Statistics for Environmental Engineers

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Moving Average Chart

Moving average charts are useful when the single observations themselves are used. If the process has operated at a constant level with constant variance, the moving average gives essentially the same information as the average of several replicate observations at time t.

The moving average chart is based on the average of the к most recent observations. The quantity to be plotted is:


The central control line is the average for a period when the process performance is in stable control.

The control limits are at distances ±3-C assuming single observations at each interval.


Exponentially Weighted Moving Average Chart

The exponentially weighted moving average (EWMA) chart is a plot of the weighted sum of all previous observations:

Vt — (1 — A)lA!yt-t


The EWMA control chart is started with V0 — T, where T is the target or long-term average. A convenient updating equation is:

Vt — (1 — A) yt + AVt-1

The weight Я is a value less than 1.0, and often in the range 0.1 to 0.5. The weights decay exponentially from the current observation into the past. The current observation has weight 1 — Я, the previous has weight (1 — Я)Я, the observation before that (1 — Я)Я2, and so on. The value of Я determines the weight placed on the observations in the EWMA. A small value of Я gives a large weight to the current observation and the average does not remember very far into the past. A large value of Я gives a weighted average with a long memory. In practice, a weighted average with a long memory is dominated by the most recent four to six observations.

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