Question

What does Chebyshev's rule say about the percentage of observations in any data set that lie within
a.$1.25$ standard deviations to either side of the mean?
b. $3.5$standard deviations to either side of the mean?

Short answer

a. Atlest 36%of data lies within 1.25 standard deviations to either side of the mean by using chebshev's rule with $k=1.25$

b.Atlest $91.84\%$of data lies within 1.25 standard deviations to either side of the mean by using chebshev's rule with $k=3.5$

Step-by-step solution

Part (a) Step 1. Given information

Given standard deviation$1.25$

Part (a) Step 2. Finding 1.25 standard deviations to either side of the mean 

Find the minimum percentage by using chebyshev's rule when $k=1.25$

$$\begin{gathered} \left(1-\frac{1}{k^2}\right)100\%=\left(1-\frac{1}{{\left(1.25\right)}^2}\right)100\% \\ =\left(1-\frac{1}{\left(1.5625\right)}\right)100\% \\ =\left(\frac{0.5625}{1.5625}\right) \\ =36\% \end{gathered}$$

Thus atlest 36%of data lies within 1.25 standard deviations to either side of the mean by using chebshev's rule with$k=1.25$

Part (b) Step 1. Given information 

Given standard deviation$=3.5$

Part (b) Step 2. Finding 3.5 standard deviations to either side of the mean 

Find the minimum percentage by using chebyshev's rule when 3.5

$$\begin{gathered} \left(1-\frac{1}{k^2}\right)100\%=\left(1-\frac{1}{{\left(3.5\right)}^2}\right)100\% \\ =\left(1-\frac{1}{12.25}\right)100\% \\ =\left(\frac{11.25}{12.25}\right)100\% \\ =91.84\% \end{gathered}$$

Thus atlest 91.84% of data lies within 3.5 standard deviations to either side of the mean by using chebshev's rule with$k=3.5$