Q. 3.115 - Elementary Statistics

Question

A quantitative data set has size $50$. At least how many observations lie within three standard deviation to either side of the mean $?$

Step-by-Step Solution

Answer

$45$ observations lie within three standard deviation to either side of the mean.

Step1: Given information.

We have been given that,

data size= $50$, $K=3$

We need to find the number of observation lie within three standard deviation of the mean.

Step2: Simplify.

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

Now,

Using Chebyshev's Rule with $k=3$,we know that at least

    $100\left(1-\frac{1}{K^2}\right)\%=100\left(1-\frac{1}{3^2}\right)\%=100\left(\frac{8}{9}\right)\%=89\%$

is within three standard deviation from the mean.

 

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

               $89\%\times 50=.89\times 50=45$

Thus, at least $45$ observations lie within three standard deviation of the mean.