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Gjonathan
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Tyro
Asked: 2021-04-09T13:16:09+00:00 In: ,

Find the mean, variance and standard deviation

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A population N= 5  consist of the following values 10, 11, 12, 13, and 14. Find the mean variance and standard deviation using a random variable of size n=2

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  2. acedstud
    This answer was edited.

    The table below shows the possible samples of size 2 with their means written in parenthesis

    Draw 1/Draw 2 10 11 12 13 14
    10 10, 10   (10) 10, 11   (10.5) 10, 12   (11) 10, 13   (11.5) 10, 14   (12)
    11 11, 10   (10.5) 11, 11   (11) 11, 12   (11.5) 11, 13   (12) 11, 14   (12.5)
    12 12, 10   (11) 12, 11   (11.5) 12, 12   (12) 12, 13   (12.5) 12, 14   (13)
    13 13, 10   (11.5) 13, 11   (12) 13, 12   (12.5) 13, 13   (13) 13, 14   (13.5)
    14 14, 10   (12) 14, 11   (12.5) 14, 12   (13) 14, 13   (13.5) 14, 14   (14)

    The table below shows the distribution of the means of the samples of size 2.

    \bar{x} Frequency (f) f\bar{x} \bar{x}-\mu (\bar{x}-\mu)^2 f(\bar{x}-\mu)^2
    10 1 10 -2 4 4
    10.5 2 21 -1.5 2.25 4.5
    11 3 33 -1 1 3
    11.5 4 46 -0.5 0.25 1
    12 5 60 0 0 0
    12.5 4 50 0.5 0.25 1
    13 3 39 1 1 3
    13.5 2 27 1.5 2.25 4.5
    14 1 14 2 4 4
    \sum{f}=25 \sum{f\bar{x}}=300 \sum{f(\bar{x}-\mu)^2}=25 
    mean = \mu_{\bar{x}}=\dfrac{\sum{f\bar{x}}}{\sum{f}}=\dfrac{300}{25}=12

    Variance = \sigma^2_{\bar{x}}=\dfrac{\sum{f(\bar{x}-\mu)^2}}{\sum{f}}=\dfrac{25}{25}=1

    Standard deviation = \sqrt{\text{Variance}}=\sqrt{1}=1.

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