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Chebyshev's Theorem
The Russian mathematician P. L. Chebyshev (1821- 1894)
discovered that the fraction of observations falling between two
distinct values, whose differences from the mean have the same
absolute value, is related to the variance of the population.
Chebyshev's Theorem gives a conservative estimate to the above
percentage.
For any population or sample, at least (1 - (1 / k)2)
of the observations in the data set fall within k standard
deviations of the mean, where k ³ 1.
Using the concept of z scores, we can restate Chebyshev's
Theorem to say that for any population or sample, the proportion
of all observations, whose z score has an absolute value less
than or equal to k, is no less than
(1 - (1 / k2 )). For k = 1, this theorem states that
the fraction of all observations having a z score between -1 and
1 is (1 - (1 / 1))2 = 0; of course, this is not a very
helpful statement. But for k ³ 1,
Chebyshev's Theorem provides a lower bound to the proportion of
measurements that are within a certain number of standard
deviations from the mean. This lower bound estimate can be very
helpful when the distribution of a particular population is
unknown or mathematically intractable.
EX Chebyshev's Theorem can be utilized for the following
values of k:
k = 1.5 1 - (1 / 1.52) = 0.5556 of all observations
fall within 1.5s of m.
k = 2.0 1 - (1 / 2.02) = 0.7500 of all observations
fall within 2.0s of
m.
k = 2.5 1 - (1 / 2.52) = 0.8400 of all observations
fall within 2.5s of
m.
k = 3.0 1 - (1 / 3.02) = 0.8889 of all observations
fall within 3.0s of m.
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