The Challenge of Numerical Stability in Variance Calculation

Derive the single-pass 'textbook' formula for variance from its statistical definition.

Describe the two-pass algorithm for computing variance, which serves as a stable baseline.

Explain the concept of catastrophic cancellation as a source of error in floating-point arithmetic.

Analyze why the single-pass variance formula is numerically unstable, particularly for datasets with a large mean and small variance.