Soo King Lim - 1 - 1.0 Cumulative Sum Control Chart .................................................. 3 1.1 Tabular or Algorithmic CUSUM for Monitoring the Mean of the Process ...................................................................................... 4 1.2 V-Mask for Monitoring the Mean of the Process ...................... 6 1.3 Standardized Cumulative Sum Control Chart ........................ 10 1.4 ARL of Cumulative Sum Control Chart .................................. 11
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Soo King Lim
- 1 -
1.0 Cumulative Sum Control Chart .................................................. 3
1.1 Tabular or Algorithmic CUSUM for Monitoring the Mean of
the Process ...................................................................................... 4
1.2 V-Mask for Monitoring the Mean of the Process ...................... 6
1.3 Standardized Cumulative Sum Control Chart ........................ 10
1.4 ARL of Cumulative Sum Control Chart .................................. 11
Soo King Lim
- 2 -
Figure 1: Showing the deviation of xi value from targeted
iC value,
iC value,
decision interval, and counter values ..................................................... 5
Figure 2: CUSUM control chart plotted with data shown in Fig. 1 ...................... 6
Figure 3: Illustration of V-mask parameters .......................................................... 7
Figure 4: An illustration of V-mask showing out of control point ........................ 8
Figure 5: Calculated CUSM values for the sample batches .................................. 9
Figure 6: The V-mask results placed (a) at batch number 4, (b) at batch number
8, (c) at batch number 10, and (d) at batch number 12 ........................ 10
Figure 7: ARL for CUSUM with k = 0.5 and h = 4 and h = 5 ............................. 12
Figure 8: ARL of CUSUM chart for a given standardized k = 0.5 and h = 4 and
5, and ARL of Shewart X control chart .............................................. 13
Soo King Lim
- 3 -
1.0 Cumulative Sum Control Chart
A cumulative sum CUSUM control chart is a time-weighted control chart that
displays the cumulative sum of the deviation of each sample value from the target
value. Owing to the factor that it is cumulative, a minor drifting in the process mean
will lead to steadily increasing or decreasing cumulative deviation value. It was
developed by E. S. Page of the University of Cambridge. Cumulative sum CUSUM
control chart has been shown to be more efficient in detecting small shift in the mean
of a process. In analyzing the average run length ARL, CUSUM control chart shows
a better result than Shewhart control chart when it is desired to detect shift in the
mean by less than two standard deviations. However, CUSUM control chart is
relatively slow to respond to large shift and hard to detect and analyze special trend
patterns.
Let’s collect n sample batches, each of sample size m, and calculate the mean
of each sample batch
m
j
iji
1
xx . The cumulative sum CUSUM control chart is
formed by plotting one of the following quantities.
n
i
in
1
0ˆxC (1)
where 10 CˆxC nin .
or standardized cumulative sum
n
i
in
i1
0
x
' ˆx1
C (2)
where '
1
x
0' Cˆx
C -ni
n
i
. Cn is called cumulative sum up to and including the ith
sample. 0̂ is the estimate of the in-control mean and ix
is the known or estimated
of standard deviation of the sample mean ix . The choice of which of these two
quantities is plotted is usually determined by the statistical software package. In
either case, as long as the process remains in control centered at 0̂ , the CUSUM
plot will show variation in a random pattern centered about zero. If the mean of the
process shifts upward, the plotted CUSUM points will eventually drift upward and