8/12/2019 Chebyshevs Theorem (1)

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Chebyshevs Theorem:

P.L. Chebyshev Russian Mathematician who lives from 1821 1894. e was a

!rofessor at the "niversity of #t. Petersbur$% where he &i& a $reat &eal of im!ortant wor'

in both !ure an& a!!lie& mathematics. (he most sur!risin$ as!ect of Chebyshev)s

theorem is that it a!!lies to any an& all &istributions of &ata values.

Chebyshev)s (heorem* +or any set of &ata ,either !o!ulation or sam!le- an& for any

constant k$reater than 1% the !ro!ortion of the &ata that must lie within kstan&ar&&eviations on either si&e of the mean is at least

1 /1/ k2

0n or&inary wor&s% Chebyshev)s (heorem says the followin$ about sam!le or !o!ulation

&ata*

1- #tart at the mean.2- ac' off kstan&ar& &eviations below the mean an& then a&vance kstan&ar&

&eviations above the mean.- (he fractional !art of the &ata in the interval &escribe& will be at least 1 13k2

,we assume ' 1-.

Minimal Percenta$e of 5ata +allin$ within k #tan&ar& 5eviations of the Mean*

k 2 3 4 5 10

,1 13k2- 6 177 : 88.9 9.8 9; 99

8/12/2019 Chebyshevs Theorem (1)

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17 17; 114 1 111 1;2 148 119 127 144

a- Calculate the sam!le mean an& sam!le stan&ar& &eviation.b- "se Chebyshev)s (heorem to fin& an interval centere& about the mean in

which you woul& e?!ect : of the years to fall.

c- "se Chebyshev)s (heorem to fin& an interval centere& about the mean inwhich you woul& e?!ect 9.8 of the years to fall.

&- "se Chebyshev)s (heorem to fin& an interval centere& about the mean in

which you woul& e?!ect 99 of the years to fall.

- ase& on the followin$ &ata% answer the Duestions below*

89 4 97 82 48 92 47 2 4

: 4 89 : 7 98 24 : 87 9:8 97 : 98 74 : 89 7 2 :8

97 4 7 :4 8 9 :8 9 4 79

: 48 : 9 82

a- Calculate the sam!le mean an& sam!le stan&ar& &eviation.

b- "se Chebyshev)s (heorem to fin& an interval centere& about the mean inwhich you woul& e?!ect : of the years to fall.

c- "se Chebyshev)s (heorem to fin& an interval centere& about the mean in

which you woul& e?!ect 88.9 of the years to fall.

&- "se Chebyshev)s (heorem to fin& an interval centere& about the mean inwhich you woul& e?!ect 9; of the years to fall.

4- ase& on the followin$ &ata% answer the Duestions below*4; 8;2 418 9; 28 94 89 ;1: 89 ;1: 98

;4 8: ; 849 2;: 98; :8 94 4 982

4 94 792 941 87 491 4 98 98 42

a- Calculate the sam!le mean an& sam!le stan&ar& &eviation.b- "se Chebyshev)s (heorem to fin& an interval centere& about the mean in

which you woul& e?!ect : of the years to fall.

c- "se Chebyshev)s (heorem to fin& an interval centere& about the mean in

which you woul& e?!ect 88.9 of the years to fall.&- "se Chebyshev)s (heorem to fin& an interval centere& about the mean in

which you woul& e?!ect 9; of the years to fall.