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Sample standard deviation proof

  1. Jan 10, 2012 #1
    1. The problem statement, all variables and given/known data

    Let [itex]x_{1},...,x_{n}[/itex] be n observations. If [itex]y_{1},...,y_{n}[/itex] is another set of observations s.t. [itex]y_{i}=ax_{i}+b[/itex] , prove that [itex]s_{y}=|a|s_{x}[/itex] .

    3. The attempt at a solution


    Attempt at a proof: Since [itex]\bar{y}=a\bar{x} +b[/itex] then [itex]\bar{x}=(\bar{y}-b)/a[/itex] and [itex]s_{x}=\sqrt{\frac{1}{n-1}\sum(x_{i}-\frac{\bar{y}-b}{a})}[/itex]. This is where I get stuck. Any ideas?
     
  2. jcsd
  3. Jan 10, 2012 #2
    First of all, you should know that standard deviation is independent of translation (change of origin), but is affected by scale. See if you can knock out the proofs of each separately, and then you should be able to put them together. Also, begin your proofs by using the variance, not the standard deviation.
     
  4. Jan 10, 2012 #3

    lanedance

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    and it will probably be easier to calculate [tex] s_y [/tex] directly and compare with the form of [tex] s_x [/tex]
     
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