Estimating standard deviation from wheighted sampleThe standard deviation is given by ∑ ( x i − x ) 2 n , however when we estimate the standart deviation from a sample, the best estimation is ∑ ( x i − x ) 2 n − 1 How do I have to adjust the standarddeviation if I want to wheight my samples?I.e. the standard deviation would be ∑ w ( x i ) ( x i − x ) 2 ∑ w ( x i ) , if I had the entire data set. What is the correct estimation of the standard deviation, if I'm only given a subsample of the population?

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Estimating standard deviation from wheighted sampleThe standard deviation is given by                     ∑        (                  x          i                −        x                  )          2                    n      , however when we estimate the standart deviation from a sample, the best estimation is                     ∑        (                  x          i                −        x                  )          2                            n        −        1            How do I have to adjust the standarddeviation if I want to wheight my samples?I.e. the standard deviation would be                     ∑        w        (                  x          i                )        (                  x          i                −        x                  )          2                            ∑        w        (                  x          i                )            , if I had the entire data set. What is the correct estimation of the standard deviation, if I&#039;m only given a subsample of the population?

Teagan Raymond · Accepted Answer

We know that irrational numbers never repeat by combining the following two facts:every rational number has a repeating decimal expansion, andevery number which has a repeating decimal expansion is rational.Together these facts show that a number is rational if and only if it has a repeating decimal expansion.Decimal expansions which don&#039;t repeat are easy to construct; other answers already have examples of such things.The irrationality of a number by examining its decimal expansion. While it is true that an irrational number has a non-repeating decimal expansion, you don&#039;t need to show a given number has a non-repeating decimal expansion in order to show it is irrational. In fact, this would be very difficult as we would have to have a way of determining all the decimal places. Instead, we use the fact that an irrational number is not rational; in fact, this is the definition of an irrational number (note, the definition is not that it has a non-repeating decimal expansion, that is a consequence). In particular, to show a number like       2   is irrational, we show that it isn&#039;t rational. This is a much better approach because, unlike irrational numbers, rational numbers have a very specific form that they must take, namely       a    b   where   a and   b are integers,   b non-zero. The standard proofs show that you can&#039;t find such   a and   b so that       2    =      a    b   thereby showing that       2   is not rational; that is,       2   is irrational.

The standard deviation is given by sqrt{(sum(x_i-x)^2)/(n)}, however when we estimate the standart deviation from a sample, the best estimation is sqrt{(sum(x_i-x)^2)/(n-1)}

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