Apr-Jun.12 - the Nitie

5.0 COMPUTATION OF ARLs
The ARLs of the traditional (Shewhart) are
calculated at various levels of correlation i.e. at lowest
limits (L). Following tables gives the ARLs at various
and L.
Behavior of the traditional (Shewhart) control chart
when the data is i.i.d. from the normal distribution of
mean of 0 and standard deviation of 1. Table 1 shows
the ARLs of traditional chart at various width of
control limits (L).
Table 1 ARLs of the
chart for random i.i.d. data
Shift
(in L = 2.9 L= 2.95 L=3.0 L=3.05 L=3.1
mean)
0.0 285.7 312.5 371 400.0 555.6
0.2 250.0 263.2 294.1 303.0 312.5
0.4 208.3 181.8 270.3 227.3 285.7
0.5 144.9 142.9 155.2 178.6 169.5
0.6 99.0 114.9 120.5 133.3 111.1
0.8 66.2 68.0 76.3 99.0 74.1
1.0 38.3 38.2 44.1 46.5 51.0
1.2 26.4 27.7 30.2 29.5 31.4
1.4 16.1 17.5 19.2 20.0 20.3
1.5 13.1 13.7 14.9 17.9 15.7
1.6 11.0 12.0 12.3 13.3 13.4
1.8 7.7 7.9 9.0 9.4 9.5
2.0 5.7 5.9 6.3 6.7 7.1
2.5 3.1 3.0 3.2 3.4 3.5
3.0 1.9 1.9 2.0 2.1 2.1
3.5 1.4 1.4 1.5 1.5 1.5
4.0 1.2 1.2 1.2 1.2 1.2
Since the in-control ARL of traditional chart is 370.
So, ARLs at different levels of correlations are calculated
by keeping the in-control ARL of approximately 370.
Control limits are adjusted in such a way that the incontrol
ARL of approximately 370 may be maintained.
Table 2 shows the ARLs of the traditional chart at
Shift
(in
mean)
Table 2 ARLs of the
correlation(r) of 0.1, 0.2 and 0.3
L=
2.95
r= 0.1 r= 0.2 r= 0.3
L=3.0 L = 2.9 L=
2.95
L = 2.9 L=
2.95
L=3.0
L=3.1
0.0 368.3 384.6 322.5 370.37 312.5 322.6 368.5 381.4
0.2 283.0 263.7 222.2 290.6 188.7 243.2 294.1 284.2
0.4 168.6 217.4 172.4 175.4 166.7 129.9 200.0 232.1
0.5 119.3 153.9 117.5 126.6 113.6 117.7 161.3 179.4
0.6 92.6 123.5 102.0 105.3 105.3 102.0 119.1 131.1
0.8 69.9 72.5 57.5 69.9 69.4 67.1 73.5 84.2
1.0 34.5 40.5 36.0 39.5 37.6 44.8 44.8 51.3
1.2 26.3 30.2 21.2 30.2 24.1 26.3 30.8 38.0
1.4 16.6 18.1 15.0 17.4 16.3 20.3 21.2 27.7
1.5 13.6 15.1 12.5 14.8 15.2 15.3 18.3 21.1
1.6 10.8 12.9 10.6 12.2 12.3 12.8 13.9 17.3
1.8 8.0 8.8 8.0 8.5 8.4 9.3 10.3 12.6
2.0 5.7 6.3 5.8 6.4 6.3 6.8 7.0 9.2
2.5 3.1 3.3 3.0 3.3 3.3 3.4 3.7 3.9
3.0 1.9 2.0 1.9 2.0 2.0 2.1 2.2 2.5
3.5 1.4 1.5 1.4 1.5 1.5 1.5 1.6 1.6
4.0 1.2 1.2 1.2 1.2 1.2 1.2 1.3 1.4
Table 3 shows the ARLs of the traditional
chart at
Table 3 ARLs of the
correlation(r) of 0.5 and 0.7
r= 0.5 r= 0.7
Shift
(in L = 2.9 L= 2.95 L=3.0 L = 2.9 L= 2.95 L=3.0 L=3.1
mean)
0.0 303.0 369.6 400 303.0 303.0 370.1 416.7
0.2 238.1 313.3 344.8 243.9 243.9 322.5 384.6
0.4 175.4 227.7 256.4 178.6 212.8 243.9 285.7
0.5 113.6 163.9 163.9 144.9 126.6 191.7 222.2
0.6 92.6 126.7 108.7 128.2 120.5 141.5 196.1
0.8 63.7 104.2 76.9 95.2 102.0 118.9 156.3
1.0 42.6 51.3 46.1 47.0 60.6 68.1 82.6
1.2 28.7 37.5 35.5 33.9 41.8 43.9 55.6
1.4 19.8 23.3 21.3 26.5 26.4 28.6 37.0
1.5 15.7 21.5 19.5 20.5 20.9 24.6 29.9
1.6 13.4 17.1 16.7 16.3 18.0 22.1 27.5
1.8 9.3 12.2 11.7 13.3 13.6 12.4 19.9
2.0 7.3 8.6 8.4 9.3 9.5 11.2 13.7
2.5 3.4 4.4 4.3 4.9 5.6 5.1 7.0
3.0 2.4 2.4 2.7 2.9 3.2 3.0 3.8
3.5 1.7 1.8 1.8 2.0 2.1 2.3 2.5
4.0 1.3 1.4 1.4 1.5 1.6 1.6 1.7
Vol. 36, No. 2, April-June, 2012
39