4	1,5	0,7286	1,6281	0,9213	1,0854	0	2,266	0	2,0877	2,0588	0,4857	0,8798	0	4,6982	0	2,2821
5	1,3416	0,5768	1,4273	0,94	1,0638	0	2,089	0	1,9636	2,3259	0,4299	0,8641	0	4,9182	0	2,1145
6	1,2247	0,4832	1,2871	0,9515	1,0509	0,0304	1,9696	0,0289	1,8742	2,5344	0,3946	0,848	0	5,0785	0	2,0038
7	1,1339	0,4193	1,1819	0,9594	1,0424	0,1177	1,8823	0,1129	1,8058	2,7044	0,3698	0,8332	0,2047	5,204	0,0757	1,9243
8	1,0607	0,3725	1,0991	0,965	1,0362	0,1851	1,8149	0,1786	1,7514	2,8472	0,3512	0,8198	0,3877	5,3067	0,1362	1,8638
9	1	0,3367	1,0317	0,9693	1,0317	0,2391	1,7609	0,2318	1,7068	2,97	0,3367	0,8078	0,5465	5,3935	0,184	1,816
10	0,9487	0,3083	0,9754	0,9727	1,0281	0,2837	1,7163	0,2759	1,6694	3,0775	0,3249	0,7971	0,6864	5,4687	0,223	1,777
11	0,9045	0,2851	0,9274	0,9754	1,0253	0,3213	1,6787	0,3134	1,6373	3,1729	0,3152	0,7873	0,8109	5,5348	0,2556	1,7444
12	0,866	0,2658	0,8859	0,9776	1,023	0,3535	1,6465	0,3456	1,6095	3,2585	0,3069	0,7785	0,923	5,5939	0,2833	1,7167
13	0,8321	0,2494	0,8495	0,9794	1,021	0,3816	1,6184	0,3737	1,5851	3,336	0,2998	0,7704	1,0247	5,6472	0,3072	1,6928
14	0,8018	0,2354	0,8173	0,981	1,0194	0,4062	1,5938	0,3985	1,5634	3,4068	0,2935	0,763	1,1177	5,6958	0,3281	1,6719
15	0,7746	0,2231	0,7885	0,9823	1,018	0,4282	1,5718	0,4206	1,544	3,4718	0,288	0,7562	1,2032	5,7405	0,3466	1,6534
16	0,75	0,2123	0,7626	0,9835	1,0168	0,4479	1,5521	0,4405	1,5265	3,532	0,2831	0,7499	1,2823	5,7817	0,363	1,637
17	0,7276	0,2028	0,7391	0,9845	1,0157	0,4657	1,5343	0,4585	1,5106	3,5879	0,2787	0,7441	1,3557	5,82	0,3779	1,6221
18	0,7071	0,1943	0,7176	0,9854	1,0148	0,4818	1,5182	0,4748	1,496	3,6401	0,2747	0,7386	1,4243	5,8558	0,3913	1,6087
19	0,6882	0,1866	0,6979	0,9862	1,014	0,4966	1,5034	0,4898	1,4826	3,689	0,2711	0,7335	1,4885	5,8894	0,4035	1,5965
20	0,6708	0,1796	0,6797	0,9869	1,0132	0,5102	1,4898	0,5036	1,4703	3,7349	0,2677	0,7287	1,5489	5,921	0,4147	1,5853
21	0,6547	0,1733	0,6629	0,9876	1,0126	0,5228	1,4772	0,5163	1,4589	3,7783	0,2647	0,7242	1,6058	5,9509	0,425	1,575
22	0,6396	0,1675	0,6473	0,9882	1,012	0,5344	1,4656	0,5281	1,4483	3,8194	0,2618	0,7199	1,6596	5,9791	0,4345	1,5655
23	0,6255	0,1621	0,6327	0,9887	1,0114	0,5452	1,4548	0,5391	1,4383	3,8583	0,2592	0,7159	1,7107	6,006	0,4434	1,5566
24	0,6124	0,1572	0,6191	0,9892	1,0109	0,5553	1,4447	0,5493	1,4291	3,8953	0,2567	0,7121	1,7591	6,0316	0,4516	1,5484
25	0,6	0,1526	0,6063	0,9896	1,0105	0,5648	1,4352	0,5589	1,4203	3,9306	0,2544	0,7084	1,8053	6,056	0,4593	1,5407

Donde:

\[ A = \frac{3}{\sqrt{n}} \]
\[ A_2 = \frac{3}{\sqrt{n}\:d_2} \]
\[ A_3 = \frac{3}{\sqrt{n}\:c_4} \]
\[ c_4 = \sqrt{\frac{2}{n-1}}\:\frac{\Gamma \left( \frac{n}{2} \right)}{\Gamma \left( \frac{n -1}{2} \right)} \]
\[ \frac{1}{c_4} = \sqrt{\frac{n -1}{2}}\:\frac{\Gamma \left( \frac{n -1}{2} \right)}{\Gamma \left( \frac{n }{2} \right)} \]
\[ B_3 = 1 - 3\:\frac{\sqrt{1 - c_{4}^{2}}}{c_{4}} \]
\[ B_4 = 1 + 3\:\frac{\sqrt{1 - c_{4}^{2}}}{c_{4}} \]
\[ B_{5} = c_{4} - 3\:\sqrt{1 - c_{4}^{2}} \]
\[ B_{6} = c_{4} + 3\:\sqrt{1 - c_{4}^{2}} \]
\[ D_{1} = d_{2} - 3 \, d_{3} \]
\[ D_{2} = d_{2} + 3 \, d_{3} \]
\[ D_{3} = 1 - 3\,\frac{d_{3}}{d_{2}} \]
\[ D_{4} = 1 + 3\,\frac{d_{3}}{d_{2}} \]
Si \(n > 25\) \[ c_{4} \cong \frac{4\left ( n -1 \right )}{4\,n - 3} \] \[ B_{3} \cong 1 - \frac{3}{c_{4} \sqrt{2\left ( n -1 \right )}} \] \[ B_{4} \cong 1 + \frac{3}{c_{4} \sqrt{2\left ( n -1 \right )}} \] \[ B_{5} \cong c_{4} - \frac{3}{\sqrt{2\left ( n -1 \right )}} \] \[ B_{6} \cong c_{4} + \frac{3}{\sqrt{2\left ( n -1 \right )}} \]