1. What is Chebyshev's Rule?

Chebyshev's Rule is a statistical theorem that provides a lower bound on the proportion of observations that fall within a specified number of standard deviations from the mean, regardless of the shape of the distribution.

2. How Does the Calculator Work?

The calculator uses Chebyshev's formula:

Where:

• \( k \) — Number of standard deviations from the mean (dimensionless, must be ≥1)

Explanation: The rule states that for any distribution, at least \( 1 - \frac{1}{k^2} \) of the data values lie within k standard deviations of the mean.

3. Importance of Chebyshev's Rule

Details: This rule is particularly valuable because it applies to any probability distribution with defined mean and variance, making it a universal tool for understanding data dispersion.

4. Using the Calculator

Tips: Enter the k value (number of standard deviations from the mean). The value must be greater than or equal to 1.

5. Frequently Asked Questions (FAQ)

Q1: What does k represent in Chebyshev's rule?
A: k represents the number of standard deviations from the mean. It must be a value greater than or equal to 1.

Q2: How accurate is Chebyshev's rule?
A: Chebyshev's rule provides a conservative lower bound. The actual proportion of data within k standard deviations is often higher than the rule suggests.

Q3: When is Chebyshev's rule most useful?
A: It's most useful when the distribution shape is unknown or non-normal, as it provides a guaranteed minimum proportion.

Q4: What are the limitations of Chebyshev's rule?
A: The rule can be overly conservative for well-behaved distributions (like normal distributions) where empirical rules provide more accurate estimates.

Q5: Can k be less than 1?
A: No, k must be ≥1. For k < 1, the proportion would be negative, which doesn't make sense in this context.