1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sum of deviations

  1. Oct 3, 2016 #1
    1. The problem statement, all variables and given/known data
    The average value of N measurements of a quantity ##v_i## is defined as
    $$ \langle v \rangle \equiv \frac {1}{N} \sum_{i=1}^Nv_i = \frac {1}{N}(v_1 + v_2 + \cdots v_N)$$
    The deviation of any given measurement ##v_i## from the average is of course ##(v_i - \langle v \rangle)##. Show mathematically that the sum of all the deviations is zero; i.e. show that
    $$\sum_{i=1}^Nv_i(v_i - \langle v \rangle) = 0$$
    2. Relevant equations
    ##?##
    3. The attempt at a solution
    I understand that this is simply describing an average, but I am not sure how to express this mathematically. It makes sense to me that the sum of the deviations would be zero.
     
  2. jcsd
  3. Oct 3, 2016 #2

    Mark44

    Staff: Mentor

    Your formula above is incorrect, as it has an extra ##v_i##.
    The sum of the deviations is
    $$\sum_{i = 1}^N (v_i - \bar{v})$$
    Here ##\bar{v}## is the mean of the measurements ##v_i##.
    Simply expand the summation.
     
    Last edited: Oct 4, 2016
  4. Oct 4, 2016 #3
    $$((v_1 - \langle v \rangle) + (v_2 - \langle v \rangle) + \cdots (v_N - \langle v \rangle))$$
    $$((v_1 - \frac {v_1 + v_2 + \cdots v_N}{N}) + (v_2 - \frac {v_1 + v_2 + \cdots v_N}{N}) + \cdots (v_N - \frac {v_1 + v_2 + \cdots v_N}{N}))$$
    Is this sufficient to "show mathematically" that the sum of all the deviations is zero? I am just not sure what I am being asked to do.
     
  5. Oct 4, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You are being asked to show that the summation you wrote above evaluates to ##0## for any possible inputs ##v_1, v_2, \ldots, v_N##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Sum of deviations
  1. Mean deviation (Replies: 2)

  2. Standard deviation (Replies: 3)

Loading...