Quality and Reliability Methods - SAS

106 Cumulative Sum Control Charts Chapter 6
Formulas for CUSUM Charts
Two-Sided Cusum Schemes
If the cusum scheme is two-sided, the cumulative sum S t plotted for the t th subgroup is
S t = S t - 1 +z t
for t = 1, 2,..., n. Here S 0 =0, and the term z t is calculated as
z t
= ( X t – μ 0
) ⁄ ( σ ⁄ n t
)
where X is the t th subgroup average, and n t is the t th t
subgroup sample size. If the subgroup samples consist
of individual measurements x t , the term z t simplifies to
z t = (x t – μ 0 )/σ
Since the first equation can be rewritten as
S t
t t
= z i
= ( X i – μ 0
) ⁄ σ
Xi
i = 1 i = 1
the sequence S t cumulates standardized deviations of the subgroup averages from the target mean μ 0 .
In many applications, the subgroup sample sizes n i are constant (n i = n), and the equation for S t can be
simplified.
S t
t
= ( 1 ⁄ σ ) X
( X i – μ 0
) =
i = 1
( n ⁄ σ) ( X i – μ 0
)
i = 1
In some applications, it might be preferable to compute S t as
t
S t
= ( X i – μ 0
)
i = 1
t
which is scaled in the same units as the data. In this case, the procedure rescales the V-mask parameters h
and k to h' = hσ ⁄ n and k' = kσ ⁄ n , 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 μ 0 , the points will not exhibit a trend since
positive and negative displacements from μ 0 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.