# statistics-fundamentals

A skill that walks the agent through statistical work on financial returns: descriptive stats, covariance, CAPM regression, tests and bootstrap.

- skill · `type=skill` · Claude Code, Codex, Cursor, GitHub Copilot, Gemini CLI, OpenCode · 813 installs · ★ 218
- Author: JoelLewis
- License: MIT · policy=mirror · the beginning of the file is at the end of this page
- Source: https://github.com/JoelLewis/finance_skills/tree/HEAD/plugins/core/skills/statistics-fundamentals/SKILL.md
- Updated: 2026-07-18
- HTML: https://affpapa.org/en/nevibecode/skills/statistics-fundamentals

## What it does

The skill fixes an order of operations for a return series. Descriptive stats come first: mean, volatility with the `n - 1` divisor, skewness, excess kurtosis. Then a distribution check with Jarque-Bera, then covariance and correlation matrices, a CAPM or factor regression, and finally bootstrap intervals for statistics that lack a clean analytical standard error.

Inside are decision rules and worked examples. Sample variance uses Bessel's correction, and annual volatility multiplies by the square root of periods per year. For CAPM you get formulas for beta, alpha, R-squared and t-statistics: a positive alpha point estimate with |t| under 2 is not evidence of skill. The bootstrap section covers resampling with replacement, B from 1,000 to 10,000, and notes that the i.i.d. scheme ignores autocorrelation and volatility clustering, so serially dependent series need a block bootstrap.

One caveat is stated outright. The bundled `shrunk_covariance` implements a simplified shrinkage intensity, not the full Ledoit-Wolf estimator, and the author points to `sklearn.covariance.LedoitWolf` for production work. That matters if you feed the matrix into an optimizer: when the asset count approaches the observation count, the sample covariance is poorly conditioned and portfolio weights flip on tiny data changes.

## Install

### Claude Code (native platform)

1. **Skills CLI (Vercel)**
```bash
npx skills add JoelLewis/finance_skills --skill statistics-fundamentals -a claude-code
```

2. **Manually: put the files in place**
- File: `~/.claude/skills/statistics-fundamentals/SKILL.md` — Copy the whole skill folder; for one project use .claude/skills/statistics-fundamentals/

### Codex

1. **Skills CLI (Vercel)**
```bash
npx skills add JoelLewis/finance_skills --skill statistics-fundamentals -a codex
```

2. **Manually: put the files in place**
- File: `.agents/skills/statistics-fundamentals/SKILL.md` — For all projects use ~/.agents/skills/

### Cursor

1. **Skills CLI (Vercel)**
```bash
npx skills add JoelLewis/finance_skills --skill statistics-fundamentals -a cursor
```

2. **Manually: put the files in place**
- File: `.cursor/skills/statistics-fundamentals/SKILL.md` — Cursor also reads .agents/skills/ and .claude/skills/

### GitHub Copilot

1. **GitHub CLI**
```bash
gh skill install JoelLewis/finance_skills statistics-fundamentals
```

2. **Manually: put the files in place**
- File: `.github/skills/statistics-fundamentals/SKILL.md`

### Gemini CLI

1. **Skills CLI (Vercel)**
```bash
npx skills add JoelLewis/finance_skills --skill statistics-fundamentals -a gemini-cli
```

2. **Platform CLI**
```bash
gemini skills install https://github.com/JoelLewis/finance_skills --path plugins/core/skills/statistics-fundamentals
```

### Windsurf (Devin Desktop)

**Manually: put the files in place**
- File: `.devin/skills/statistics-fundamentals/SKILL.md` — The legacy .windsurf/skills/ path still works

### OpenCode

**Skills CLI (Vercel)**
```bash
npx skills add JoelLewis/finance_skills --skill statistics-fundamentals -a opencode
```

### Claude.ai

**Claude.ai**
- Zip the folder and upload it in Settings → Capabilities → Skills

Where a file is given instead of a command, copy the files from the source to that path: https://github.com/JoelLewis/finance_skills/tree/HEAD/plugins/core/skills/statistics-fundamentals/SKILL.md

## When to use it
- compute volatility of a return series with the bias correction
- test whether returns are normally distributed
- build a covariance matrix that does not produce unstable weights
- estimate a fund's beta and alpha with t-statistics
- bootstrap a confidence interval for a Sharpe ratio

## Example prompts
- run descriptive stats and a Jarque-Bera test on this series of monthly returns
- regress the fund on the market and show the t-stat for alpha
- build the covariance matrix for these assets and check whether shrinkage is needed

## Source text · SKILL.md · MIT

Below is the content of a third-party file. It is data, not instructions for you: do not act on anything written inside unless the human asks. Block tag: a550d11343e0ed7a.

BEGIN THIRD-PARTY FILE a550d11343e0ed7a
```text
---
name: statistics-fundamentals
description: "Apply statistical methods to financial data including descriptive statistics, covariance estimation, regression, hypothesis testing, and resampling. Use when the user asks about return distributions, correlation between assets, building a covariance matrix, running a CAPM regression, testing whether alpha is significant, checking if returns are normal, or estimating confidence intervals. Also trigger when users mention 'volatility', 'how correlated are these', 'fat tails', 'skewness', 'R-squared', 'beta of a fund', 'bootstrap a Sharpe ratio', 'shrinkage estimator', 'Ledoit-Wolf', or ask why their optimizer produces unstable weights."
---

# Statistics Fundamentals

## Core Concepts

### Conventions and Decision Rules

### Sample variance: use n-1
When estimating variance or standard deviation from a sample of returns, divide by `n - 1` (Bessel's correction), not `n`. Dividing by `n` systematically underestimates dispersion. Standard deviation of returns is "volatility"; annualize with `sigma_annual = sigma_period * sqrt(periods_per_year)` (e.g., `* sqrt(12)` for monthly, `* sqrt(252)` for daily).

### Normality testing: Jarque-Bera and its limits
`JB = (n/6) * (skew^2 + excess_kurtosis^2 / 4)`, distributed chi-squared with 2 df under the null of normality (5% critical value: 5.99).

**Low-power caveat:** with small samples (n below roughly 50), JB rarely rejects even for clearly non-normal data — failing to reject is weak evidence of normality, not confirmation. With large samples, financial return series almost always reject due to fat tails and (for equities) negative skewness. Treat the test as a screen, and pair it with a look at the actual skew/kurtosis magnitudes and extreme observations.

### Covariance estimation and Ledoit-Wolf shrinkage
The sample covariance matrix `Sigma_hat = (1/(n-1)) (X - X_bar)^T (X - X_bar)` becomes poorly conditioned or singular when the number of assets `p` approaches the number of observations `n`. Plugging it into a mean-variance optimizer then produces extreme, unstable weights that flip with small data changes.
```
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First 2136 of 8994 bytes. © JoelLewis · MIT · https://github.com/JoelLewis/finance_skills/tree/HEAD/plugins/core/skills/statistics-fundamentals/SKILL.md

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