Eii Tii

where k is the number of segments with different process levels and Hi is the number of observations in each sample.

3. Estimation of S by moving-ranges of "a" successive data points. Use differences of successive observations as if they were ranges of n observations. A plot of S for group size a versus a will indicate if there is between-sample variation. If the plot is flat, the between-sample variation is insignificant. This approach should not be used if there is a trend in data. If there are missing observations, all groups containing them should be excluded from computations.

The procedure for estimating S by moving-ranges is:

1. Calculate moving-ranges of size a, a = 2,3, • • •, using 25 to 100 observations.

MRt =| max(xi) — min(xj) | , i = (t — a + 1), t. (6.29)

2. Calculate the mean of the ranges for each a

3. Divide the result of Step 2 by d2 (for each a).

4. Tabulate and plot results for all a.

Process Level Monitoring by Moving-Average Charts

In a moving-average (MA) chart, the averages of consecutive groups of size a are computed and plotted. The control limit computations are based on these averages. Several original data points at the start and end of the chart are excluded, since there is not enough data to compute the moving-average at these times. MA charts detect small drifts efficiently (better than x chart). However, they respond slowly to sudden shifts in level and the MA generates autocorrelation in computed values.

Procedure For Chart Development

The procedure is outlined for m samples of size n. For individual measurements, let n = 1.

1. Compute the sample averages xu i = l,m.

2. Compute the moving average Mt of span a at time t as

3. Compute the variance of Mt

Hence, cr = Sjcw/afn or a — MRjdzy/n, using MR for R. 4. Compute the control limits with the centerline at x:

In general, the span a and the magnitude of the shift to be detected are inversely related.

Spread Monitoring by Moving-Range Charts

In a moving-range chart, the range of two consecutive sample groups of size a are computed and plotted. For a > 2,

MRt =) max(xi) — min(xi) | , i — (t — a + l),t (6.33)

The computation procedure is:

2. Obtain estimates of MR and a = MR/d-t by using the moving-ranges MRt of length a. For a total of m samples:

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