Jensen's Inequality, explained.
Jensen's inequality compares the function of an expectation with the expectation of a function: convexity and concavity determine the inequality's direction under suitable integrability conditions.
Why it happens
Two equally likely inputs are five minus spread and five plus spread, with spread zero to four. Compare their mean transformed value with the transform of five under x², √x or x. The square is convex, square root concave and linear function gives equality.
Jensen's inequality compares the function of an expectation with the expectation of a function: convexity and concavity determine the inequality's direction under suitable integrability conditions.
Read the result
Track the constant mean input of five while changing spread. The average transformed output responds to curvature even though the input mean does not move.
A worked example
The same mean through three functions
Use inputs two and eight, whose mean is five, and apply the square function to obtain an average of thirty-four versus twenty-five at the mean.
Switch to square root and compare the average transformed inputs with the square root of five; the inequality reverses.
Switch to the linear function, where both calculations return five for every permitted spread.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Two equally likely inputs are five minus spread and five plus spread, with spread zero to four. Compare their mean transformed value with the transform of five under x², √x or x. The square is convex, square root concave and linear function gives equality.
Where this idea is useful
A practical use
A process with a curved response can change its average output when variability changes even though average input stays fixed.
A common misconception
“Applying a function to an average always gives the average outcome.”
Averages commute with linear functions, but curvature generally breaks that equality. The inequality's direction requires a justified convex or concave response.
What this explanation leaves out
- These positive inputs and three functions are exact authored examples. They do not identify a real system's curvature, which must be justified before applying the inequality to business, risk or biology.
Can the inequality prove that variability helps a real business?
Only after establishing the relevant response function and constraints. The theorem does not discover a real process's curvature from its average alone.
Which nonlinear step lies between the average input you report and the average outcome you care about?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.