Two-Sided CUSUM Charts

If the CUSUM chart is two-sided, the cumulative sum St plotted for the tth subgroup is as follows:

St = St - 1 +zt

t = 1, 2,..., n. Here S0=0, and the term zt is calculated as follows:

is the tth subgroup average, and nt is the tth subgroup sample size. If the subgroup samples consist of individual measurements xt, the term zt simplifies to the following computation:

zt = (xt – μ0)/σ

The first equation can be rewritten as follows:

where the sequence St cumulates standardized deviations of the subgroup averages from the target mean μ0.

In many applications, the subgroup sample sizes ni are constant (ni = n), and the equation for St can be simplified, as follows:

In some applications, it might be preferable to compute St as follows:

which is scaled in the same units as the data. In this case, the procedure rescales the V-mask parameters h and k to

and

, respectively. Some authors use the symbols F for k' and H for h'.

If the process is in control and the mean μ is at or near the target μ0, the random walk model applies. Therefore, the points might wander away from zero, but they will not exhibit a large trend since positive and negative displacements from μ0 tend to cancel each other. If μ shifts in the positive direction, the points exhibit an upward trend, and if μ shifts in the negative direction, the points exhibit a downward trend.