R-Tool — Statistical Research Workbench

A browser-native inferential statistics environment with exact distributional computations, multivariate visualization, and AI-augmented interpretation. Zero server-side dependencies. Zero data exfiltration. Every computation executes deterministically in the client runtime.

Motivation

Most browser-based "statistics tools" approximate significance via hard-coded threshold tables or offload computation to remote APIs that introduce latency, privacy exposure, and vendor lock-in. R-Tool eliminates all three by implementing the full inferential pipeline — from raw data ingestion through regularized special functions to final p-value resolution — entirely within a single-page client application. The result is a workbench that behaves like a local R or SPSS session but deploys as a static web asset.

Capabilities

Data Ingestion

Flexible multi-modal input supporting heterogeneous datasets:

• Delimited Import — Paste or drag-and-drop CSV with automatic column type inference (numeric vs. categorical) via heuristic classification at an 80% numeric threshold
• Schema-First Manual Entry — Define column names and types a priori, then populate observations row-by-row with type-enforced validation
• Programmatic Column Detection — Quoted fields, mixed delimiters, and sparse entries are handled by a streaming character-level parser with quote-state tracking

name,age,group,score,hours,satisfaction,region
Alice,28,A,82,12,4.2,North
Bob,35,B,67,8,3.1,South
Carol,22,A,91,15,4.7,North
...

Upon ingestion, the engine partitions columns into numeric and categorical sets, auto-selects default axis mappings, and initializes the correlation inclusion vector.

Descriptive Statistics

Each numeric variable produces a full summary vector:

Statistic	Method
Arithmetic Mean	$\bar{x} = \frac{1}{n}\sum x_i$
Sample Standard Deviation	Bessel-corrected ($n - 1$ denominator)
Median	Midpoint of sorted array; averaged middle pair for even $n$
Min / Max	Extrema of observed range
Q1 / Q3	Linear interpolation at positions $0.25(n-1)$ and $0.75(n-1)$ in the sorted sample
Skewness	Adjusted Fisher-Pearson: $\frac{n}{(n-1)(n-2)} \sum\left(\frac{x_i - \bar{x}}{s}\right)^3$
Excess Kurtosis	$\frac{n(n+1)}{(n-1)(n-2)(n-3)} \sum\left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)}$

Example output for a variable score with $n = 15$:

Mean: 82.67  |  SD: 14.23  |  Med: 85.00
Min: 54.00   |  Max: 95.00
Q1: 61.50    |  Q3: 90.50
Skew: -0.41  |  Kurt: -1.18

A negative skew indicates left-tail elongation; a negative excess kurtosis (platykurtic) suggests lighter tails than the Gaussian reference.

Inferential Testing

Four parametric tests, each producing exact p-values from continuous distribution functions — not discretized lookup tables:

Welch's Independent Samples t-Test

Compares means of two groups without assuming equal variances. Degrees of freedom via the Welch-Satterthwaite approximation:

$$df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}}$$

Reports: $t$-statistic, $df$, two-tailed $p$, and Cohen's $d$ (pooled SD denominator) with qualitative magnitude (small/medium/large per Cohen 1988).

Pearson Product-Moment Correlation

Computes $r$, then transforms to a $t$-statistic for significance testing:

$$t = r\sqrt{\frac{n - 2}{1 - r^2}}, \quad df = n - 2$$

Reports: $r$, $r^2$, $t$, and two-tailed $p$.

Ordinary Least Squares Regression

Simple bivariate OLS with slope significance:

$$\hat{\beta}_1 = r \cdot \frac{s_y}{s_x}, \quad \hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}$$

Slope $t$-statistic derived from the same $r \to t$ transformation. Reports: $\hat{\beta}_0$, $\hat{\beta}_1$, slope $t$, slope $p$, $r$, $R^2$, and the fitted equation.

