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Chebyshev's Theorem Calculator

Chebyshev’s Theorem (Chebyshev’s Inequality) states that regardless of a distribution’s underlying shape or skewness, a guaranteed minimum fraction of observations must fall within k standard deviations of the mean.

Mean value of the distribution.

Standard deviation of the distribution.

Number of standard deviations away from the mean (must be strictly greater than 1).

Calculated Result
≥ 75%

Minimum Within Interval

Guaranteed Minimum Data Share

≥ 75%

Interval Range

70 to 130

Multiplier (k)

2 standard deviations

Maximum Tail Probability

≤ 25.00%

Calculation Breakdown

  1. Chebyshev's Inequality FormulaP(|X - μ| < kσ) ≥ 1 - (1 ÷ k²) = 1 - (1 ÷ 2²) = 1 - 0.2500 = 75%.
  2. Interval Computation[μ - kσ, μ + kσ] = [100 - 2(15), 100 + 2(15)] = [70, 130].

Chebyshev's Guaranteed Minimum Data vs k Multiplier

Interactive visualization based on your current inputs

Min Data (%)
0.023477094k = 1.5 SDsk = 2.0 SDsk = 2.5 SDsk = 3.0 SDsk = 4.0 SDsMultiplier (k SDs)Guaranteed Minimum (%)

What Is the Chebyshev's Theorem Calculator?

A Chebyshev’s Theorem calculator provides universal mathematical lower bounds on probability spread.

How Does the Chebyshev's Theorem Calculator Work?

Applies Pafnuty Chebyshev’s probability inequality bounding tail probabilities via second moments.

Chebyshev's Theorem Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

P(|X - μ| < kσ) ≥ 1 - (1 ÷ k²) · Interval: [μ - kσ, μ + kσ]

Calculates the guaranteed minimum lower bound of data contained within the interval for any distribution.

How to Use the Chebyshev's Theorem Calculator

  1. Enter the mean, standard deviation, and standard deviation multiplier k.

Step-by-Step Example Calculation

Mean 100, SD 15 with k = 2

Input Values:

mean:100
sd:15
kStdDevs:2

Understanding Your Result

Shows guaranteed minimum percentage within the calculated interval and maximum allowable tail probability.

Factors That Affect the Result

  • The multiplier k is the sole factor determining the theoretical percentage bound.

When Should You Use This Calculator?

  • Quality assurance with non-normal data, finance risk budgeting, and real-world skewed data modeling.

Assumptions & Limitations

  • Applies to any probability distribution with finite variance.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Mathematical theorem proven by Pafnuty Chebyshev.

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