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MACHART Statement |
The following notation is used in this section:
A_{i} | uniformly weighted moving average for the i^{ th} subgroup |
w | span parameter (number of terms in moving average) |
process mean (expected value of the population of measurements) | |
process standard deviation (standard deviation of the population of measurements) | |
x_{ij} | j^{ th} measurement in i^{ th} subgroup, with j =1, 2, 3, ..., n_{i} |
n_{i} | sample size of i^{ th} subgroup |
mean of measurements in i^{ th} subgroup. If n_{i}=1, then the subgroup mean reduces to the single observation in the subgroup. | |
weighted average of subgroup means | |
inverse standard normal function |
The following table presents the formulas for the limits:
Table 21.19: Limits for Moving Average ChartControl Limits |
LCL = |
UCL = |
Probability Limits |
LCL = |
UCL = |
These formulas assume that the data are normally distributed. If standard values and are available for and , respectively, replace with and replace with in Table 21.19. Note that the limits vary with both n_{i} and i.
If the subgroup sample sizes are constant (n_{i}=n),
the formulas
for the control limits
simplify to
If you specify the ASYMPTOTIC option, constant control limits
with the following values are displayed:
For asymptotic probability limits, replace k with in these equations. You can display asymptotic limits by specifying the ASYMPTOTIC option.
You can specify parameters for the moving average limits as follows:
You can use Table 21.20 and Table 21.21 to find a combination of k and w that yields a desired ARL for an in-control process () and for a specified shift of .
Table 21.20: Average Run Lengths for One-Sided Uniformly Weighted Moving Average Charts
w (span) | ||||||||
k | 2 | 3 | 4 | 5 | 6 | 8 | 10 | |
2.0 | 0.00 | 51.58 | 60.97 | 70.58 | 80.18 | 89.78 | 108.65 | 127.47 |
2.0 | 0.25 | 25.01 | 26.47 | 28.00 | 29.33 | 30.76 | 33.08 | 35.18 |
2.0 | 0.50 | 13.41 | 13.31 | 13.40 | 13.69 | 14.01 | 14.66 | 15.17 |
2.0 | 0.75 | 8.00 | 7.75 | 7.78 | 7.97 | 8.15 | 8.60 | 9.06 |
2.0 | 1.00 | 5.27 | 5.20 | 5.29 | 5.45 | 5.67 | 6.15 | 6.69 |
2.0 | 1.50 | 2.90 | 3.03 | 3.24 | 3.50 | 3.73 | 4.23 | 4.66 |
2.0 | 2.00 | 2.04 | 2.27 | 2.51 | 2.73 | 2.95 | 3.32 | 3.65 |
2.0 | 2.50 | 1.68 | 1.91 | 2.11 | 2.31 | 2.48 | 2.78 | 3.04 |
2.0 | 3.00 | 1.46 | 1.68 | 1.85 | 2.01 | 2.16 | 2.40 | 2.63 |
