Quality & Six Sigma SPC
Tabular CUSUM Control Chart Formula: C+ / C- Cumulative Sum & h-Sigma Shift Alarm
Detect sustained 0.5-sigma to 1.5-sigma process drifts 3x–4x faster than Shewhart 3-sigma charts using Page’s Two-Sided Tabular CUSUM recursion C_i+ = MAX(0, x_i - (mu_0 + k*sigma) + C_{i-1}+).
Catch persistent 1-sigma mean shifts that hide inside +-3-sigma Shewhart limits!
A standard Shewhart Individuals chart only looks at one point at a time, requiring ~44 samples on average (ARL_1 = 43.9) to catch a sustained +1.0*sigma process drift! E. S. Page’s Tabular Cumulative Sum (CUSUM) accumulates deviations beyond reference slack K = k * sigma (typically k = 0.5 to target a 1-sigma shift): Upper CUSUM C_i^+ = MAX(0, x_i - (mu_0 + k*sigma) + C_{i-1}^+) and Lower CUSUM C_i^- = MAX(0, (mu_0 - k*sigma) - x_i + C_{i-1}^-)!
With decision interval H = h * sigma (typically h = 4 or 5, giving in-control ARL_0 ~ 168 to 465), a sustained +1-sigma shift triggers C_i^+ > H in ~8–10 samples!
| CUSUM Parameter (ISO 7870-4 / Montgomery) | Symbol & Value | Computed Step Result |
|---|---|---|
| Current Observation x_i & Target mu_0 | x_i = 101.8, mu_0 = 100.0 | Raw deviation = +1.80 (inside +-3.0 Shewhart) |
| Process Sigma & Reference Slack k | sigma = 1.0, k = 0.5 (K = 0.5) | Net excess above 100.5 = +1.30 |
| Prior C_{i-1}^+ & Decision Limit h | C_{i-1}^+ = 3.1, h = 4.0 (H = 4.0) | C_i^+ = 4.40 > 4.00 -> Out-of-Control Upward Shift! |
Updated Upper Tabular CUSUM Statistic C_i^+
Updates C_i^+ = MAX(0, B2 - (B3 + B5*B4) + B6) with non-negative reset at zero.
=IF(OR(B4 <= 0, B5 <= 0, B6 < 0, B7 <= 0), "Invalid CUSUM Parameters", ROUND(MAX(0, B2 - (B3 + B5 * B4) + B6), 4))Decision Interval Alarm Status Against H = h * sigma
Compares updated C_i^+ against H = B7 * B4 to trigger shift investigation.
=IF(MAX(0, B2 - (B3 + B5 * B4) + B6) > B7 * B4, "Out-of-Control Upward Mean Shift", "In Control")Try the example
Use the original values or change them to test your understanding.