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 the either side of the mean?

Short answer

a.$36\%$of the observations lie within $1.25$standard deviations to either side of the mean.

b. $91.836\%$of the observations lie within $3.5$standard deviations to either side of the mean.

Step-by-step solution

Part(a) Step 1: Given information

We have been given that $k=1.25$.

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$1-\frac{1}{{(1.25)}^2}=1-\frac{1}{1.5625}=\frac{0.5625}{1.5625}=36\%$ of the observations lie within $1.25$ standard deviations to either side of the mean.

Part(b) step 1: Given information 

We have been given that $k=3.5$.

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$1-\frac{1}{{(3.5)}^2}=1-\frac{1}{12.25}=\frac{11.25}{12.25}=91.836\%$of the observations lie within $3.5$standard deviations to either side of the mean.