2.0 | 4.00 | 1.20 | 1.38 | 1.52 | 1.64 | 1.75 | 1.94 | 2.10 |
2.0 | 5.00 | 1.06 | 1.18 | 1.31 | 1.41 | 1.50 | 1.65 | 1.79 |
2.5 | 0.00 | 179.92 | 204.43 | 230.32 | 259.32 | 287.08 | 339.71 | 394.43 |
2.5 | 0.25 | 72.62 | 71.56 | 72.48 | 72.93 | 73.40 | 75.54 | 77.47 |
2.5 | 0.50 | 33.67 | 30.13 | 28.54 | 27.49 | 26.93 | 26.29 | 26.03 |
2.5 | 0.75 | 17.28 | 15.01 | 13.91 | 13.42 | 13.13 | 13.00 | 13.10 |
2.5 | 1.00 | 9.94 | 8.66 | 8.20 | 8.01 | 7.96 | 8.24 | 8.63 |
2.5 | 1.50 | 4.43 | 4.13 | 4.21 | 4.39 | 4.64 | 5.17 | 5.69 |
2.5 | 2.00 | 2.65 | 2.77 | 3.03 | 3.29 | 3.54 | 4.01 | 4.43 |
2.5 | 2.50 | 1.98 | 2.24 | 2.50 | 2.74 | 2.95 | 3.32 | 3.67 |
2.5 | 3.00 | 1.70 | 1.95 | 2.17 | 2.37 | 2.55 | 2.86 | 3.14 |
2.5 | 4.00 | 1.37 | 1.59 | 1.76 | 1.90 | 2.03 | 2.28 | 2.49 |
2.5 | 5.00 | 1.15 | 1.35 | 1.51 | 1.62 | 1.73 | 1.92 | 2.08 |
3.0 | 0.00 | 792.24 | 867.57 | 963.95 | 1051.77 | 1150.79 | 1345.96 | 1539.75 |
3.0 | 0.25 | 269.28 | 244.26 | 231.50 | 226.25 | 220.89 | 209.87 | 204.74 |
3.0 | 0.50 | 104.18 | 83.86 | 72.84 | 65.43 | 60.85 | 54.62 | 50.34 |
3.0 | 0.75 | 45.69 | 34.45 | 28.79 | 25.69 | 23.66 | 21.24 | 20.15 |
3.0 | 1.00 | 22.73 | 16.74 | 14.20 | 12.89 | 12.12 | 11.52 | 11.45 |
3.0 | 1.50 | 7.65 | 6.16 | 5.70 | 5.64 | 5.75 | 6.23 | 6.78 |
3.0 | 2.00 | 3.77 | 3.49 | 3.63 | 3.89 | 4.17 | 4.71 | 5.20 |
3.0 | 2.50 | 2.46 | 2.63 | 2.90 | 3.18 | 3.43 | 3.88 | 4.28 |
3.0 | 3.00 | 1.96 | 2.23 | 2.50 | 2.74 | 2.95 | 3.33 | 3.65 |
3.0 | 4.00 | 1.57 | 1.81 | 2.00 | 2.18 | 2.34 | 2.62 | 2.87 |
3.0 | 5.00 | 1.30 | 1.55 | 1.72 | 1.85 | 1.97 | 2.20 | 2.40 |
3.5 | 0.00 | 4275.15 | 4536.99 | 4853.63 | 5168.75 | 5485.97 | 6088.03 | 6613.01 |
3.5 | 0.25 | 1281.12 | 1078.59 | 964.86 | 886.26 | 830.03 | 751.66 | 684.98 |
3.5 | 0.50 | 413.30 | 294.47 | 235.00 | 197.27 | 169.50 | 136.01 | 115.48 |
3.5 | 0.75 | 153.50 | 98.31 | 73.49 | 59.29 | 50.49 | 40.45 | 34.53 |
3.5 | 1.00 | 63.68 | 39.34 | 29.37 | 24.06 | 20.88 | 17.70 | 16.12 |
3.5 | 1.50 | 15.84 | 10.44 | 8.50 | 7.78 | 7.47 | 7.51 | 7.97 |
3.5 | 2.00 | 6.06 | 4.73 | 4.49 | 4.61 | 4.86 | 5.43 | 6.01 |
3.5 | 2.50 | 3.27 | 3.13 | 3.34 | 3.63 | 3.92 | 4.45 | 4.91 |
3.5 | 3.00 | 2.31 | 2.54 | 2.83 | 3.11 | 3.36 | 3.80 | 4.19 |
3.5 | 4.00 | 1.77 | 2.02 | 2.25 | 2.45 | 2.64 | 2.97 | 3.27 |
3.5 | 5.00 | 1.48 | 1.74 | 1.91 | 2.06 | 2.21 | 2.48 | 2.71 |
Table 21.21: Average Run Lengths for Two-Sided Uniformly Weighted Moving Average Charts
