Q. 3.113 - Elementary Statistics

Question

A quantitative data set has size $80$. At least how many observations lie within two standard deviation of the either side$?$ 

Step-by-Step Solution

Answer

$80$ observations lie within two standard deviation of the either side .

Step1: Given information.

We have been given that,

data size=$80$ 

We need to find the number of observations lie within two standard deviation of either side of the mean.

Step2: Simplify.

Chebyshev's Rule: At least $100\left(1-\frac{1}{K^2}\right)\%$  of the data values lie within $k$ standard deviation from the mean $\left(k>1\right)$.

 

Using Chebyshev's Rule with $k=2$,we know that at least

$100\left(1-\frac{1}{K^2}\right)\%=100\left(1-\frac{1}{2^2}\right)\%=100\left(1-\frac{1}{4}\right)\%=100\left(\frac{3}{4}\right)\%=75\%$

is within $2$ standard deviation from the mean.

The number of the observations within $2$ standard deviation of the mean is the product of the percentage and the sample size $n$.

                          $75\%\times 80=0.75\times 80= 60$

Thus ,at least $80$ observations lie within two standard deviation of the mean.