# Chebyshev's Theorem Calculator

$$P \ ( \ | \ X \ - \ E \ ( \ X \ ) \ | \ \geqslant \ k \ ) \ \leqslant \ \frac{σ^2}{k^2}$$

$$P \ ( \ | \ X \ - \ E \ ( \ X \ ) \ | \ \geqslant \ k \ ) \ \leqslant \ \frac{1}{k^2}$$

$$P \ ( \ | \ X \ - \ μ \ | \ < \ kσ \ ) \ \geqslant \ 1 \ - \ \frac{1}{k^2}$$

$$\sqrt{\frac{1}{1 \ - \ P \ ( x \ - \ μ \ < \ kσ )}}$$

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An online Chebyshev's theorem calculator determines the probabilities of an event being far from its expected value with detailed information. This helps you to find out how much percentage of the population is in the range of standard deviation.

## Statement:

This theorem is stated as:

“The minimum proportion of observation that is in the range of a specific number of the standard deviation from the mean”.

This theorem is also known as Chebyshev's inequality and no more their range than a certain fraction of values can be more than a certain distance from the mean. Chebyshev's inequality calculator indicates how to use this theorem to find the probability of an arbitrary distribution.

## Chebyshev’s Theorem Formula:

If the mean μ and the standard deviation σ of the data set are known then the 75% to 80 % points lie in between two standard deviations. The probability that x is within the K standard deviation is determined by the following formula:

Pr ( ∣X − μ∣ < kσ ) ≥ 1 − 1 / k^2

Where:

• P denoted the probability of an event
• X is the random variable that shows a sample
• k is the number of standard deviations greater than 1
• σ represents the variance

This is applicable to any distribution and for normal distribution use the empirical probability.

## Practical Example:

Accept that you are sitting watching a football match. You made a bet that your favorite team will score 5 goals in the first half and everyone expects only 2 goals.

So all you want is that they don't score 3 points more or less than that expected value. So we take into account k = 4 and σ² = 15. This means that your team has had its ups and downs in the last few matches, and getting 5 points or 6 points is not out of the question.

### Solution:

Given Data:

Number of standard deviation = 4
Variance σ² = 15

For this 2-number formula parameter following Chebyshev's theorem formula is determined by Chebyshev's theorem calculator with mean and standard deviation.

$$P \ ( \ | \ X \ - \ E \ ( \ X \ ) \ | \ \geqslant \ k \ ) \ \leqslant \ \frac{1}{k^2}$$

$$P \ ( \ | \ X \ - \ E \ ( \ X \ ) \ | \ \geqslant \ 4 \ ) \ \leqslant \ \frac{15}{4^2}$$

$$P \ ( \ | \ X \ - \ E \ ( \ X \ ) \ | \ \geqslant \ 4 \ ) \ \leqslant \ 0.063$$

With the mean and the standard deviation, the probability is determined by Chebyshev's rule calculator in between 6.25%. The estimated divergence of the probability from the expected value will be 16, and E(X) is almost 0.063.

## Working of Chebyshev's Theorem Calculator:

Chebyshev calculator is a dynamic tool that finds the percentage of the population that lies in between a certain amount of standard deviation.

Input:

• Choose the parameter of the formula
• Insert the value of bound k and variance σ²
• Tap the “Calculate” button

Output:

Our dynamic Chebyshev's theorem calculator functions well and gives you fast calculations so you can see what the result will be.

• Divergence of the probability from an expected value
• Complete calculations in simple steps

### What are the Percentages of Chebyshev's Theorem?

• 75% of data indicates the -2s to 2s standard deviations of the mean
• 89% of data shows the -3s to 3s standard deviations of the mean
• 95% of data represents the -4s to 4s standard deviations of the mean

### What Does “k” Equal in Chebyshev's Theorem?

In this theorem, the k value indicates the number of standard deviations. It describes at least 1-1/K 2 falls in this range and their value is always greater than 1.

## References:

From the source Wikipedia: Chebyshev's inequality, Statement, Example, Sharpness of bounds, Finite samples.

### Shaun Murphy

I'm graduated in biomedical and electrical engineering. Specialization in sports and medical topics but will gladly tackle everything you throw at him. He is a sailor, hiker, and motorcyclist in his free time.