18.3. Chebyshev’s Theorem 623
meanO.¢/Figure 18.1 The standard deviation of a distribution indicates how wide the
“main part” of it is.
Definition 18.3.4. Thestandard deviation,R, of a random variable,R, is the
square root of the variance:
RWWDp
VarŒRçDq
ExŒ.R ExŒRç/^2 ç:So the standard deviation is the square root of the mean of the square of the
deviation, or theroot mean squarefor short. It has the same units —dollars in our
example —as the original random variable and as the mean. Intuitively, it measures
the average deviation from the mean, since we can think of the square root on the
outside as canceling the square on the inside.
Example18.3.5.The standard deviation of the payoff in Game B is:
BDp
VarŒBçDp
2;004;0021416:The random variableBactually deviates from the mean by either positive 1001
or negative 2002; therefore, the standard deviation of 1416 describes this situation
reasonably well.
Intuitively, the standard deviation measures the “width” of the “main part” of the
distribution graph, as illustrated in Figure 18.1.
It’s useful to rephrase Chebyshev’s Theorem in terms of standard deviation which
we can do by substitutingxDcRin (18.1):
Corollary 18.3.6.LetRbe a random variable, and letcbe a positive real number.
PrŒjR ExŒRçjcRç1
c^2