Chebyshev's Theorem Calculator

A Chebyshev’s Rule calculator is a tool that uses Chebyshev's inequality to calculate the minimum proportion of a data set that falls within a specified number of standard deviations from the mean.

What is Chebyshev’s Theorem?

Chebyshev's Theorem is also known as Chebyshev's Inequality. It provides a way to calculate the minimum proportion of values that lie within a specified number of standard deviations from the mean in any data set. 

Formula of Chebyshev’s Theorem

You can calculate the Chebyshev's Inequality by using the following formula:

P (|X - μ| ≥ Kσ) ≤ 1/K2

Where: 

• μ = Mean of the data set

• σ = Standard deviation of the data set.

• k = Number of standard deviations from the mean.

• P = Proportion of the data set.

How to Calculate Chebyshev’s Theorem?

In this section, we’ll explain how to find the proportion using Chebyshev’s rule.

Example

Evaluate the minimum proportions of the mean to the standard deviations using Chebyshev’s rule if K is 8.

Solution

K = 9

Apply the formula of:

• P (|X - μ| ≥ Kσ) ≤ 1/K2

• ​P (∣ X − μ ∣ < Kσ) = 1 − 1 / (8)2

• ​P (∣ X − μ ∣ < kσ) = 1 – 1 / 64

• P (∣ X − μ ∣ < Kσ) ​= 1 − 0.0156

• P (∣ X − μ ∣ < Kσ) = 0.9844

Decision

Using Chebyshev's Theorem, we can conclude that a minimum of 98.437% of the data points are within 8 standard deviations from the mean. If you want to learn more about standard deviation, use our standard deviation calculator.

Example 

Suppose a company wants to understand the minimum proportion of employees' ages within 2.5 standard deviations from the mean. The mean age of the employees is 35 years, and the standard deviation is 5 years. Use Chebyshev's Theorem to find the answer.

Solution

1. Identify the values

• Mean (μ) = 35

• Standard deviation (σ) = 5

• Number of standard deviations (k) = 2.5

2. Apply Chebyshev Theorem Formula

Here k = 2.5, μ = 35 and σ = 5

Calculate the minimum proportion

P (|X -35|< 2.5 × 5) ≥ 1- 1/ (2.5)2

P (|X -35|< 12.5) ≥ 1- 1/ (6.25)

Compute 1/ (6.25) = 0.16

Therefore, P (|X -35|< 12.5) ≥ 1- 0.16

P (|X -35|< 12.5) ≥ 0.84

Conclusion

This means that at least 84% of the data (employees' ages in this case) lies within 2.5 standard deviations from the mean. Specifically, the ages are between 35−12.5=22.5 and 35+12.5=47.5 years.

Frequently Asked Questions

1. How do you know when to use Chebyshev’s Theorem?

Chebyshev's Theorem is used when you need to estimate data spread without knowing the distribution shape. It provides a minimum proportion of data within a specified number of standard deviations from the mean, applicable to any distribution.

2. Does Chebyshev’s Theorem apply to all samples? 

Yes, Chebyshev’s Theorem can apply to all possible data sets. It explains the minimum proportion of the measurements within 1, 2, or more standard deviations of the mean.

3. What is the Chebyshev’s Theorem valid for? 

Chebyshev’s Theorem is valid for any probability distribution, regardless of its shape. It provides a general bound on the proportion of data within a specified number of standard deviations from the mean. 

This theorem is useful for understanding data spread when the exact distribution is unknown or when the data does not follow a specific distribution pattern.