Question

Compare Chebyshev’s rule and empirical rule, calculate the proportion of observations within k standard deviations of the mean k = 2, 3, using both rules.

Answer 1

According to  Chebyshev’s rule , proportion of observation that lie within k std dev away from mean given by 1-1/k²

So,

for k=2 ,

according to Chebyshev's rule, proportion of observations within 2 standard deviations of the mean = 1-1/2² = 1-1/4 = 3/4 = 75%

and according to empirical rule, proportion of observations within 2 standard deviations of the mean = 95%

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for k=3,

according to Chebyshev's rule, proportion of observations within 3 standard deviations of the mean = 1-1/3² = 1-1/9 = 8/9 = 88.89%

and according to empirical rule, proportion of observations within 3 standard deviations of the mean = 99.7%