w (span) | ||||||||
k | 2 | 3 | 4 | 5 | 6 | 8 | 10 | |
2.0 | 0.00 | 25.46 | 29.62 | 33.94 | 38.08 | 42.35 | 51.20 | 59.48 |
2.0 | 0.25 | 20.43 | 22.38 | 24.21 | 25.87 | 27.35 | 30.08 | 32.33 |
2.0 | 0.50 | 12.73 | 12.80 | 13.02 | 13.29 | 13.57 | 14.19 | 14.84 |
2.0 | 0.75 | 7.87 | 7.68 | 7.71 | 7.86 | 8.03 | 8.44 | 8.90 |
2.0 | 1.00 | 5.24 | 5.14 | 5.22 | 5.40 | 5.59 | 6.09 | 6.60 |
2.0 | 1.50 | 2.90 | 3.02 | 3.24 | 3.48 | 3.71 | 4.19 | 4.63 |
2.0 | 2.00 | 2.04 | 2.26 | 2.51 | 2.73 | 2.94 | 3.31 | 3.63 |
2.0 | 2.50 | 1.67 | 1.91 | 2.12 | 2.30 | 2.47 | 2.77 | 3.03 |
2.0 | 3.00 | 1.46 | 1.67 | 1.85 | 2.01 | 2.15 | 2.40 | 2.63 |
2.0 | 4.00 | 1.20 | 1.38 | 1.52 | 1.64 | 1.75 | 1.94 | 2.10 |
2.0 | 5.00 | 1.06 | 1.19 | 1.31 | 1.41 | 1.50 | 1.65 | 1.79 |
2.5 | 0.00 | 89.48 | 101.24 | 114.35 | 127.74 | 140.88 | 166.98 | 192.93 |
2.5 | 0.25 | 63.12 | 64.91 | 67.00 | 68.75 | 69.84 | 72.22 | 74.49 |
2.5 | 0.50 | 32.46 | 29.54 | 28.20 | 27.33 | 26.72 | 25.92 | 25.72 |
2.5 | 0.75 | 17.28 | 14.97 | 13.85 | 13.29 | 13.02 | 12.81 | 12.98 |
2.5 | 1.00 | 9.94 | 8.61 | 8.16 | 7.99 | 8.01 | 8.23 | 8.63 |
2.5 | 1.50 | 4.42 | 4.14 | 4.20 | 4.38 | 4.62 | 5.16 | 5.67 |
2.5 | 2.00 | 2.65 | 2.77 | 3.03 | 3.29 | 3.54 | 4.00 | 4.43 |
2.5 | 2.50 | 1.99 | 2.24 | 2.50 | 2.73 | 2.95 | 3.33 | 3.65 |
2.5 | 3.00 | 1.69 | 1.95 | 2.17 | 2.37 | 2.54 | 2.86 | 3.14 |
2.5 | 4.00 | 1.37 | 1.59 | 1.76 | 1.90 | 2.04 | 2.27 | 2.49 |
2.5 | 5.00 | 1.15 | 1.35 | 1.51 | 1.63 | 1.73 | 1.92 | 2.09 |
3.0 | 0.00 | 397.12 | 436.27 | 481.16 | 527.14 | 574.05 | 667.68 | 762.89 |
3.0 | 0.25 | 245.51 | 228.67 | 222.75 | 216.07 | 213.79 | 207.03 | 201.71 |
3.0 | 0.50 | 103.15 | 83.49 | 72.47 | 65.67 | 60.67 | 53.93 | 50.30 |
3.0 | 0.75 | 45.56 | 34.25 | 29.01 | 25.72 | 23.59 | 21.12 | 19.93 |
3.0 | 1.00 | 22.68 | 16.81 | 14.19 | 12.92 | 12.18 | 11.54 | 11.48 |
3.0 | 1.50 | 7.68 | 6.14 | 5.71 | 5.65 | 5.77 | 6.23 | 6.77 |
3.0 | 2.00 | 3.74 | 3.49 | 3.63 | 3.88 | 4.17 | 4.71 | 5.21 |
3.0 | 2.50 | 2.46 | 2.63 | 2.90 | 3.18 | 3.43 | 3.89 | 4.29 |
3.0 | 3.00 | 1.96 | 2.23 | 2.50 | 2.73 | 2.95 | 3.32 | 3.66 |
3.0 | 4.00 | 1.57 | 1.81 | 2.00 | 2.18 | 2.34 | 2.62 | 2.88 |
3.0 | 5.00 | 1.30 | 1.55 | 1.72 | 1.85 | 1.97 | 2.20 | 2.40 |
3.5 | 0.00 | 2217.61 | 2372.09 | 2567.27 | 2775.06 | 2983.70 | 3398.08 | 3810.50 |
3.5 | 0.25 | 1186.27 | 1027.67 | 940.30 | 875.91 | 826.53 | 744.59 | 676.61 |
