Statistics (M4)

The Statistics module performs distribution analysis on the radial deviations collected across sections. While M1 summarizes each section with a single RMS value, M4 examines the full population of per-point deviations to reveal distributional characteristics, outliers, and directional biases.

Distribution analysis

The module takes the per-point deviation data from TTunnelDeviationEngine (converted to millimeters) and computes:

Central tendency and spread

Statistic

Description

Mean

Average deviation across all points (mm). Positive = net overbreak.

Standard Deviation

Spread of deviations around the mean

Skewness

Asymmetry of the distribution. Positive skew = long tail toward overbreak.

Excess Kurtosis

Tail heaviness relative to normal distribution. Positive = more outliers than expected.

Percentiles

The full percentile table (P0 through P100) is computed from the sorted deviation values. Key percentiles for engineering interpretation:

Percentile

Typical use

P5

Lower bound of the practical range – deviations below this are outliers

P25 (Q1)

First quartile – 25% of points are below this deviation

P50 (Median)

Central value – more robust than mean when outliers are present

P75 (Q3)

Third quartile – 75% of points are below this deviation

P95

Upper bound of the practical range – deviations above this are outliers

Tip

If the mean and median differ significantly, the distribution is skewed. A mean much larger than the median indicates a few large overbreak values pulling the average up. In this case, the median is a better indicator of typical excavation quality.

Histogram

The deviation values are binned into a frequency histogram (default 20 bins). Each bin records:

  • Low and high boundary values (mm)

  • Count of points in the bin

  • Relative frequency (0.0–1.0)

The histogram reveals the shape of the deviation distribution:

  • Symmetric, narrow peak: Well-controlled excavation with uniform overbreak

  • Right-skewed: Most sections are close to the template, with occasional large overbreak

  • Bimodal: Two distinct excavation regimes (possibly different ground conditions or equipment)

  • Heavy left tail: Significant underbreak population requiring attention

Normal distribution fit

If deviations follow a Gaussian (normal) distribution, the mean and standard deviation fully characterize the population. The skewness and kurtosis values indicate departures from normality:

  • Skewness near 0 and kurtosis near 0: approximately normal

  • Significant positive skewness: log-normal or Weibull distribution may be more appropriate

  • High positive kurtosis: heavy tails with more extreme values than a normal distribution predicts

Box plot by zone

The module computes box plot statistics (minimum, Q1, median, Q3, maximum, mean, count) separately for six angular zones around the tunnel perimeter:

Zone

Angle range

Location

Invert

330–30 degrees

Bottom

Left Wall

30–90 degrees

Lower left

Left Shoulder

90–150 degrees

Upper left

Crown

150–210 degrees

Top

Right Shoulder

210–270 degrees

Upper right

Right Wall

270–330 degrees

Lower right

Note

Angles are measured clockwise from the invert (bottom = 0 degrees), following the internal convention of the deviation engine. The crown is at 180 degrees.

Per-zone statistics enable directional bias detection:

  • Crown median significantly higher than walls: Normal in drill-and-blast – crown overbreak is expected due to gravity and blast gas effects

  • Asymmetric left/right: Systematic drilling offset or geological structure

  • High wall variance: Inconsistent excavation at the walls, possibly due to support installation interference

  • Invert outliers: Floor irregularities (if clip level is not enabled)

Tip

Compare zone medians across multiple sections to determine whether a directional bias is systematic (present in all sections) or localized (only at specific chainages). Combine with the M3 longitudinal heatmap for spatial context.

Practical workflow

  1. Run the full analysis from the Analytics dashboard

  2. Export to XLSX for the tabular per-section metrics

  3. Review the histogram shape in the statistics output:

    • If approximately normal, the mean and standard deviation provide a complete summary

    • If skewed, focus on percentiles and median instead

  4. Check per-zone box plots for directional biases

  5. Cross-reference zones with high variance against the M3 heatmap to locate problem chainages