Chebyshev's Theorem Calculator
Calculate probability bounds for data spread using Chebyshev's inequality
Calculate Chebyshev's Inequality
The mean or expected value of the distribution
Standard deviation: 0.000
Maximum deviation from expected value
Example: Employee Ages
Scenario
Company data: Employee ages
Mean age: 35 years
Standard deviation: 5 years (variance = 25)
Question: What proportion of employees have ages within 2.5 standard deviations?
Solution
Using k = 2.5 standard deviations:
P(|X - 35| ≥ 2.5 × 5) ≤ 1/(2.5)² = 1/6.25 = 0.16
Result: At least 84% of employees have ages within 2.5 standard deviations (between 22.5 and 47.5 years)
Key Concepts
Universal Applicability
Works for any probability distribution
Conservative Bound
Provides minimum guarantees
Data Spread
Measures concentration around mean
Applications
Quality control in manufacturing
Risk assessment in finance
Biostatistics and medical research
Performance analysis in sports
Understanding Chebyshev's Theorem
What is Chebyshev's Theorem?
Chebyshev's theorem (also known as Chebyshev's inequality) is a fundamental result in probability theory that provides bounds on the probability that a random variable deviates from its expected value by more than a specified amount. Unlike many statistical rules, it applies to any probability distribution.
Key Properties
- •Distribution-free: Works for any shape of distribution
- •Conservative: Provides minimum guarantees
- •Universal: Requires only mean and variance
Two Forms of the Inequality
Form 1: Absolute Bound
P(|X - E(X)| ≥ k) ≤ σ²/k²
Probability of deviating by at least k units from the mean
Form 2: Standard Deviations
P(|X - E(X)| ≥ kσ) ≤ 1/k²
Probability of deviating by at least k standard deviations
Note: The complement gives the minimum proportion of data within the specified range
Common Values and Interpretations
| Standard Deviations (k) | Maximum Probability Outside | Minimum Probability Inside |
|---|---|---|
| 1.5 | ≤ 44.44% | ≥ 55.56% |
| 2 | ≤ 25% | ≥ 75% |
| 2.5 | ≤ 16% | ≥ 84% |
| 3 | ≤ 11.11% | ≥ 88.89% |