3.5 | 0.50 | 411.69 | 295.62 | 232.68 | 195.65 | 169.21 | 135.73 | 116.06 |
3.5 | 0.75 | 152.52 | 97.33 | 72.30 | 58.98 | 50.59 | 40.22 | 34.71 |
3.5 | 1.00 | 64.03 | 39.46 | 29.18 | 24.08 | 20.80 | 17.54 | 16.16 |
3.5 | 1.50 | 15.83 | 10.36 | 8.47 | 7.73 | 7.46 | 7.56 | 8.00 |
3.5 | 2.00 | 6.05 | 4.71 | 4.49 | 4.61 | 4.85 | 5.44 | 6.00 |
3.5 | 2.50 | 3.27 | 3.12 | 3.34 | 3.64 | 3.92 | 4.44 | 4.91 |
3.5 | 3.00 | 2.32 | 2.54 | 2.83 | 3.11 | 3.36 | 3.80 | 4.19 |
3.5 | 4.00 | 1.77 | 2.02 | 2.25 | 2.46 | 2.65 | 2.97 | 3.26 |
3.5 | 5.00 | 1.49 | 1.74 | 1.91 | 2.06 | 2.21 | 2.48 | 2.71 |
For example, suppose you want to construct a two-sided moving average chart with an in-control ARL of 100 and an ARL of 9 for detecting a shift of . Table 21.21 shows that the combination w=3 and k=2.5 yields an in-control ARL of 101.24 and an ARL of 8.61 for .
Note that you can also use Table 21.20 and Table 21.21 to evaluate an existing moving average chart (see Example 21.2).
The following SAS program computes the average run length for a two-sided moving average chart for various shifts in the mean. This program can be adapted to compute averages run lengths for various combinations of k and w.
data sim; drop span delta time j y x; span=4; do shift=0,.25,.5,.75,1,1.5,2,2.5,3,4,5; do j=1 to 50000; do time=1 to 15000; if time<=100 then delta=0; else delta=shift; y=delta+rannor(234); if time<span then x=.; else x=(y+lag1(y)+lag2(y)+lag3(y))/span; if time>=101 and abs(x)>3/sqrt(span) then leave; end; arl=time-100; output; end; end; proc means; class shift; run;
In the preceding program, the size of the span w (SPAN) is 4 and the shifts in the mean are introduced to the variable (Y) after the first 100 observations. The first DO loop specifies shifts of various magnitude, the second DO loop performs 50000 simulations for each shift, and the third DO loop counts the run length (TIME), that is, the number of samples observed before the control chart signals. A large upper bound (15000) for TIME is specified so that the run length is uncensored.
The program can be generalized for various span sizes by assigning
a different value for the variable SPAN
and changing the expression for X appropriately.
Optionally, you can compute the ARL for a one-sided chart
by changing the
limits, that is, x>3/sqrt(span)
.
This was the technique used to construct
Table 21.20 and Table 21.21.
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