# Statistics for Environmental Engineers

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i

Hk) = k X y i = k, k + 1,…,n

j=i-k+1

Thus, a seven-day moving average (MA7) uses the latest seven daily values, a ten-day average (MA10) uses 10 points, and so on. Each data point is given equal weight in computing the average.

As each new observation is made, the summation will drop one term and add another term, giving the simple updating formula:

yi(k) = yii (k) + kyi — kyi-k = yii (k) + k (yi — yi-k)

By smoothing random fluctuations, the moving average sharpens the focus on recent performance levels. Figure 4.2 shows the MA7 and MA30 moving averages for some PCB data. Both moving averages help general trends in performance show up more clearly because random variations are averaged and smoothed.

Observation

FIGURE 4.2 Seven-day and thirty-day moving averages of PCB data.

The MA7, which is more reflective of short-term variations, has special appeal in being a weekly average. Notice how the moving average lags behind the daily variation. The peak day is at 260, but the MA7 peaks three to four days later (about k/2 days later). This does not diminish its value as a smoother, but it does limit its value as a predictor. The longer the smoothing period (the larger k), the more the average will lag behind the daily values.

The MA30 highlights long-term changes in performance. Notice the lack of response in the MA30 at day 255 when several high PCB concentrations occurred. The MA30 did not increase by very much — only from 25 ^g/L to about 40 ^g/L—but it stayed at the 40 ^g/L level for almost 30 days after the elevated levels had disappeared. High concentrations of PCBs are not immediately harmful, but the chemical does bioaccumulate in fish and other organisms and the long-term average is probably more reflective of the environmental danger than the more responsive MA7.

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