How far can values stray?

You run a web server. The average response time is 200ms. Your boss asks: "What fraction of requests take more than 1 second?" You don't know the full distribution — just the mean.

Can you still say something useful?

What do you think?

Average response time is 200ms. Without knowing anything else, what's the most requests that could exceed 1000ms?

Markov's inequality

Markov's Inequality

If $X \geq 0$ and $a > 0$ : $P(X \geq a) \leq \Large\frac{E[X]}{a}$

That's it. One line. The only requirement: $X$ is non-negative.

The intuition

If the average is $\mu$ , then values can't mostly be far above $\mu$ — there wouldn't be enough "mass" left for the average to work out. Markov makes this precise.

Proof (three lines)

Proof of Markov's Inequality

E[X] = \int_0^\infty x f(x)\, dx \geq \int_a^\infty x f(x)\, dx

Drop the part from 0 to a — that only decreases the integral.

Step 1 of 3

A non-negative RV has E[X] = 8. What does Markov say about P(X ≥ 40)? (decimal, e.g. 0.42)

Is Markov's bound always tight? E.g., if X is always exactly 8, what is the actual P(X ≥ 8)? (whole number)

Chebyshev's inequality

Markov only uses the mean. If you also know the variance, you can do much better.

Chebyshev's Inequality

For any random variable $X$ with mean $\mu$ and variance $\sigma^2$ , and for any $k > 0$ : $P(|X - \mu| \geq k\sigma) \leq \Large\frac{1}{k^2}$

Equivalently: $P(|X - \mu| \geq a) \leq \sigma^2 / a^2$ .

Proof from Markov

Chebyshev from Markov

P(|X - \mu| \geq k\sigma) = P((X - \mu)^2 \geq k^2 \sigma^2)

Square both sides (both are non-negative).

Step 1 of 3

Comparing bounds to reality

Inequality Explorer

Inequality

k (number of σ): 2.0

Exact

4.550%

Chebyshev bound

25.00%

Bound / Exact

5.5×

Bound vs. Exact

Exact

4.55%

Bound

25.0%

Toggle between Markov and Chebyshev. Watch how the bound compares to the exact probability. Chebyshev is always tighter because it uses more information (the variance), but both are conservative — that's the price of distribution-free guarantees.

$k$ (σ's from mean)	Chebyshev bound	Normal exact	Bound / Exact
1	100%	31.7%	3.2×
2	25%	4.6%	5.5×
3	11.1%	0.27%	41×
4	6.25%	0.006%	988×

The bounds get looser as $k$ grows. That's not a bug — Chebyshev must hold for every distribution. The Normal is well-behaved; the bound accounts for the worst-case shape.

When to use which

Situation	Use
Only know E[X], X ≥ 0	Markov
Know E[X] and Var(X)	Chebyshev
Know the distribution	Compute exactly!

Markov requires $X \geq 0$ . Chebyshev works for any distribution with finite variance. Neither requires knowing the distribution's shape.

What do you think?

Exam scores have mean 75, SD 10. At most what fraction scored below 55 or above 95?

Enter a decimal, e.g. 0.42

Applications

Quality control

A machine produces bolts with mean diameter 10mm and SD 0.1mm. Chebyshev guarantees at most $1/k^2$ fraction deviate by more than $k \times 0.1$ mm.

Financial risk

A portfolio has expected return 8% and SD 15%. Markov: P(loss > 20%) needs care (shift by mean). Chebyshev: P(|return − 8%| > 30%) ≤ 1/4.

E[X] = 50, Var(X) = 25. Chebyshev bound for P(|X − 50| ≥ 15)? (decimal to 3 places, e.g. 0.456)

Non-negative X with E[X] = 10. Markov bound for P(X ≥ 50)? (decimal, e.g. 0.42)

Can Chebyshev's bound ever exceed 1? E.g., for k = 0.5? (1 for yes, 0 for no)

See how tight (or loose) these bounds are across different distributions:

Bound Tightness Explorer

Distribution

Inequality

Threshold a = 3.0: 3.0

0.58

Markov bound μ/a True probability

P(X ≥ a) exact

0.049787

Markov bound

0.333333

Looseness ratio

6.7×

The bound is 6.7× the true value — useful for guarantees but not precise.

Summary

Inequality	Statement	Requires
Markov	$P(X \geq a) \leq E[X]/a$	$X \geq 0$
Chebyshev	$P(\\|X - \mu\\| \geq k\sigma) \leq 1/k^2$	Finite variance
Key idea	Bound tail probabilities without knowing the distribution

Markov and Chebyshev are blunt instruments — and that's the point. They work for any distribution meeting their minimal assumptions. When you know more about the distribution, you can do better. When you don't, these are your safety net.

What's next

Chebyshev says the sample average probably stays near the true mean. The Law of Large Numbers makes this precise: the average converges to the mean as the sample grows.