Linear Algebra Done Right · Problem 6A.13
Inner Product Spaces — Sheldon Axler
Mathematics — Linear Algebra · inner-products-and-norms
Problem
Show that the square of an average is less than or equal to the average of the squares. More precisely, show that if , then the square of the average of is less than or equal to the average of .Intuitive explanation
This is the Cauchy–Schwarz inequality in disguise, but it's easiest to see through the lens of variance. The key insight is that variance is always non-negative: when you square deviations from the mean, their average can never be negative. Expanding that constraint algebraically will give you exactly the inequality you need—the "extra" terms that appear when you square the mean are precisely what get absorbed by the variance.
Intuitive explanation
This is the Cauchy–Schwarz inequality in disguise, but it's easiest to see through the lens of variance. The key insight is that variance is always non-negative: when you square deviations from the mean, their average can never be negative. Expanding that constraint algebraically will give you exactly the inequality you need—the "extra" terms that appear when you square the mean are precisely what get absorbed by the variance.