Compute and Explain Mean and Variance

# Compute and Explain Mean and Variance The preserved report describes a short numeric-array exercise that computes mean and variance, followed by a deeper discussion. It does not state whether variance is population or sample variance, how empty or non-finite inputs behave, or how floating-point results are compared. Explain the questions you would settle before implementation. Then, as an explicit practice assumption, analyze a nonempty finite array using population variance. ### Constraints & Assumptions - For the worked analysis only, let `n >= 1`, use population variance, and assume finite real inputs. - The practice convention divides squared deviations by `n`; a sample-variance contract would instead use `n - 1` where defined. - No bitwise cross-language equality is assumed for floating-point results. - Empty inputs, non-finite values, numeric precision, and answer tolerance must be agreed before coding. ### Clarifying Questions to Ask - Does variance mean population variance or unbiased sample variance? - What should happen for an empty array or a one-element sample? - Are inputs integers or floating-point values, and what magnitude is expected? - Must results match within a tolerance, follow one mandated operation order, or be returned exactly as rational values? ```hint Fix the statistical and numeric contracts separately The denominator defines the estimator; the operation order and comparison policy define how a floating-point implementation can be tested. ``` ### What a Strong Answer Covers - Correct formulas for the agreed mean and variance convention - A one-pass or two-pass algorithm with its numerical trade-offs - Edge cases such as singleton, empty, large-offset, and non-finite data - Why valid floating-point algorithms can differ in their last bits across languages - Time and auxiliary-space complexity ### Follow-up Questions 1. When is Welford's online update preferable to a direct one-pass formula? 2. How would you make outputs exactly comparable if every input were an integer? 3. Why can `E[x^2] - E[x]^2` lose substantial precision?