TF ErrorAna PropOfErr
Instrumental and Statistical Uncertainties
P=68% = Probability that a measurement of a Gaussian variant will lie within 1
of the meanExample of cosmic counting experiments. Is the variation statistical?
Date | Time (hrs) | Coincidence Counts | Mean Coinc/Hour | |||
---|---|---|---|---|---|---|
9/12/07 | 20.5 | 330 | 2233 | 108.93 | 47 | |
9/14/07 | 21 | 330 | 1582 | 75.33 | 40 | |
10/3/07 | 21 | 330 | 2282 | 108.67 | 48 | |
10/4/07 | 21 | 330 | 2029 | 96.62 | 45 | |
10/15/07 | 21 | 330 | 2180 | 103.81 | 47 | |
10/18/07 | 21 | 330 | 2064 | 98.52 | 45 | |
10/23/07 | 21 | 330 | 2003 | 95.38 | 45 | |
10/26/07 | 21 | 330 | 1943 | 92.52 | 44 |
Taylor Expansion
A quantity which is calculated using quantities with known uncertainties will have an uncertainty based upon the uncertainty of the quantities used in the calculation.
To determine the uncertainty in a quantity which is a function of other quantities, you can consider the dependence of these quantities in terms of a tayler expansion
Consider a calculation of a Table's Area
The mean that the Area (A) is a function of the Length (L) and the Width (W) of the table.
The Taylor series expansion of a function f(x) about the point a is given as
For small values of x (x << 1) we can expand the function about 0 such that
The talylor expansion of a function with two variables
about the average of the two variables is given by
or
The term
represents a small fluctuation of the function from its average
if we ignore higher order terms in the Taylor expansion ( this means the fluctuations are small).Based on the Definition of Variance
We can write the variance of the area
where
is defined as the Covariance between and .Weighted Mean and variance
If each observable (
) is accompanied by an estimate of the uncertainty in that observable ( ) then weighted mean is defined asThe variance of the distribution is defined as
Date | Time (hrs) | Coincidence Counts | Coinc/Hour | ||
---|---|---|---|---|---|
9/6/07 | 18 | 45 | 1065 | 59.2 | |
9/7/07 | 14.66 | 45 | 881 | 60.1 | |
9/9/07 | 43 | 60 | 1558 | 36.23 | |
9/12/07 | 20.5 | 330 | 2233 | 108.93 | |
9/13/07 | 21 | 315 | 2261 | 107.67 | |
9/14/07 | 21 | 330 | 1582 | 75.33 | |
9/18/07 | 21 | 300 | 1108 | 52.8 | |
9/19/07 | 21 | 300 | 1210 | 57.62 | |
9/20/07 | 21 | 300 | 1111 | 52.69 | |
9/21/07 | 21 | 300 | 1012 | 57.62 | |
9/26/07 | 21 | 315 | 1669 | 79.48 | |
9/27/07 | 21 | 315 | 1756 | 83.29 | |
9/29/07 | 24.5 | 315 | 2334 | 95.27 | |
10/3/07 | 21 | 330 | 2282 | 108.67 | |
10/4/07 | 21 | 330 | 2029 | 96.62 | |
10/10/07 | 21 | 315 | 1947 | 92.71 | |
10/15/07 | 69 | 330 | 2180 | 31.59 | |
10/18/07 | 21 | 330 | 2064 | 98.52 | |
10/23/07 | 21 | 330 | 2003 | 95.38 | |
10/26/07 | 21 | 330 | 1943 | 92.52 | |
11/2/07 | 21 | 330 | 2784 | ||
11/5/07 | 69 | 330 | 10251 | 148.57 | |
11/16/07 | 21 | 30 | 3581 | 170.52 |