Calculate Probability Bounds for Any Distribution
Calculate minimum probability that data falls within k standard deviations of the mean using Chebyshev's Theorem. Works for any distribution shape, making it universally applicable in statistics.
Determine the minimum probability within k standard deviations.
Use our Chebyshev's Theorem Calculator to find the minimum probability that a data point lies within a specified number of standard deviations from the mean. This tool is essential for understanding statistical distributions and variability.
Chebyshev's Theorem is a fundamental concept in statistics that provides a minimum probability for a random variable lying within a certain number of standard deviations from the mean, regardless of the distribution's shape.
The theorem states that for any real number k > 1:
P(|X - μ| < kσ) ≥ 1 - (1 / k²)
Where:
Our Chebyshev's Theorem Calculator simplifies this process, providing quick and accurate results for your statistical analysis.
Remember, Chebyshev's Theorem provides a conservative estimate; actual probabilities may be higher depending on the data distribution.
Chebyshev's Theorem states: for any distribution, at least (1 - 1/k²) × 100% of data falls within k standard deviations of the mean. Examples: k=2: at least 75% within 2σ, k=3: at least 89% within 3σ. Unlike empirical rule (68-95-99.7), Chebyshev's works for ANY distribution, not just normal.
Use Chebyshev's Theorem when: distribution shape is unknown, data is highly skewed, outliers are present, you need conservative estimates, or dealing with small samples. Common in: quality control, risk assessment, financial analysis, and scientific research where normality can't be assumed.
Limitations: provides minimum bounds (actual percentage usually higher), not useful for k ≤ 1, less precise than normal distribution results when data is normal. For normal distributions, use empirical rule for tighter bounds. Chebyshev's is valuable precisely because it makes no distribution assumptions.
