Difference between Standard Deviations?
Hi
In some of my statistics books, they use two versions of the standard deviation for a data set {x} with N measurements. The first version is
^2}<br />
)
where mu is the mean of {x}.
The second version is
^2}<br />
)
where f_i is where we put the mean, so it is a function (this is what it says in my book). This version they use to estimate the standard deviation of {x}, if it is not known beforehand. Unfortunately my book is not very explicit about:
1) What f really is
2) What the difference is between the "normal" way of writing the standard deviation (top one) and the lower one. I thought that in the lower one, the data points do not come from 1 distribution, but rather have their own -- whereas in the top formula, all data comes from 1 single distribution. But I am not sure.
I hope someone will help me by shedding light on these two questions.
Best,
Niles.
Re: Difference between Standard Deviations?
The structure looks the same to me, but they are different as the subscript in 
implies the potential for different values to be subtracted. When expanding the sums we have:
^2} =\frac{1}{N}((x_1-\mu)^2+(x_2-\mu)^2+(x_3-\mu)^2+\dots ))
^2} = \frac{1}{N}((x_1-f_1)^2+(x_2-f_2)^2+(x_3-f_3)^2+\dots))
Re: Difference between Standard Deviations?
Quote:
Originally Posted by
pickslides 
The structure looks the same to me, but they are different as the subscript in
implies the potential for different values to be subtracted. When expanding the sums we have:
Thanks, do you know if I am correct about saying that it might be because the data in #1 comes from the same distribution, and each point in #2 has its own distribution?
Re: Difference between Standard Deviations?
I'm not sure what you are trying to say here.
Re: Difference between Standard Deviations?
So in the first version, all the data points {x} originate from one single distribution with the same mean, whereas in #2 each data points originates from its own "unique" distribution with a unique mean?
Re: Difference between Standard Deviations?
Quote:
Originally Posted by
Niles_M So in the first version, all the data points {x} originate from one single distribution with the same mean, whereas in #2 each data points originates from its own "unique" distribution with a unique mean?
O.K now I understand what you are saying.
For example in the second equation 
came from a distribution with a mean . So we would be finding the standard deviation of a set of points from different distributions (or even different variables).
I need to consider this more, but my initial thinking is - is equation 2 then really finding a standard deviation at all?
Re: Difference between Standard Deviations?
Quote:
Originally Posted by
pickslides O.K now I understand what you are saying.
I need to consider this more, but my initial thinking is - is equation 2 then really finding a standard deviation at all?
My question exactly! The context in which it is presented in my book is: We look at a problem Mp = d, where d is our data, p our parameters and M our "observation matrix" (as mentioned here: Inverse problem - Wikipedia, the free encyclopedia) -- basically an overdetermined problem.
The book talks about an example where we have a vector of data d whose standard deviation we don't know. Then we try and estimate it, and the estimate is
_i)^2}<br />
)
which has the form of version #2, and this is where my question came from: How can they estimate a standard deviation like this?
