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Cumulative Sum Chart (CUSUM)

CUSUM charts are constructed by calculating and plotting a cumulative sum based on the data.  Let X1, X2, …, X24 represent 24 data points.  From this, the cumulative sums S0, S1, …, S24 are calculated.  Notice that 24 data points leads to 25 (0 through 24) sums.  The cumulative sums are calculated as follows:

1. First calculate the average:

   

2. Start the cumulative sum at zero by setting S0 = 0.

3. Calculate the other cumulative sums by adding the difference between current value and the average to the previous sum, i.e.:

   

 for i=1,2,…,24.

The cumulative sum is not the cumulative sum of the values.  Instead it is the cumulative sum of differences between the values and the average.  Because the average is subtracted from each value, the cumulative sum also ends at zero (S24=0).

How does one interpret a CUSUM chart?  Suppose that during a period of time the values are all above average.  The amounts added to the cumulative sum will be positive and the sum will steadily increase.  A segment of the CUSUM chart with an upward slope indicates a period where the values tend to be above average.  Likewise a segment with a downward slope indicates a period of time where the values tend to be below the average.