Question

A quantitative data set has mean $30$and standard deviation $4$. At least what percentage of the observations lie between $10$and$50$$?$

Short answer

$96\%$ of the observations lie between$10$and$50$.

Step-by-step solution

Given information.

We have been given that,

Mean =$30$,Standard deviation= $4$

We need to find the percentage of observations lie between$10$and$50$.

Simplify.

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

Let us determine the number of standard deviation that $10$and $50$are from the mean,

$$\begin{gathered} \frac{10-mean}{s\tan darddeviation}=\frac{10-30}{4}=-5 \\ \frac{50-mean}{s\tan darddeviation}=\frac{50-30}{4}=5 \end{gathered}$$

Thus, $10$and $50$are $5$standard deviation from the mean.

Using Chebyshev's Rule with$k=5$,we know that at least

$100\left(1-\frac{1}{K^2}\right)\%=100\left(1-\frac{1}{5^2}\right)\%=100\left(1-\frac{1}{25}\right)\%=96\%$

is within $5$standard deviation from the mean.