Applied Statistics and Probability for Engineers

5-14

This result is interpreted as follows. The probability that a random variable differs from its

mean by at least c standard deviations is less than or equal to 1c^2. Note that the rule is useful

only for c> 1.

For example, using c= 2 implies that the probability that anyrandom variable differs

from its mean by at least two standard deviations is no greater than 14. We know that for a

normal random variable, this probability is less than 0.05. Also, using c3 implies that the

probability that any random variable differs from its mean by at least three standard deviations

is no greater than 19. Chebyshev’s inequalityprovides a relationship between the standard

deviation and the dispersion of the probability distribution of any random variable. The proof

is left as an exercise.

Table S5-1 compares probabilities computed by Chebyshev’s rule to probabilities com-

puted for a normal random variable.

EXAMPLE S5-8 The process of drilling holes in printed circuit boards produces diameters with a standard

deviation of 0.01 millimeter. How many diameters must be measured so that the probability is

at least 89 that the average of the measured diameters is within 0.005 of the process mean

diameter ?

Let X 1 , X 2 ,... , Xnbe the random variables that denote the diameters of nholes. The aver-

age measured diameter is Assume that the X’s are independent

random variables. From Equation 5-40, Consequently, the

standard deviation of is (0.01^2 n)^1 ^2. By applying Chebyshev’s inequality to ,

Let c= 3. Then,

Therefore,

P 10 X 031 0.01^2 n 21
22  8
9

P 10 X 0  31 0.01^2 n 21
22  1
9

P 10 X 0 c 1 0.01^2 n 21
22  1
c^2

X X

E 1 X 2  and V 1 X 2 0.01^2 n.

X 1 X 1 X 2 p Xn 2
n.

For any random variable Xwith mean and variance ^2 ,

for c> 0.

P 10 X 0  c 2  1
c^2

Chebyshev's

Inequality

Table S5-1 Percentage of Distribution Greater than cStandard

Deviations from the Mean

Chebyshev’s Rule Normal

c for any Probability Distribution Distribution

1.5 less than 44.4% 13.4%

2 less than 25.0% 4.6%

3 less than 11.1% 0.27%

4 less than 6.3% 0.01%

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