Use Chebyshev s Inequality to Describe Any Set of Data
-If nothing is known about the shape of a distribution, what percentage of the observations fall within 3 standard deviations of the mean
A) at least $89 \%$
B) at most $11 \%$
C) approximately $99.7 \%$
D) approximately $0.3 \%$

Question

Use Chebyshev s Inequality to Describe Any Set of Data
-If nothing is known about the shape of a distribution, what percentage of the observations fall within 3 standard deviations of the mean 
A) at least $89 \%$
B) at most $11 \%$
C) approximately $99.7 \%$
D) approximately $0.3 \%$

Quizplus · Accepted Answer

Chebyshev's Inequality states that for any set of data, regardless of the distribution shape, at least $1 - \frac{1}{k^2}$ of the data falls within $k$ standard deviations from the mean. For $k = 3$, this gives at least $1 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9}$, which is approximately $88.89\%$, rounded to at least $89\%$.

Use Chebyshev S Inequality to Describe Any Set of Data 89%89 \%89%

Use Chebyshev S Inequality to Describe Any Set of Data $89 \%$