Page 226 - Applied Statistics And Probability For Engineers

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5-14

Chebyshev's
2
Inequality For any random variable X with mean and variance ,
P10 X 0 c 2 1
c 2

for c > 0.

This result is interpreted as follows. The probability that a random variable differs from its
2
mean by at least c standard deviations is less than or equal to 1 c . Note that the rule is useful
only for c > 1.
For example, using c = 2 implies that the probability that any random variable differs
from its mean by at least two standard deviations is no greater than 1 4. We know that for a
normal random variable, this probability is less than 0.05. Also, using c 3 implies that the
probability that any random variable differs from its mean by at least three standard deviations
is no greater than 1 9. Chebyshev’s inequality provides 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 8 9 that the average of the measured diameters is within 0.005 of the process mean
diameter ?
Let X 1 , X 2 , . . . , X n be the random variables that denote the diameters of n holes. The aver-
age measured diameter is X 1X X p X 2
n. Assume that the X’s are independent
n
1
2
2
random variables. From Equation 5-40, E1X 2 and V1X 2 0.01
n. Consequently, the
2
1 2
standard deviation of X is (0.01 n) . By applying Chebyshev’s inequality to , X
2 1
2 2
P10 X 0 c10.01
n2 2 1
c
Let c = 3. Then,
2 1
2
P10 X 0 310.01
n2 2 1
9
Therefore,

2 1
2
P10 X 0 310.01
n2 2 8
9

Table S5-1 Percentage of Distribution Greater than c Standard
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%