Chebyshev inequality intuitive explanation

 Idea in plain words

Line up two lists of numbers that both go up, like

• heights: $a_1 \le a_2 \le \dots \le a_n$

• backpack weights: $b_1 \le b_2 \le \dots \le b_n$

Now pair the first height with the first weight, the second with the second, etc.

• Product of averages $\displaystyle \left(\frac{a_1+\cdots+a_n}{n}\right)\left(\frac{b_1+\cdots+b_n}{n}\right)$ is like saying:
  “Pretend every student has the average height and carries the average backpack. What would the ‘typical’ height×weight be?”

• Average of products $\displaystyle \frac{a_1b_1+a_2b_2+\cdots+a_nb_n}{n}$ is what you actually get when the shorter students carry the lighter backpacks and the taller students carry the heavier ones.

Chebyshev’s inequality says: when both lists rise together (small with small, big with big),

$$\text{average of products} \;\;\ge\;\; \text{product of averages.}$$

So matching big-with-big makes the average product at least as large as using “average × average.”

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Why this makes sense (intuition)

When the lists are both increasing, the “ups” happen together:

• A bigger $a$ is paired with a bigger $b$.

• A smaller $a$ is paired with a smaller $b$.

So each pair is either small×small or big×big. That boosts the total compared to mixing big with small, where the large and small cancel each other out. (This is the same reason why pairing big-with-small gives the smallest total—called the rearrangement principle.)

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Quick example

$a=[1,3,5]$, $b=[2,4,6]$

• Product of averages: $\left(\frac{1+3+5}{3}\right)\left(\frac{2+4+6}{3}\right)=3\times4=12.$

• Average of products: $\frac{1\cdot2+3\cdot4+5\cdot6}{3}=\frac{2+12+30}{3}=\frac{44}{3}\approx 14.67.$

Indeed, $14.67 \ge 12$.

If we mismatched on purpose (big-with-small): $b=[6,4,2]$,
$\frac{1\cdot6+3\cdot4+5\cdot2}{3}=\frac{6+12+10}{3}=\frac{28}{3}\approx 9.33,$
which is smaller than the product of averages.

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One-sentence takeaway

When two lists rise together, pairing them in order (small with small, big with big) makes the average product at least as large as “average × average.” That’s Chebyshev’s sum inequality.