If "k" > 1 and the distribution of "X" is symmetric then we have  Equality holds if and only if "Z" = −"k", 0 or "k" with probabilities , and respectively. An extension to a two-sided inequality is also possible. Let "u", "v" > 0. Then we have  An alternative method of obtaining sharper bounds is through the use of semivariances (partial variances). The upper ("σ"+2) and lower ("σ"−2) semivariances are defined as  where "m" is the arithmetic mean of the sample and "n" is the number of elements in the sample. The variance of the sample is the sum of the two semivariances:  In terms of the lower semivariance Chebyshev's inequality can be written  Putting  Chebyshev's inequality can now be written  A similar result can also be derived for the upper semivariance. If we put  Chebyshev's inequality can be written  Because "σ"u2 ≤ "σ"2, use of the semivariance sharpens the original inequality. If the distribution is known to be symmetric, then  and  This result agrees with that derived using standardised variables. Selberg derived an inequality for "P"("x") when "a" ≤ "x" ≤ "b". To simplify the notation let  where  and  The result of this linear transformation is to make "P"("a" ≤ "X" ≤ "b") equal to "P"(|"Y"| ≤ "k"). The mean ("μ""X") and variance ("σ""X") of "X" are related to the mean ("μ""Y") and variance ("σ""Y") of "Y":  With this notation Selberg's inequality states that  These are known to be the best possible bounds. Cantelli's inequality due to Francesco Paolo Cantelli states that for a real random variable ("X") with mean ("μ") and variance ("σ"2)  where "a" ≥ 0. This inequality can be used to prove a one tailed variant of Chebyshev's inequality with "k" > 0  The bound on the one tailed variant is known to be sharp. To see this consider the random variable "X" that takes the values  Then E("X") = 0 and E("X"2) = "σ"2 and P("X" < 1) = 1 / (1 + "σ"2).