Question

What does Chebyshev's rule say about the percentage of observations in any data set that lie within

a. four standard deviations to either side of the mean?

b.$2.5$ standard deviations to the either side of the mean?

Short answer

a. $93.75\%$of the observations lie within $4$standard deviations to either side of the mean.

b. $84\%$of the observations lie within $2.5$standard deviations to either side of the mean.

Step-by-step solution

Part(a) Step 1: Given information

We have been given that $k=4$.

We need to find the percentage of observations in any data set that lie within four standard deviations to either side of the mean.

Part(a) Step 2: Explanation 

According to Chebyshev's rule, for any quantitative data set and any real number $k$greater than or equal to $1$, at least $1-\frac{1}{k^2}$of the observations lie within $k$standard deviations to either side of the mean, that is, $\overbrace{x}-ks$and $\overbrace{x}+ks$.

By Chebyshev's rule,

At least $1-\frac{1}{4^2}=1-\frac{1}{16}=\frac{15}{16}=93.75\%$of the observations lie within$4$standard deviations to either side of the mean.

Part(b) step 1: Given information 

We have been given that $k=2.5$.

We need to find the percentage of observations in any data set that lie within four standard deviations to either side of the mean.

Part(a) Step 2: Explanation 

According to Chebyshev's rule, for any quantitative data set and any real number $k$greater than or equal to $1$, at least $1-\frac{1}{k^2}$ of the observations lie within $k$standard deviations to either side of the mean, that is, $\overbrace{x}-ks$and $\overbrace{x}+ks$.

By Chebyshev's rule,

At least $1-\frac{1}{{(2.5)}^2}=1-\frac{1}{6.25}=\frac{5.25}{6.25}=84\%$of the observations lie within $2.5$standard deviations to either side of the mean.