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Mean Squared Deviation

  1. Jan 6, 2013 #1
    Could someone explain the meaning of "Mean Squared Deviation"?

    Also, in <x1+x2+..xn>
    what is the meaning of the pointy brackets <..> ?
  2. jcsd
  3. Jan 6, 2013 #2

    I like Serena

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    Welcome to PF, zeeshahmad!

    Hmm, didn't I see you somewhere else? :wink:

    The term "mean squared deviation" is a bit ambiguous and can mean 2 things.
    I'll try to explain.

    Suppose you have a sample of n measurements x1, x2, ..., xn.

    Then the sum of squared deviations, often abbreviated as SS is:
    $$SS = \sum (x_i - \bar x)^2$$
    where ##\bar x## is the mean.

    This set of measurements come with a "degrees of freedom", abbreviated DF.
    For a "normal" repeated measurement, we have:
    $$DF = n - 1$$

    In statistics, when the term "mean squared deviation" is used, it usually means:
    $$MS = {SS \over DF}$$
    This is exactly the variance (or squared standard deviation) of the sample.

    However, taken literally, "mean squared deviation" means just the average of the squared deviations, which is:
    $$SS \over n$$

    I can't tell you what <x1+x2+..xn> means.
    Do you have a context for that?
    Last edited: Jan 6, 2013
  4. Jan 6, 2013 #3
    Nice posting to you again :approve:
    Actually I have got the lecture notes, in which it tells the meaning, but I don't understand it:

    "Consider a distribution with average value μ and standard deviation σ from which a sample measurements are taken, i.e.

    [itex]\mu = \left\langle x \right\rangle[/itex]
    [itex]\sigma^2 = \left\langle x^2 \right\rangle - {\left\langle x \right\rangle}^2[/itex]

    "where the brackets <..> mean an average with respect to the whole distribution."
  5. Jan 6, 2013 #4

    I like Serena

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    Ah, I see what you mean.
    <...> as you show it, is also called the "expected value".
    The expected valueof a variable X is also written as EX or E(X).

    If the variable x can take only specific values ##x_i## with an associated chance of ##p_i##, then in general, the expectation of a function f(x) is:
    $$\langle f(x) \rangle = \sum f(x_i)p_i$$
    Or if x is a continous variable, it is:
    $$\langle f(x) \rangle = \int f(x)p(x)dx$$
    where p(x) is the so called probability density function.

    So <x1+x2+...xn> would be the expected value of the sum.
    This is equal to <x1>+<x2>+...+<xn>.
  6. Jan 6, 2013 #5
    Thankyou for the detailed explanation
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