Chebyshev's inequality

CFA level I / Quantitative Methods: Basic Concepts / Statistical Concepts and Market Returns / Chebyshev's inequality

Chebyshev's inequality gives the proportion of values within k standard deviations of the mean. 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/k2 for all k>1.

Its advantage is its generality as it applied to all discrete and continuous data regardless of the shape of the distribution.

Example 10: Proportion from Chebyshev's inequality

The mean and standard deviation of annual returns on a stock is 10.5 percent and 18.0 percent respectively. The mean and standard deviation has been calculated from 24 data points.

(a) What are the end points of the interval that must contain at least 75 percent of annual returns according to Chebyshev's inequality?

(b) What is the minimum number of observations that must lie in the interval computed above, according to Chebyshev's inequality?

Solution:

(a) According to Chebyshev's inequality, at least 75 percent of the observations must lie within two standard deviations from the mean. (1-1/22 = 1-0.25 = 0.75). Therefore, the interval is from 10.5 - 2*18.0 to 10.5+2*18.0 or from -25.5 percent to 46.5 percent.

(b) According to Chebyshev's inequality, at least 75 percent observations must lie between -25.5 percent and 46.5 percent. 75 percent of 24 observations is 18. So, at least 18 observations must lie between -25.5 percent and 46.5 percent.

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