What is Chebyshev's inequality?

Chebyshev's inequality states that for any random variable with mean μ and variance σ²: P(|X - μ| ≥ a) ≤ σ² / a². It bounds the probability of being far from the mean using only mean and variance.

How does Chebyshev relate to standard deviations?

Setting a = kσ gives P(|X - μ| ≥ kσ) ≤ 1/k². This means at least 75% of values are within 2σ of the mean (k=2 gives 1/4), and at least 89% within 3σ (k=3 gives 1/9).

Why does Chebyshev show two shaded regions?

Chebyshev bounds deviations in BOTH directions from the mean: P(X μ+a). The red shading shows both tails, representing values more than 'a' away from μ in either direction.

Is Chebyshev tighter than Markov?

Generally yes, because Chebyshev uses more information (both mean and variance). Markov only requires E[X]. The extra variance information allows Chebyshev to produce tighter bounds, especially for distributions with small variance.

Why is the actual probability often much smaller than Chebyshev's bound?

Chebyshev must work for ANY distribution with given mean and variance. The bound equals the worst case. Well-behaved distributions like Normal concentrate much more tightly around the mean than Chebyshev guarantees.

Chebyshev Inequality Visualization

Chebyshev Inequality Visualizer

P(|X - μ| ≥ a) ≤ σ² / a²

P(|X - 10| ≥ 3) ≤ 4 / 9.0 = 0.444 = 44.4%

Actual: 13.4%

Distribution Type

Mean μ: 10

Variance σ²: 4

Distance from mean (threshold a): 3

Chebyshev Inequality: For any distribution with mean μ and variance σ²: P(|X - μ| ≥ a) ≤ σ² / a²

Red areas show P(X < μ-a) + P(X > μ+a). Bound: 44.4%, Actual: 13.4%

Getting Started with the Chebyshev Visualizer

Understanding the Chebyshev Bound

The k Standard Deviations Form

Using the Control Sliders

Two-Tailed vs One-Tailed Bounds

Comparing Bound to Actual Probability

Why Chebyshev's Inequality Matters

Related Tools and Concepts

Visualizing Chebyshev's Inequality

Chebyshev's inequality bounds the probability of deviating from the mean: P(|X - μ| ≥ a) ≤ σ² / a². This tool visualizes the two-tailed bound across nine distributions, showing both the Chebyshev bound and actual probability. Adjust mean, variance, and deviation threshold to explore how bounds behave.

Getting Started with the Chebyshev Visualizer

This tool demonstrates Chebyshev's inequality, which bounds the probability of deviating from the mean using variance. The visualization shows a probability distribution with both tails highlighted in red.

The left panel displays the PDF (for continuous) or PMF (for discrete distributions). A green dashed line marks the mean μ, and red dashed lines mark μ-a and μ+a. The red shaded regions represent the actual probability of being more than a away from the mean.

The top panel shows the Chebyshev bound formula and compares it to the actual two-tailed probability. Adjust mean, variance, and deviation threshold to explore when the bound is tight versus loose.

Understanding the Chebyshev Bound

Chebyshev's inequality states:

P(|X - \mu| \geq a) \leq \frac{\sigma^2}{a^2}

This bounds the probability of being at least a units away from the mean μ, in either direction. The bound uses two parameters:

• μ = EX], the [expected value
• σ² = Var(X), the variance

The bound decreases quadratically as a increases. Doubling the threshold reduces the bound by a factor of 4. This quadratic relationship makes Chebyshev tighter than Markov for large deviations.

The visualization shows both tails simultaneously, representing P(X < μ-a) + P(X > μ+a).

The k Standard Deviations Form

Chebyshev is often expressed in terms of standard deviations. Setting a = kσ gives:

P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}

This yields memorable bounds:

• k = 2: At least 75% of values within 2σ of mean (bound = 1/4 = 25% outside)
• k = 3: At least 89% within 3σ (bound = 1/9 ≈ 11% outside)
• k = 4: At least 94% within 4σ (bound = 1/16 ≈ 6% outside)
• k = 5: At least 96% within 5σ (bound = 1/25 = 4% outside)

These percentages are worst-case guarantees. Actual distributions like Normal concentrate much more tightly—99.7% of Normal values fall within 3σ, far better than Chebyshev's 89% guarantee.

Using the Control Sliders

Three sliders control the distribution and bound:

Mean (μ) Slider (green): Sets the expected value from 5 to 20. The distribution centers on this value, and the green dashed line moves accordingly.

Variance (σ²) Slider (purple): Sets the variance from 1 to 16. Higher variance spreads the distribution wider, affecting both the shape and the Chebyshev bound.

Deviation Threshold (a) Slider (red): Sets the distance from mean from 0.5 to 10. The red dashed lines at μ-a and μ+a move, changing the shaded tail regions.

Key experiments:

• Fix μ and a, then increase σ²: The bound increases (loosens)
• Fix μ and σ², then increase a: The bound decreases quadratically
• Compare Normal vs Exponential at identical settings

Two-Tailed vs One-Tailed Bounds

Chebyshev bounds two-tailed probability: deviations in BOTH directions from the mean. The visualization shows this with red shading on both the left tail (X < μ-a) and right tail (X > μ+a).

For symmetric distributions like Normal, these tails are equal. For asymmetric distributions like Exponential, one tail dominates.

If you need a one-tailed bound, Chebyshev gives:

P(X \geq \mu + a) \leq \frac{\sigma^2}{a^2}

But this is looser than necessary because the two-tailed bound includes both sides. For strictly one-tailed bounds, Markov's inequality applied to appropriate transformations may be tighter.

Comparing Bound to Actual Probability

The information panel displays:

• Bound: Chebyshev's upper bound σ²/a²
• Actual: True probability P(|X - μ| ≥ a)

Typical observations across distributions:

Normal distribution: Chebyshev is very conservative. At 2σ, bound is 25% but actual is about 4.5%. At 3σ, bound is 11% but actual is about 0.3%.

Uniform distribution: Chebyshev can be exact at the distribution boundaries. The uniform distribution is one of the "worst cases" for Chebyshev.

Exponential distribution: Asymmetric, so left tail contributes differently than right tail. Bound is moderately loose.

The gap demonstrates that Chebyshev guarantees apply to all distributions, including pathological ones that concentrate probability at exactly ±a from the mean.

Why Chebyshev's Inequality Matters

Chebyshev's inequality is fundamental because:

Distribution-free: Works for ANY distribution with finite mean and variance. No shape assumptions required.

Tighter than Markov: Uses variance information for quadratic improvement. The 1/k² decay is much faster than Markov's 1/k.

Theoretical cornerstone: Used to prove the weak law of large numbers, convergence of sample means, and consistency of estimators.

Practical applications:
• Quality control: Setting tolerance limits based on process variance
• Finance: Bounding portfolio deviations from expected return
• Statistics: Constructing distribution-free confidence intervals

The tradeoff is conservatism. When you know more about your distribution (e.g., it's Normal), distribution-specific bounds are much tighter.

Related Tools and Concepts

Chebyshev's inequality connects to other probability concepts and tools:

Theory Pages:

• Chebyshev Inequality covers complete theory and proofs

• Markov Inequality is the weaker precursor

• Variance is a key input for Chebyshev

• Expected Value provides the center point

Other Visualizations:

• Markov Visualizer shows one-tailed bounds

• Variance Visualizer demonstrates spread concepts

• Distribution Visualizers display full PDFs and CDFs

Related Topics:

• Probability Distributions covers the distributions used here

• Central Limit Theorem relies on concentration arguments