Q. 3.112 - Elementary Statistics

Question

A quantitative data set has size  $40$. At least how many observations lie within two standard deviation to either side of the mean $?$

Step-by-Step Solution

Answer

At least $30$ observations lie within two standard deviation .

Step1: Given information

We have been given that,

data size= $40$

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 is within $k$ standard deviation from the mean $\left(k>1\right)$.

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

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

is within $2$ standard deviation rom the mean.

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

                  $75\%\times 40 = 30$

Thus, at least $30$ observations lie within $2$ standard deviation of the mean.