Probability/Quantile Charts
Distribution-free control charts using probability plots and quantiles for non-normal data.
Chart Overview & Distribution-Free Monitoring
What they monitor:
Distribution-free control charts that use empirical quantiles without assuming specific distributions. Ideal for non-normal data or unknown process distributions. Rather than centering limits on a mean and standard deviation, these charts track order statistics (e.g., median, quartiles) directly estimated from the data.
Applications:
- • Non-normal process monitoring
- • Financial risk assessment
- • Environmental data analysis
- • Reliability engineering
Key Features:
- • Distribution-free approach
- • Uses empirical quantiles
- • Robust to outliers
- • Flexible control limits
Empirical Quantiles & Control Limit Formulas
Empirical Quantile Estimation:
For a subgroup ordered as x(1)≤x(2)≤…≤x(n), the p-th quantile is estimated by linear interpolation between order statistics.
Rank position for the p-th empirical quantile
= subgroup sample size
= target quantile (e.g., 0.5 for the median)
Distribution-Free Control Limits:
Empirical control limits for the tracked quantile
= baseline p-th quantile from Phase I data
= desired false-alarm probability
UCL/LCL are set at the alpha/2 and 1-alpha/2 empirical percentiles of the baseline quantile statistic's sampling distribution, obtained by bootstrap or order-statistic theory rather than a Normal-theory formula.
Key Assumptions
- • No specific parametric distribution is assumed
- • Observations are independent and identically distributed within Phase I
- • Sufficient baseline data to estimate tail quantiles reliably
Reading the Probability Chart
Example Chart & Signal Interpretation
Worked Example:
A skewed cycle-time process is monitored using the subgroup median. Baseline Phase I data yields a median of 10.1 with empirical 2.5th/97.5th percentile limits of 9.6 and 10.8. A subgroup median of 9.4 falls below the LCL, signaling a shift.
In Control Signals
- • Subgroup quantile stays within empirical limits
- • Random fluctuation with no trend
- • Stable spread between tracked quantiles
Out of Control Signals
- • Quantile statistic beyond UCL/LCL
- • Sustained drift in tracked quantile
- • Widening or narrowing interquartile spread
sixsigmacenter.com Implementation Guide
Platform Features
- • Automatic Quantile Estimation: Computes order statistics and interpolated quantiles per subgroup
- • Bootstrap Limits: Derives distribution-free control limits without normality assumptions
- • Flexible Tracking: Supports median, quartile, or custom percentile monitoring
Preparing Your Data for Upload
- • One column holding the measured characteristic, one row per measured unit — do not pre-summarize into medians or quartiles.
- • A subgroup identifier column so the platform knows which readings belong to the same sample; keep the file in production time order.
- • Numeric values only in the measurement column: no units suffix, no "OK", no blank-as-zero. Blank cells are treated as missing, not as zero.
- • Record the measurement resolution you actually read to. Quantile charts inherit the resolution of the raw data.
Minimum 25 to 30 baseline subgroups for stable empirical quantile limits — more than a Shewhart chart needs, because no distributional shape is assumed.
Example Data Layout for Probability Chart
Quantile-based monitoring without distributional assumptions. Sample Size Recommendation: Minimum 25-30 subgroups for reliable empirical quantile estimation and control limit calculation in distribution-free monitoring.
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.