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Chebyshev inequality calculator comes up with a simple interface that requires easier steps:

Put the value of

**K**in the given field. It should be greater than 1Hit the

**Calculate**button to get the result.Use the

**Reset**button to clear the input fields and results.

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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.

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.

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.

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.

**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.