Latin Hypercube Sampling (LHS) is a stratified sampling technique that divides each factor's range into $N$ equal intervals and places exactly one sample point in each interval — ensuring marginal uniformity for every factor while maintaining good space-filling properties in the full-dimensional space.

How LHS Works

• Stratification: Divide each factor range into $N$ equal probability intervals.
• Random Placement: Place one point randomly within each interval for each factor.
• Permutation: Randomly permute the assignments across factors to create the design.
• Optimization: Optimized LHS (MaxiMin, correlation-minimizing) improves multi-dimensional uniformity.

Why It Matters

• Marginal Coverage: Guarantees that every region of each variable is sampled — no gaps.
• Efficient: Provides better coverage than random sampling with the same number of points.
• Standard Practice: The default sampling method for computer experiments, sensitivity analysis, and Monte Carlo studies.

LHS is fair sampling across all dimensions — ensuring that every factor's range is evenly covered regardless of sample size.