Question

According to Chebyshev’s rule what percent of the data are within 2.5 times standard deviation from the mean in a population? What assumption is necessary for the population for the answer to be valid?

Answer 1

Let X be a random variable defined on population F(.).

Now to use Chebyshev's inequality some condition must satisfied,

1) X be a integrable random variable.

2) X has finite and constant mean .

3) Variance of X, is positive and finite.

Then for a constant K, inequality is

In our problem we have given that 2.5 times of Standard deviation

Then

That is,

so 84% chance of getting with 2.5 times SD.