One-Way ANOVA

Partitions total variance into between-group and within-group components:

$$F = \frac{MS_{between}}{MS_{within}}, \quad p \text{ from } F(df_1, df_2)$$

Reports: $F$, $df_{between}$, $df_{within}$, $MS_{between}$, $MS_{within}$, and exact $p$.

P-Value Engine

The core of the inferential pipeline is a numerically stable implementation of the regularized incomplete beta function $I_x(a, b)$, evaluated via Lentz's continued fraction algorithm (Numerical Recipes, Ch. 6):

1. Log-Gamma — Lanczos approximation ($g = 7$, 9-term series) with reflection formula for $z < 0.5$
2. Continued Fraction — Modified Lentz's method with $\epsilon = 3 \times 10^{-14}$ convergence tolerance and 200-iteration ceiling
3. Symmetry Exploitation — Automatic argument swap when $x > (a+1)/(a+b+2)$ to ensure rapid convergence

From $I_x(a,b)$, both the Student's $t$ CDF and the Fisher-Snedecor $F$ CDF are derived:

$$P(T \leq t) = 1 - \tfrac{1}{2},I_{\frac{df}{df + t^2}}!\left(\tfrac{df}{2}, \tfrac{1}{2}\right)$$

$$P(F \leq f) = I_{\frac{d_1 f}{d_1 f + d_2}}!\left(\tfrac{d_1}{2}, \tfrac{d_2}{2}\right)$$

All p-values are displayed to four decimal places, with values below $10^{-4}$ rendered as < .0001.

Visualization

Five chart types rendered via Chart.js (scatter, histogram, bar, line) and custom SVG (box plot):

Chart	Use Case	Encoding
Scatter	Bivariate numeric relationships	Optional categorical grouping with color-coded series
Histogram	Univariate distributional shape	Configurable bin count (2–30)
Bar	Group means comparison	Categorical X, numeric Y (mean aggregation)
Line	Sequential / time-series trends	Cubic Bezier interpolation ($\tau = 0.3$) with area fill
Box Plot	Distributional comparison across groups	Custom SVG: whiskers (min/max), IQR box (Q1–Q3), median rule, per-group $n$ annotation

Box plot example — score by group with groups A and B:

Group A (n=8):  Min=82, Q1=85.5, Med=88.5, Q3=91.5, Max=95
Group B (n=7):  Min=54, Q1=56.5, Med=60.0, Q3=65.0, Max=72

Each box renders proportionally within a shared Y-axis domain with 5% padding, axis tick marks, and group labels.

Correlation Matrix

Interactive Pearson $r$ matrix for all selected numeric variables. Cells are chromatically encoded:

• Diagonal — $r = 1.00$ (blue tint)
• Strong positive ($r > 0.50$) — green
• Moderate positive ($r > 0.20$) — light green
• Negative ($r < -0.20$) — orange
• Weak ($|r| \leq 0.20$) — neutral grey

Column inclusion is toggled via checkboxes; a "Select All" control is provided.

Azura Interpretation Layer

An on-demand AI-generated statistical narrative synthesized from the current dataset state. Activated explicitly via a Generate Analysis button (no auto-computation) to preserve user agency and prevent stale interpretations.

The summary includes:

• Dataset dimensionality ($N$, variable counts by type)
• Top-3 variable descriptives
• Strongest pairwise correlation from the active matrix
• Most recent test result
• Statistical power advisory ($N < 30$ caveat)

A thumbs-up / thumbs-down feedback mechanism persists ratings to localStorage keyed by a deterministic hash of the dataset, enabling per-dataset feedback continuity across sessions.

Stack

Layer	Technology
Framework	Next.js 14 (App Router, static export)
Visualization	Chart.js + react-chartjs-2 + custom SVG
Statistical Engine	Pure TypeScript — no external stats libraries
Readings	react-markdown
Persistence	Browser localStorage (feedback ratings only)

Part of Mental Wealth Academy

This application is a standalone module extracted from the Mental Wealth Academy platform. It operates independently with no shared backend, database, or authentication requirements.