CUSUM Chart

Cumulative Sum Chart for detecting small persistent shifts in process mean.

Chart Overview

What is a CUSUM Chart?

The Cumulative Sum (CUSUM) chart is a sequential analysis technique that accumulates deviations of sample values from a target value. Unlike traditional control charts that plot individual observations, CUSUM charts plot cumulative sums, making them extremely sensitive to small but persistent shifts in the process mean. They excel at detecting gradual process shifts that might go unnoticed in other control charts.

Key Characteristics:

• • Purpose: Detect small, persistent shifts in process mean (0.5σ to 2σ)
• • Sensitivity: Excellent for gradual drift and systematic changes
• • Data Type: Continuous numerical data or subgroup averages
• • Best For: Processes where small shifts are critical to detect quickly
• • Memory Effect: Accumulates information from all previous observations

Typical Applications:

• • Chemical processes requiring tight control of concentration
• • Financial monitoring for fraud detection
• • Clinical trials and medical monitoring
• • Precision manufacturing with strict tolerances
• • Pharmaceutical production quality control
• • Decision making in sequential analysis

Statistical Foundation

Tabular CUSUM:

The tabular CUSUM uses two one-sided cumulative sums to detect upward and downward shifts separately.

Upper CUSUM (detects upward shifts)

= observation at time i

= target (in-control) mean

= reference value (allowable slack)

Lower CUSUM (detects downward shifts)

Signal when C⁺ᵢ > H or C⁻ᵢ > H, where K = allowable slack and H = decision interval

Parameter Guidelines:

• K = 0.5σ: Good balance for 1σ shifts

• H = 4σ: ARL₀ ≈ 168 (in-control)

• H = 5σ: ARL₀ ≈ 465 (conservative)

• Adjust based on desired false alarm rate

V-Mask Alternative:

• Sᵢ: Cumulative sum of deviations

• Lead distance: d = 2/k

• Half-angle: tan(θ) = k/2

• Signal when Sᵢ touches V-mask arms

Average Run Length Properties:

In-Control Performance

ARL₀ represents average runs before false alarm under stable conditions

Out-of-Control Detection

ARL₁ represents average runs to detect various shift magnitudes

Reading the CUSUM Chart

Interpretation Guidelines

Process Control Status:

In Statistical Control:

• • Both C⁺ and C⁻ remain below decision interval H
• • CUSUM values fluctuate near zero
• • No sustained upward or downward trends
• • Process mean remains close to target μ₀
• • Random variation within expected limits

Out-of-Control Signals:

• • C⁺ > H indicates upward shift in process mean
• • C⁻ > H indicates downward shift in process mean
• • Persistent trends in cumulative sum values
• • Gradual drift away from zero line
• • Sustained accumulation in one direction

Parameter Tuning Guidelines:

Reference Value (K):

• • Smaller K: More sensitive, faster detection
• • Larger K: Less sensitive, fewer false alarms
• • K = 0.5σ: Good compromise for most applications
• • Choose based on shift size of greatest concern

Decision Interval (H):

• • Smaller H: Faster detection, more false alarms
• • Larger H: Slower detection, fewer false alarms
• • H = 4σ: Typical choice (ARL₀ ≈ 168)
• • H = 5σ: Conservative choice (ARL₀ ≈ 465)

Practical Applications:

• Quality Monitoring: Detect gradual quality deterioration before customer impact

• Process Control: Monitor chemical processes for concentration drift

• Financial Analysis: Detect systematic changes in financial metrics

• Healthcare: Monitor patient vital signs for gradual changes

• Manufacturing: Control precision processes with tight specifications

CUSUM Reset and Maintenance:

When to Reset:

• • After investigating and correcting special causes
• • When process changes are implemented
• • After confirmed out-of-control signals
• • During process capability studies

Ongoing Management:

• • Regular parameter review and optimization
• • Documentation of all signals and actions
• • Performance monitoring through ARL analysis
• • Integration with corrective action systems

Implementation Guide

Example Data Layout for CUSUM Chart

Individual measurements with target value. C+ and C- columns show cumulative sums for upper and lower CUSUM calculations with K=0.5. Sample Size Recommendation: Minimum 20-25 baseline samples to establish target and reference values, then continuous monitoring for detecting small persistent shifts.

This is how your spreadsheet or CSV file should be structured before you upload it. Match the column names, keep one row per observation, and leave the analysis to the platform.