Average of a Curve ≠ Curve of the Average

2026-06-06 · Source: DataMListic · Field: Science & Research — Mathematics & Computational Sciences · Depth: Novice, quick

Summary

Jensen's inequality describes a fundamental mathematical principle for convex functions, stating that the expected value of a function f(x) is always greater than or equal to f of the expected value. This concept is intuitively explained using a convex, upward-bending curve. When two points on such a curve are connected by a straight line, or chord, the midpoint of this chord represents the average of the curve's heights at those two points. Crucially, this midpoint always lies above the curve itself at the corresponding middle input. The point on the curve at the average input is termed the "curve of the average," while the chord's midpoint is the "average of the curve." The persistent difference between these two points precisely illustrates Jensen's inequality, highlighting that these two values are distinct.

Key takeaway

For research scientists or data analysts interpreting expected values in non-linear systems, recognize that applying a convex function before averaging will yield a different result than averaging first and then applying the function. This distinction, formalized by Jensen's inequality, is crucial for accurate model design, risk assessment, and understanding how transformations impact aggregate metrics. Always consider the order of operations when dealing with averages and non-linear transformations.

Key insights

Jensen's inequality states the average of a convex function's output is not the output of its average input.

Principles

• For convex functions, E[f(X)] ≥ f(E[X])
• A chord connecting two points on a convex curve always lies above the curve.

Topics

• Jensen's Inequality
• Convex Functions
• Expected Value
• Mathematical Inequalities
• Non-linear Transformations

Best for: AI Student, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by DataMListic.