Chebyshev's Theorem Calculator
Created By : Vaibhavi Kumari
Reviewed By : Rajashekhar Valipishetty
Last Updated : Mar 20, 2023
Utilize this Chebyshev's Theorem Calculator or chebyshev 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. There are various calculators regarding this concept like chebyshev's inequality calculator, chebyshev's theorem calculator with mean and standard deviation, chebyshev's rule calculator, chebyshev calculator, chebyshevs theorem calculator, chebyshev rule calculator, and chebyshev inequality calculator.
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.