How this calculation works
A small persistent process shift can be hard to see in isolated observations. A one-sided cumulative-sum chart accumulates evidence in one chosen direction after subtracting a reference allowance. This page follows a standardized tabular CUSUM through the entered observation sequence. It shows the first decision-limit crossing and continues the calculation without automatically resetting after a signal.
C0=0; Cup=max(0,Cprevious+z−k); Cdown=max(0,Cprevious−z−k); signal if C>hA worked example
Four observations of one, with target zero, sigma one and k equal to 0.5, add 0.5 at each step. The upward accumulator is 0.5, one, 1.5 and two. With h equal to one, the first strict crossing is the third point, not the second. A downward chart on the same positive sequence stays at zero.
Good to know
Independent constant-variance observations and a valid baseline are assumed. Subgroup means require their own standard-error scaling and cannot automatically use the individual sigma field unchanged. No baseline fitting, reset policy, multiple-chart false-alarm adjustment or production-control approval is included. Investigate signals using the actual process and measurement context.
Reference: NIST — tabular CUSUM charts (opens in a new tab) ↗
What this calculation answers
A small persistent process shift can be hard to see in isolated observations. A one-sided cumulative-sum chart accumulates evidence in one chosen direction after subtracting a reference allowance.
This page follows a standardized tabular CUSUM through the entered observation sequence. It shows the first decision-limit crossing and continues the calculation without automatically resetting after a signal.
The EWMA control chart with finite startup limits calculator also remembers earlier observations but uses exponential weighting rather than a one-sided accumulated allowance.
Use the right input conventions
Observation order represents time and must be preserved. The target mean and individual-observation sigma come from a separately established baseline, not from the same monitored sequence by default.
Reference k and decision h are in standardized sigma units. Upward and downward charts use different signed increments; selecting one direction does not simultaneously evaluate a two-sided alarm rule.
Interpret the result
The accumulator never falls below zero because evidence opposite to the chosen direction erases the current run of accumulated evidence. A point signals only when C is strictly greater than the entered h.
The maximum statistic and first crossing describe the entered chart. They do not supply an average run length, false-alarm calibration or proof that a process has become unstable for a known reason.
Work through the example
Four observations of one, with target zero, sigma one and k equal to 0.5, add 0.5 at each step. The upward accumulator is 0.5, one, 1.5 and two.
With h equal to one, the first strict crossing is the third point, not the second. A downward chart on the same positive sequence stays at zero.
Where the model stops
Independent constant-variance observations and a valid baseline are assumed. Subgroup means require their own standard-error scaling and cannot automatically use the individual sigma field unchanged.
No baseline fitting, reset policy, multiple-chart false-alarm adjustment or production-control approval is included. Investigate signals using the actual process and measurement context.
A few questions, answered
Does equality with h signal in this implementation?
No. The displayed rule is a strict crossing C greater than h.
What changes in the comparison?
Reverse the monitored shift direction on the same positive observations.