Q. 3.146 - Elementary Statistics

Question

How many standard deviations to either side of the mean must we go to ensure that for any data set, at least $95\%$ of the observations lie within? 

Step-by-Step Solution

Answer

At least $95\%$ of the data lies within $k=4.47$ standard deviations of the mean.

Step 1: Given Information

We have been given that at least $95\%$ of the observation lie within.

Step 2: Simplification

We know Chebyshev's Rule says that at least $100(1-\frac{1}{k^2})\%$ of the data values is within k standard deviations from the mean $(k>1)$.

We are interested in finding the number of standard deviations k such that $100(1-\frac{1}{k^2})\%$ becomes $95\%$.

So, $100(1-\frac{1}{k^2}) = 95$

On simplifying, we get 

$k^2=20$

$k= 2\sqrt{5}$

$k=4.47$

Thus Chebyshev's rule then tells us that at least $95\%$ of the data lies within $k=4.47$ standard deviations of the mean.