THE CUSUM METHOD
Cumulative Sum (CUSUM) charts have a range of applications, but they are most useful for identifying when an event—typically
a change in average—took place. The principle behind a CUSUM chart is that it displays a running total of the differences
from an average or target value. One common use of the CUSUM approach is in golf where a scorecard records if a player is
below par or ahead of par for each hole, and the cumulative difference during the course of a game.
Interpreting the chart depends on an examination of the slopes of the lines. The vertical position and the vertical scale
are irrelevant here. If a process is running on target, the CUSUM plot is a horizontal line; if the process average is high,
then the line is rising; if the process average is low, then the line has a downward slope. It is easy for us to see a change
in slope of a line and thus recognize when an event occurred. The other advantage of a CUSUM chart is that it smoothes a response
so that trends are easier to see.
Software programs are available for CUSUM plotting but it is not difficult to perform the calculations in Excel. Table 1 shows
an extract from an Excel spreadsheet that was set up to calculate the CUSUM values for these data.
Table 1. Extract from an Excel spreadsheet that was set up to calculate the CUSUM values.
- Column A shows the date for each batch
- Column B contains the raw data for the contaminant concentration values. Cell B2 is the arithmetic average of all the data,
not just those in the extract shown here.
- Column C shows the CUSUM values. The first cell in the CUSUM sequence here is equal to the first actual raw data value.
- The second point for the CUSUM takes the previous CUSUM point (cell C3), adds the new data value (cell B4), and subtracts
the fixed average (cell B2).
- The third and subsequent CUSUM points are similarly based on the previous CUSUM point, the new data value, and the fixed average.
- Column D shows the formulae that are used in column C.
Table 1 shows that the CUSUM score is decreasing steadily. This happens because all the data values are below the average.
Similarly, a golfer's cumulative par score would get lower and lower if he were playing below par at each hole. Figure 5 shows
the raw data and the CUSUM scores for concentration, in which the downward trend in the CUSUM line at the start of the year
is quite obvious. Starting in August, the raw data become higher and are mostly above the average; on the CUSUM plot this
is shown as a sharp change in the slope of the line, which is now rising.
Figure 5. Combined raw data and CUSUM for batch leakage