Question

A quantitative data set has size $60$. At least how many observations lie within three standard deviation to either side of the mean $?$

Short answer

$53$observations lie within three standard deviation to either side of the mean.

Step-by-step solution

Given information.

We have been given that ,

data size = $60$, $K=3$

We need to find the number of observations to either side of the mean.

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 $k=3$, 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 from the mean is the product of the percentage and sample size $n$.

$89\%\times 60=53$

Thus, at least $53$observations lie within three standard deviation of the mean.