Q. 8 - Elementary Statistics

Question

What does Chebyshev's rule say about the percentage of observations in any data set that lie within

(a) six standard deviations to either side of the mean? 

(b) 1.5 standard deviations to either side of the mean?

Step-by-Step Solution

Answer

(a) $97.2\%$of observations lie within $6$ standard deviations.

(b) $55.55\%$ of observations lie within $1.5$ standard deviations.

Part (a) Step 1: Given Information

We are given that standard deviations are $6$ and we have to find out the percentage of observations in any data set that lie in it.

Part (a) Step 2: Explanation

The Chebyshev's rule state that the percentage of the observation lie with in x standard deviation is given by, $(1-\frac{1}{x^2})\times 100$

where x is no of standard deviations.

and we are given that $x=6$.

Putting the value in formula,

we get, $(1-\frac{1}{6^2})\times 100=(1-\frac{1}{36})\times 100$

On solving we get $97.2\%$.

Hence, $97.2\%$ of observations lie with in the $6$ standard deviations.

Part (b) Step 1: Given Information

We are given that standard deviations are $1.5$ and we have to find out the percentage of observations in any data set that lie in it.

Part (b) Step 2: Explanation

The Chebyshev's rule state that the percentage of the observation lie with in x standard deviation is given by, $(1-\frac{1}{x^2})\times 100$

where x is no of standard deviations.

and we are given that $x=1.5$.

Putting the value in formula,

we get, $(1-\frac{1}{1.5^2})\times 100=(1-\frac{1}{2.25})\times 100$

On solving we get$55.55\%$ .

Hence, $55.55\%$of observations lie with in the $1.5$ standard deviations.