What is the best way to combine uncertainties in measurements?

[Littlepig]

Homework Statement

Suppose I have N [tex]x_{i}[/tex] measures with a given uncertainty [tex]\Delta x_{i}[/tex].

I want to have the best estimate for [tex]\bar{x}[/tex] and its uncertainty [tex]\bar{\Delta x}[/tex]


2. Homework Equations /3. The Attempt at a Solution

Well, I'm not exactly sure because or I can have a mean value of x_i and uncertainty given by standard mean and standard deviation/N formulas and I disregard the measurement uncertainties, or I use the (mean of the [tex]\Delta x_{i}[/tex])/N to the uncertainty of [tex]\bar{x}[/tex] and disregard how x_i are close to the mean [tex]\bar{x}[/tex].

Is there any way of joint both uncertainties together?

Even if you don't explain it in here, can you give me literature where you know where I find it?

Thanks,
littlepig

 

[ideasrule]

To find x-bar, you just average the individual measurements; nothing fancy about that. To find the standard deviation of this mean, you divide the standard deviation of each measurement by the square root of N.

 

[Littlepig]

To find x-bar, you just average the individual measurements; nothing fancy about that.

I agree

To find the standard deviation of this mean, you divide the standard deviation of each measurement by the square root of N.

I don't agree and I counter example with this:

assume:
Example 1:
x_1=15, delta x_1=0.2
x_2=9, delta x_2=0.1
x_3=3, delta x_3=0.2

Example 2:
x_1=9.2, delta x_1=0.2
x_2=9, delta x_2=0.1
x_3=8.8, delta x_3=0.2

By your method, both mu and sigma are the same for both examples, however, I think we both agree that example 2 should have smaller uncertainty!

 

[Littlepig]

bump, At least give me a name of a book to search on...xD please...

 

[Mindscrape]

I'm not really sure what you've done in example 1. You do want to look at the standard deviation in the mean, as ideasrule said. If you're confused see if you can find Error Analysis by Taylor.