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Math Help - Standart deviation for two groups

  1. #1
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    Standart deviation for two groups

    Hello guys,
    Here is my problem. I have two groups the first one is from 42 members and the second -36. I have the average of the two groups and their standard deviation value, but i don't have their individual results. How can I calculate the average and standard deviation for the whole 78 members.
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  2. #2
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    Quote Originally Posted by Pumata View Post
    Hello guys,
    Here is my problem. I have two groups the first one is from 42 members and the second -36. I have the average of the two groups and their standard deviation value, but i don't have their individual results. How can I calculate the average and standard deviation for the whole 78 members.
    Let group 1 have a mean \bar{x}, variance s_x^2 and be of size n. Then:

    \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \Rightarrow \sum_{i=1}^{n} x_i = n \bar{x} .... (1)

    s_x^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \bar{x}^2 \Rightarrow n\left(s_x^2 + \bar{x}^2 \right) = \sum_{i=1}^{n} x_i^2 .... (2)



    Let group 2 have a mean \bar{y}, variance s_y^2 and be of size m. Then:

    \bar{y} = \frac{\sum_{i=1}^{m} y_i}{m} \Rightarrow \sum_{i=1}^{m} y_i = m \bar{y} .... (3)

    s_y^2 = \frac{\sum_{i=1}^{m} y_i^2}{m} - \bar{y}^2 \Rightarrow m \left(s_y^2 + \bar{y}^2 \right) = \sum_{i=1}^{m} y_i^2 .... (4)



    Let the whole group (group 1 plus group 2) have a mean \bar{z} and variance s_z^2. The whole group has size n + m. Then:


    \bar{z} = \frac{\sum_{i=1}^{n+m} z_i}{n + m} = \frac{\sum_{i=1}^{n} x_i + \sum_{i=1}^{m} y_i }{n + m}.

    Substitute from (1) and (3): \bar{z} = \frac{n \bar{x} + m \bar{y}}{n+m}.



    s_z^2 = \frac{\sum_{i=1}^{n+m} z_i^2}{n+m} - \bar{z}^2 = \frac{\sum_{i=1}^{n} x_i^2 + \sum_{i=1}^{m} y_i^2}{n+m} - \bar{z}^2.

    Substitute from (2) and (4): s_z^2 = \frac{n \left( s_x^2 + \bar{x}^2 \right) + m \left( s_y^2 + \bar{y}^2 \right)}{n+m} - \bar{z}^2
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  3. #3
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    Thank you mr fantastic!
    The last formula was what I was searching for
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