Statistics for Environmental Engineers

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Cumulative deviations from T, the mean or target level of the process, are plotted on the chart. The target T is usually the average level of the process determined during some period when the process was in a stable operating condition. The deviation at time t is yt — T. At time t — 1, the deviation is yt-1 — T, and so on. These are summed from time t — 1 to the current time t, giving the cumulative sum, or Cusum:


Vt = Iyt — T)


If the process performance is stable, the deviations will vary randomly about zero. The sum of the deviations from the target level will average zero, and the cumulative sum of the deviations will drift around zero. There is no general trend either up or down.

If the mean process performance shifts upward, the deviations will include more positive values than before and the Cusum will increase. The values plotted on the chart will show an upward trend. Likewise, if the mean process performance shifts downward, the Cusum will trend downward.

The Cusum chart gives a lot of useful information even without control limits. The time when the change occurred is obvious. The amount by which the mean has shifted is the slope of the line after the change has occurred.

The control limits for a Cusum chart are not parallel lines as in the Shewhart chart. An unusual amount of change is judged using a V-Mask (Page, 1961). The V-Mask is placed on the control chart horizontally such that the apex is located a distance d from the current observation. If all previous points fall within the arms of the V-Mask, the process is in a state of statistical control.

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