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Thread: Standard Deviation - Markov or Chebyshev Theorem

  1. #1
    RSM
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    Standard Deviation - Markov or Chebyshev Theorem

    A distribution of 50 scores with a mean of -4 and a standard deviation of 5. Can have at most how many of these scores greater than 6?

    I know the following...
    n = 50
    mean = -4
    std dev = 5

    I'm thinking with a mean of -4 the std distribution doesn't meet the requirements of Markov's inequality theorem of no negative values. Therefore scores of 6 based on the Chebyshev theorem are (6-(-4))5 = 2 std deviations form the mean. Not sure where to go from here?
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  2. #2
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    Re: Standard Deviation - Markov or Chebyshev Theorem

    let $X$ be the rv representing the score

    $X>6 \Rightarrow |X - \mu| \geq 11 = \dfrac{11}{5}\sigma$

    $P[|X - \mu| \geq \dfrac{11}{5}\sigma] \leq \dfrac{1}{\left(\frac{11}{5}\right)^2} = \dfrac{25}{121}$

    $\left \lfloor 50 \dfrac{25}{121}\right \rfloor = 10$
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