# 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