an inequality on location and scale parameters

For probability distributions having an expected value and a median, the mean (i.e., the expected value) and the median can never differ from each other by more than one standard deviation. To express this in mathematical notation, let μ, m, and σ be respectively the mean, the median, and the standard deviation. Then

\left|\mu-m\right| \leq \sigma.

(There is no need to rely on an assumption that the variance exists, i.e., is finite. Unlike the situation with the expected value, saying the variance exists is equivalent to saying the variance is finite. But this inequality is trivially true if the variance is infinite.)

Proof

This proof uses Jensen's inequality twice. We have

pre> \| $\left\|\mu-m\right\| = \left\|\mathrm{E}(X-m)\right\|\!\!\!\!\!$ \| $\leq \mathrm{E}\left(\left\|X-m\right\|\right)$ \|- \| \| $\leq \mathrm{E}\left(\left\|X-\mu\right\|\right) = \mathrm{E}\left(\sqrt{(X-\mu)^2}\right)$ \|- \| \| $\leq \sqrt{\mathrm{E}((X-\mu)^2)} = \sigma.$

The first inequality comes from (the convex version of) Jensen's inequality applied to the absolute value function, which is convex. The second comes from the fact that the median minimizes the absolute deviation function

a \mapsto \mathrm{E}(\left|X-a\right|).

The third inequality comes from (the concave version of) Jensen's inequality applied to the square root function, which is concave. Q.E.D.

Alternative proof

The one-tailed version of Chebyshev's inequality is

P(X-\mu \geq k\sigma)\leq\frac{1}{1+k^2}.

Letting k = 1 gives P(X ≥ μ + σ) ≤ 1/2 and (by changing the sign of X and so μ) P(X ≤ μ − σ) ≤ 1/2. So the median is within one standard deviation of the mean.

<< Previous

Word Browser

Next >>

hirola
hyatt regency walkway collapse
john o'neill (vietnam veteran)
ulva island, new zealand
laura ingraham
gymnastics at the 1908 summer olympics
tom halk
creativity techniques
south mimms
bourne
hauraki gulf

end of line
coromandel peninsula
reginald denny
semitic pagan deities
signal (computing)
solomon ibn gabirol
the holiday plan
joe barton
velma barfield
leuciscus
topi

everson v. board of education
antineutron
vctor cervera pacheco
dulce mara sauri riancho
druid
marlon devonish
statement (programming)
vertical deflection
multi task learning
tomi ungerer
john joyce

jewish museum (bucharest)
george h. v. bulyea
national museum of art of romania
petite bourgeoisie
u.s. congressional delegations from south carolina
the last outpost
kingdom of mysore
suta dumping
visualboyadvance
curtain
expression (programming)