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How do you find the standard deviation of the weighted mean?

The weighted mean is defined: $\bar{x}_w = \frac{\sum{wx}}{\sum{w}}$

The weighted standard deviation (since it is not specified, I take it as of the distribution) is defined: $$s^2_w = \sqrt{\frac{N'\sum_{i=1}^N {w_i(x_i-\bar{x}_w)^2}}{(N'-1)\sum_{i=1}^N{w_i}}},$$ where $N'$ is the number of nonzero weights, and $\bar x_w$ is the weighted mean of the sample.

(source: http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/weightsd.pdf)

For an unweighted sample, calculating the standard deviation of the mean from the standard deviation of the distribution is described on Wikipedia here: http://en.wikipedia.org/wiki/Standard_deviation#Standard_deviation_of_the_mean

How do I calculate it for the weighted mean, and how is the expression derived?

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marked as duplicate by Jack D'Aurizio, Hakim, Davide Giraudo, Sami Ben Romdhane, 900 sit-ups a day Jun 13 at 21:26

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

The standard error of the weighted mean is not the same as the weighted standard deviation of the sample. See <math.stackexchange.com/questions/823125/…; –  Avraham Jun 13 at 16:14
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3 Answers 3

I think that your answer is

$$\frac{\sum w_i(\theta_i-\bar{\theta})(\theta_i-\bar{\theta})'}{\sum w_i}$$

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The weighted mean $\bar x=\frac{\sum\limits_{i=1}^n w_ix_i}{\sum\limits_{i=1}^n w_i}$

First, let's find its variance (suppose that $Var(x) =\sigma^2 $):

$$Var(\bar x)=Var\left(\frac{\sum\limits_{i=1}^n w_ix_i}{\sum\limits_{i=1}^n w_i}\right)=\frac{1}{\left (\sum\limits_{i=1}^n w_i \right)^2}Var\left(\sum\limits_{i=1}^n w_ix_i\right)=\frac{1}{\left (\sum\limits_{i=1}^n w_i \right)^2}\sum\limits_{i=1}^n Var\left( w_ix_i\right)=\\=\frac{\sum\limits_{i=1}^n w_i^2}{\left (\sum\limits_{i=1}^n w_i \right)^2}\sigma^2$$

Take the square root of the expression and you get standard deviation of mean.

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TO understand what the weights do, assume that the weights are all integers (if they are rational you can multiply out the LCM of the denominators). To keep it simple assume you have 3 data points with weight 2, 3 and 5.

Your mean is then $$ \frac{2 x_1 + 3 x_2 + 5 x_3}{2 + 3 + 5} $$

Now suppose you go back to your $3$ data points and want to make $x_3$ important (i.e. bias the mean towards $x_3$). One way to do this is to make up additional data by writing $x_3$ 5times. You do the same for $x_2$ except you write it $3$ times and $x_1$ two times.

To be specific, suppose that $x_1=1$, $x_2=0$, $x_3=8$.

$$ \text{Unweighted mean} = \frac{1+0+8}{3} = 3 $$ With weights, your data is now $\left[1,1, 0,0,0,8,8,8,8,8\right]$. So your weighted mean is $$ \text{Weighted mean} = \frac{1+1+0+0+0+8+8+8+8+8}{10} = 4.2$$ See how the mean shifted from $3$ towards $8$ to become $4.2$.

Weights are a way to make some data more important than others.

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