What is Chebychevs Theorem I have to solve a problem where A

What is Chebychev\'s Theorem? I have to solve a problem where \"A distribution has a mean if 80 and a standard deviation of 4.\" and the questions regarding chebychev\'s theorem are:
1. Chebychev\'s theorem assures that at least 8/9th of all scores lie between ______ and _____.
2. Chebychev\'s Theorem assures that _______ of all scores lies between 64 and 96.
Thanks.

Solution

Answer

A Russian mathematician P.L.Chebychev developed a theorem that approximated the proportion of data values lying within a given number of standard deviations of the mean regardless of whether the data is normally distributed or not. It states that,

For any data set, atleast (1 - 1/k²) of the values lies within k standard deviations either side of the mean.(k>1)

This theorem allows the determination of the least percentage of values that must lie between certain bounds identified by standard deviations.

1) Given: Mean = 80, S.d = 4.

8/9 = 89% = 1 - 1/k²

0.89 = 1- 1/k²

1 - 0.89 = 1/k²

0.11 = 1/k²

1/0.11 = k²

9.09 = k²

k = square root of 9.09 = 3.01 = 3.

Chebychev\'s theorem suggests that 89% of the scores are within +-3 standard deviations of the mean.

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