T² Chart (Multivariate Monitoring)

Monitor multiple correlated quality characteristics simultaneously using Hotelling's T² statistic.

Chart Overview & Multivariate Monitoring

What is a T² Chart?

The Hotelling T² chart monitors multiple correlated quality characteristics simultaneously. It combines information from all variables into a single statistic, providing an overall measure of process performance while accounting for correlations between variables. This approach is more effective than monitoring variables individually.

When to Use T² Charts

• • Multiple correlated quality characteristics
• • Want overall process assessment
• • Variables individually in control but jointly out of control
• • Complex manufacturing processes
• • Multivariate process monitoring

Key Advantages

• • Considers correlations between variables
• • Single chart for multiple characteristics
• • Detects subtle multivariate shifts
• • Reduces false alarm rate
• • More sensitive than individual charts

T² Statistic & Control Limits

Hotelling T² statistic

= subgroup size

= sample mean vector

= target (in-control) mean vector

= sample covariance matrix

Phase I upper control limit

= number of monitored variables

= number of baseline samples

= F-distribution critical value

Phase II replaces the leading multiplier with p(m + 1)(m - 1) / [m(m - p)].

Key Assumptions

• • Variables follow multivariate normal distribution
• • Process parameters are known or estimated from in-control data
• • Correlations between variables are stable
• • Sufficient historical data for covariance matrix estimation

Reading the T-squared Chart

Example Chart & Signal Interpretation

sixsigmacenter.com Implementation Guide

Platform Features for T² Charts

• • Automated Calculations: Handles complex matrix operations automatically
• • Covariance Matrix: Estimates and manages correlation structures
• • Decomposition Analysis: Identifies contributing variables to signals
• • Phase I/II Support: Both retrospective analysis and real-time monitoring

Preparing Your Data for Upload

• • One column per correlated characteristic and one row per sample, all measured on the same physical unit at the same time. Row alignment is what makes the covariance real.
• • Include every characteristic that moves together. Dropping one correlated variable is what a T-squared chart is meant to protect you from, and it cannot see what is not in the file.
• • No blanks in any variable column. A missing cell drops the whole row from the covariance estimate, not just that variable.
• • Keep a clean Phase I baseline block of in-control samples. The covariance matrix estimated from contaminated history hides the very shift you are watching for.

You need more rows than variables — a practical floor is 20 to 25 times the number of characteristics — or the covariance matrix is unstable and the limits are meaningless.

Example Data Layout for T² Chart

Multivariate monitoring with correlated variables combined into single T² statistic. Individual variables may appear normal while T² detects multivariate shifts. Sample Size Recommendation: Minimum 20-30 baseline subgroups with adequate observations per subgroup for stable covariance matrix estimation in multivariate control charting.

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.