About Lesson
LOS H requires us to:
calculate and interpret the proportion of observations falling within a specified number of standard deviations of the mean using Chebyshev’s inequality
a. According to Chebyshev’s Inequality, for any distribution with finite variance, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 − 1/k2for all k > 1.
b. Or, for a random variable X, with a finite mean and variance:
Thus, if the variance is small, then X is unlikely to be very far from the mean.
c. Thus, a minimum number of observations within different levels of ks, regardless of how the data are distributed, would be as follows:
Now suppose if the mean of a distribution is 0.97% and the standard deviation is 5.65. Thus, for the 75% interval that has k = 2, the range would be:
0.97 ± 2 × 5.65
i.e. 75% observations lie between -10.33% to 12.27%.