Question

A quantitative data set has mean $15$and standard deviation $2$. At least what percentage of the observations lie between $7$and $23$$?$

Short answer

$93.75\%$of the observations lie between$7$and$23$.

Step-by-step solution

Given information.

We have been given that ,

Mean = $15$, Standard deviation = $2$

We need to find the percentage of the observations that lie between$7$and$23$.

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 $\left(k>1\right)$.

Let us determine the number of standard deviation that $7$and $23$are from the mean.

$$\begin{gathered} \frac{7-mean}{s\tan darddeviation}=\frac{7-15}{2}=-4 \\ \frac{23-mean}{s\tan darddeviation}=\frac{23-15}{2}=4 \end{gathered}$$

So $7$and $23$are $4$standard deviation from the mean.

Using the Chebyshev's Rule with $k=4$, we know that at least,

$100\left(1-\frac{1}{4^2}\right)\%=100\left(1-\frac{1}{16}\right)\%=93.75\%$

is within $4$standard deviation from the mean.