Q. 3.100 - Elementary Statistics

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?

Step-by-Step Solution

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.

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.