1. The Gaussian Normal Distribution & Empirical Confidence Intervals

The normal (or Gaussian) distribution is the cornerstone of classical parametric statistics, arising ubiquitously through the Central Limit Theorem (CLT), which states that the arithmetic mean of a sufficiently large number of independent random variables will be approximately normally distributed regardless of the underlying distribution.

The probability density function (PDF) for a continuous random variable X characterized by population mean μ and standard deviation σ is defined as:

Under the standard normal standardization z = (x − μ) / σ, probabilities are computed via the cumulative distribution function Φ(z). The classical 68–95–99.7 Empirical Rule governs symmetric confidence intervals:

• μ ± 1.000σ contains 68.27% of all observations
• μ ± 1.960σ contains 95.00% of observations (μ ± 2.000σ = 95.45%)
• μ ± 3.000σ contains 99.73% of observations

2. Ordinary Least Squares (OLS) Linear Regression & Correlation

Linear regression models the conditional expectation of a continuous response variable Y given an explanatory predictor X. The Ordinary Least Squares criterion minimizes the sum of squared vertical residuals (errors) between observed points and the linear estimator:

The analytical closed-form solutions for slope β₁ and intercept β₀ are given by:

Pearson's product-moment correlation coefficient r quantifies the strength and direction of linear association (−1 ≤ r ≤ +1), while the coefficient of determination R² = r² = 1 − (SSE / SST) reflects the fraction of variance in the dependent variable explained by the regression line.

3. Chi-Square (χ²) Non-Parametric Hypothesis Testing

The Chi-Square test evaluated across categorical frequencies assesses whether observed counts deviate systematically from theoretical expected frequencies under a specified null hypothesis H₀:

In goodness-of-fit testing across k categories, degrees of freedom are df = k − 1. In contingency table tests of independence across r rows and c columns, expected cell counts are computed under the null assumption of stochastic independence as E_ij = (Row_i Total × Col_j Total) / N, with df = (r − 1)(c − 1).

If the calculated test statistic exceeds the critical threshold at significance level α (commonly α = 0.05), corresponding to p < 0.05, the null hypothesis of independence is rejected in favor of a statistically significant association.

4. Error Propagation in Quadrature & Significant Figures Rules

When physical measurements containing experimental uncertainties are combined in mathematical functions z = f(x₁, x₂, ...), the total uncertainty propagates via a first-order multivariate Taylor series expansion. For mutually uncorrelated independent variables, uncertainties add in quadrature (root-sum-square):

This general formulation yields two fundamental rules:

• Sums and Differences: Absolute errors combine in quadrature: δz = √(δA² + δB²).
• Products and Quotients: Relative fractional errors combine in quadrature: δz/|z| = √[(δA/A)² + (δB/B)²].

In scientific reporting, rounding precision is governed by significant figures: results of addition/subtraction cannot have more decimal places than the least precise term, whereas multiplication/division results retain the least number of total significant figures.

1. The Gaussian Normal Distribution & Empirical Confidence Intervals

The normal (or Gaussian) distribution is the cornerstone of classical parametric statistics, arising ubiquitously through the Central Limit Theorem (CLT), which states that the arithmetic mean of a sufficiently large number of independent random variables will be approximately normally distributed regardless of the underlying distribution.

The probability density function (PDF) for a continuous random variable X characterized by population mean μ and standard deviation σ is defined as:

Under the standard normal standardization z = (x − μ) / σ, probabilities are computed via the cumulative distribution function Φ(z). The classical 68–95–99.7 Empirical Rule governs symmetric confidence intervals:

• μ ± 1.000σ contains 68.27% of all observations
• μ ± 1.960σ contains 95.00% of observations (μ ± 2.000σ = 95.45%)
• μ ± 3.000σ contains 99.73% of observations

2. Ordinary Least Squares (OLS) Linear Regression & Correlation

Linear regression models the conditional expectation of a continuous response variable Y given an explanatory predictor X. The Ordinary Least Squares criterion minimizes the sum of squared vertical residuals (errors) between observed points and the linear estimator:

The analytical closed-form solutions for slope β₁ and intercept β₀ are given by:

Pearson's product-moment correlation coefficient r quantifies the strength and direction of linear association (−1 ≤ r ≤ +1), while the coefficient of determination R² = r² = 1 − (SSE / SST) reflects the fraction of variance in the dependent variable explained by the regression line.

3. Chi-Square (χ²) Non-Parametric Hypothesis Testing

The Chi-Square test evaluated across categorical frequencies assesses whether observed counts deviate systematically from theoretical expected frequencies under a specified null hypothesis H₀:

In goodness-of-fit testing across k categories, degrees of freedom are df = k − 1. In contingency table tests of independence across r rows and c columns, expected cell counts are computed under the null assumption of stochastic independence as E_ij = (Row_i Total × Col_j Total) / N, with df = (r − 1)(c − 1).

If the calculated test statistic exceeds the critical threshold at significance level α (commonly α = 0.05), corresponding to p < 0.05, the null hypothesis of independence is rejected in favor of a statistically significant association.

4. Error Propagation in Quadrature & Significant Figures Rules

When physical measurements containing experimental uncertainties are combined in mathematical functions z = f(x₁, x₂, ...), the total uncertainty propagates via a first-order multivariate Taylor series expansion. For mutually uncorrelated independent variables, uncertainties add in quadrature (root-sum-square):

This general formulation yields two fundamental rules:

• Sums and Differences: Absolute errors combine in quadrature: δz = √(δA² + δB²).
• Products and Quotients: Relative fractional errors combine in quadrature: δz/|z| = √[(δA/A)² + (δB/B)²].

In scientific reporting, rounding precision is governed by significant figures: results of addition/subtraction cannot have more decimal places than the least precise term, whereas multiplication/division results retain the least number of total significant figures.