Page 111 - Statistics for Environmental Engineers

L1592_frame_C12.fm Page 107 Tuesday, December 18, 2001 1:48 PM

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 y t − T. At time t − 1, the deviation is y t−1 − T,
and so on. These are summed from time t = 1 to the current time t, giving the cumulative sum, or Cusum:

t
V t ∑ ( y t – )
=
T
t =1
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.

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 infor-
mation as the average of several replicate observations at time t.
The moving average chart is based on the average of the k most recent observations. The quantity to
be plotted is:

t
V t = 1 ∑ y t
---
k
t−(k−1)
The central control line is the average for a period when the process performance is in stable control.
s
The control limits are at distances ±3 , assuming single observations at each interval.
------
k
Exponentially Weighted Moving Average Chart
The exponentially weighted moving average (EWMA) chart is a plot of the weighted sum of all previous
observations:

V t = ( 1 λ) ∑ λ y t−i
i
–
i=0
The EWMA control chart is started with V 0 = T, where T is the target or long-term average. A convenient
updating equation is:

V t = ( 1 λ)y t + λV t−1
–
 λ  λ
The control limits are ±3s ------------ .
 2 λ
–
© 2002 By CRC Press LLC