Chebyshev’s Inequality

Chebyshev’s Inequality.

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.

[Practice Problems] Over the past 240 months, an investor’s portfolio had a mean monthly return of 0.79%, with a standard deviation of monthly returns of 1.16%. According to Chebyshev’s inequality, the minimum number of the 240 monthly returns that fall into the range of −0.95% to 2.53% is closest to:

A. 80. B. 107. C. 133.

[Solutions] C

The upper limit of the range is 2.53%, which is 2.53 − 0.79 = 1.74% above the mean. The lower limit is −0.95, which is 0.79 − (−0.95) = 1.74% below the mean. k = 1.74/1.16 = 1.50 standard deviations. The proportion of observations within the interval is at least 1– 1/1.5² = 1 – 0.444 = 0.556, or 55.6%. Thus, the number of observations in the

given range is at least 240 × 55.6%, which is ≈ 133.