Chebyshev’s theorem. The Russian mathematician P.L. Chebyshev (1821-1894) showed that for any data set and any constant k greater than 1, at least 1 − ( 1 / k 2 ) of the data must lie within k standard deviations on either side of the mean A. For example, when k = 2 this says 1 − 14 = 34 (i.e., 75%) of the data must lie within two standard deviations of A (i.e., somewhere between A − 2 σ and A + 2 σ ) a. Using Chebyshev’s theorem, what percentage of a data set must lie within three standard deviations of the mean? b. How many standard deviations on each side of the mean must we take to be assured of including 99% of the data? c. Suppose that the average of a data set is A . Explain why there is no number k of standard deviations for which we can be certain that 100% of the data lies within k standard deviations on either side of the mean A .
Chebyshev’s theorem. The Russian mathematician P.L. Chebyshev (1821-1894) showed that for any data set and any constant k greater than 1, at least 1 − ( 1 / k 2 ) of the data must lie within k standard deviations on either side of the mean A. For example, when k = 2 this says 1 − 14 = 34 (i.e., 75%) of the data must lie within two standard deviations of A (i.e., somewhere between A − 2 σ and A + 2 σ ) a. Using Chebyshev’s theorem, what percentage of a data set must lie within three standard deviations of the mean? b. How many standard deviations on each side of the mean must we take to be assured of including 99% of the data? c. Suppose that the average of a data set is A . Explain why there is no number k of standard deviations for which we can be certain that 100% of the data lies within k standard deviations on either side of the mean A .
Solution Summary: The author explains that the percentage of a data set that lies within three standard deviations of the mean is 88.9%.
Chebyshev’s theorem. The Russian mathematician P.L. Chebyshev (1821-1894) showed that for any data set and any constant k greater than 1, at least
1
−
(
1
/
k
2
)
of the data must lie within k standard deviations on either side of the mean A. For example, when
k
=
2
this says
1
−
14
=
34
(i.e., 75%) of the data must lie within two standard deviations of A (i.e., somewhere between
A
−
2
σ
and
A
+
2
σ
)
a. Using Chebyshev’s theorem, what percentage of a data set must lie within three standard deviations of the mean?
b. How many standard deviations on each side of the mean must we take to be assured of including 99% of the data?
c. Suppose that the average of a data set is
A
. Explain why there is no number
k
of standard deviations for which we can be certain that 100% of the data lies within k standard deviations on either side of the mean
A
.
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