1. What is Chebyshev's Inequality?

Chebyshev's Inequality is a statistical theorem that provides a conservative estimate of the proportion of data within a certain number of standard deviations from the mean, regardless of the distribution shape.

2. How Does the Calculator Work?

The calculator uses Chebyshev's Inequality formula:

For a 75% interval, we set the probability to 0.25 (1 - 0.75):

• \( \frac{1}{k^2} = 0.25 \)
• \( k^2 = 4 \)
• \( k = 2 \)

Explanation: The interval is calculated as [μ - 2σ, μ + 2σ], which guarantees at least 75% of data falls within this range for any distribution.

3. Importance of Chebyshev's Interval

Details: Chebyshev's Inequality provides a distribution-free bound, making it valuable when the underlying distribution is unknown or non-normal.

4. Using the Calculator

Tips: Enter the population mean and standard deviation. The standard deviation must be a positive value.

5. Frequently Asked Questions (FAQ)

Q1: Why is Chebyshev's Inequality considered conservative?
A: It provides a lower bound that works for any distribution, so the actual proportion is often much higher than the guaranteed minimum.

Q2: How does this compare to the empirical rule?
A: The empirical rule (68-95-99.7) applies only to normal distributions, while Chebyshev works for any distribution but gives weaker bounds.

Q3: Can Chebyshev's Inequality be used for any percentage?
A: Yes, for any proportion 1 - 1/k², you can calculate the corresponding k value and interval.

Q4: What are the limitations of Chebyshev's Inequality?
A: It provides only a minimum guarantee and may significantly underestimate the actual proportion for well-behaved distributions.

Q5: When is Chebyshev's Inequality most useful?
A: When dealing with unknown or non-normal distributions where more specific distribution-based rules cannot be applied.