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3.4 Measures of Position
Chebyshev's Theorm
For any set of data (including populations and samples) and any
constant k > 1,
the proportion of the data that lies within k
standard deviations of the mean is at least
:
1 
As an example of this theorem let's consider the following set of
data on the travel expenses of the 12 members of a university's
physics department (in dollars)
0
0
173
378
441
733
759
857
958
985
1434
2063
The sample standard deviation of this data is 602.26. The sample mean
is 731.75. According to Chebyshev’s theorem 0.55 = 1  of the data is
within 1.5 standard deviations of the mean.
11 of 12 values are between
731.75  1.5(602.26)
and 731.75 + 1.5(602.26)
171.64
and
1635.14
Thus over 91.66 % of the data is within 1.5 standard deviations of
the mean, which is considerably more than the conservative Chebyshev
estimate of 55%.
For the bell shaped distributions called Normal
distributions, the
percent of the data within k std. devs. is higher than is guaranteed
by Chebyshev’s inequality. The following is called the
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 Fall '10
 RaySievers
 Statistics, Standard Deviation

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