Created By : Vaibhavi Kumari
Reviewed By : Rajashekhar Valipishetty
Last Updated : Mar 20, 2023

Utilize this Chebyshev's Theorem Calculator will tell you how to use chebyshev's inequality inorder to acquire the probability of arbitrary distribution. For getting instant outcomes you need to give bound and variance as inputs and then tap the calculate button.

Formula Variant
Bound(k)
Variance(σ²)
Chance at most
%

### What is Chebyshev's Theorem ?

Chebyshev's theorem evaluates that the minimum proportion of observations that decreases within a specified number of standard deviations from the mean. Chebyshev's Theorem is also known as Chebyshev's Inequality. This theorem appeals that a wide range of probability distributions.

### Chebyshev's Theorem Formula

Look at the formula which are given below about Chebyshev's Theorem.

Here,

P = probability of an event.

X = random variable.

E(X) = expected value of our event

σ² = variance of our event

k = boundary of the result

### Chebyshev's Inequality Proof

As per Chebyshev's Theorem the probability that an observation will be more than k standard deviations from the mean is almost 1/k². For Chebyshev's Theorem to be true, you need to follow only 2 conditions i.e. underlying distribution has a mean and the other is the average size of deviations away from the mean isn't infinite. To prove this inequality let us consider there exists a population of n values containing n1 values of x1 and n2 values of x2.

Let us assume the standard deviation of the population to be σ, μ is the mean, and k>1 is some positive real number.

σ2=(∑(xi−μ)2⋅ni)/n

Here the sum ranges over all distinct xi. Thus we get

σ2 > ∑(xi−μ)2⋅ni/n

Since the sum ranges only over that xi when (|xi−μ|≥kσ), we can express

σ2> ∑k2σ2⋅ni}{n} = k2σ2⋅∑ni/n = k2σ2⋅Pout,

where Pout is the proportion of the population outside k standard deviations of μ. Since we have

Pout< 1/k².

We can apply inequality to any probability distribution in which both the mean and variance can be defined. Thus we have

Pr(|X-μ| > kσ) < 1/k².

### How to Use Chebyshev's Theorem?

Example

Question: The average range of a new bike is Rs.70000 with a standard deviation of Rs.3000. what is the minimum percentage of cars that should sell between Rs.22000 and Rs.80000?

Solution

Given, mean(μ) = 70000 , Standard deviation(σ) = 3000

Here, 70000 + k(3000) = 80000

k(3000) = 80000-70000

k(3000) = 10000

k= 10000/3000 = 3.33

Formula, 1-1/K2 = 1-(1/3.32) = 1-(1/10.9) = 0.91

Therefore, chances at most is 100% as E(X) i approximately 1.

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### FAQs on Chebyshev's Theorem Calculator

1. What is Chebyshev's Theorem in probability?

Chebyshev's theorem evaluates that the minimum proportion of observations that decreases within a specified number of standard deviations from the mean. Chebyshev's Theorem is also known as Chebyshev's Inequality.

2. How do you use chebyshev's theorem?

Chebyshev's Theorem says that for any k>1, there exists atleast 1-1/k² of data depending within k standard deviations of the mean.

3. What does K equal in chebyshev's Theorem?

In chebyshev's theorem, k value represents the number of standard deviations from the mean.