Logik
Feb1-07, 04:28 PM
1. The problem statement, all variables and given/known data
I have a project for one of my class and I have been given a sheet to do the statistical analyst of my data. I am not convince this sheet is proper and I need someone to look over it it. I don't understand where my Delta R goes...
2. Relevant equations
\chi^2 =-\frac{1}{2} \sum_{i=1}^{N}{\left(\frac{y_i-ax_i}{\sigma_i^2}\right)^2} \\
\frac{\partial \chi^2}{\partial a}
= \sum_{i=1}^{N}{\frac{x_i}{\sigma_i^2}(y_i-ax_i)} = 0 \\
\sum_{i=1}^{N}{\frac{x_i y_i}{\sigma_i^2}}
= a\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} \\
a= \sum_{i=1}^{N}{\frac{x_iy_i}{x_i^2}} \\
\sigma_a^2 = \sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} S^2 \\
S^2
= \frac{1}{N-1}\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}} \\
\sigma_a^2 =
\frac{1}{N-1}\left(\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}}\r ight)\left(\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}}\right) \\
r_n^2 = \left(n-\frac{1}{2}\right)\lambda R \\
a= \sum_{n=1}^{15}{\frac{r_n^2\left(n-\frac{1}{2}\right)}{(r_n^2)^2}} \\
\sigma_a^2
= \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2} {\sigma_r^2}}\right)\left(\sum_{i=1}^{N}{\frac{(r_ n^2-a\left(n-\frac{1}{2}\right))^2}{\sigma_r^2}}\right) \\
\Psi(x) = x^2 \\
\sigma_\Psi^2 = (\frac{\partial}{\partial x}(x^2)\Delta x)^2 = 4x^2(\Delta x)^2 \\
\sigma_{r_n^2} = 4r_n^2(\Delta r)^2 \\
\sigma_a^2 = \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2} {4r_n^2}}\right)/\left(\sum_{i=1}^{N}{\frac{(r_n^2-a\left(n-\frac{1}{2}\right))^2}{4r_n^2}}\right)
3. The attempt at a solution
The error on r comes from the fact it's a measurement and we square it. I think Delta R should be there somewhere even though it is constant...
I have a project for one of my class and I have been given a sheet to do the statistical analyst of my data. I am not convince this sheet is proper and I need someone to look over it it. I don't understand where my Delta R goes...
2. Relevant equations
\chi^2 =-\frac{1}{2} \sum_{i=1}^{N}{\left(\frac{y_i-ax_i}{\sigma_i^2}\right)^2} \\
\frac{\partial \chi^2}{\partial a}
= \sum_{i=1}^{N}{\frac{x_i}{\sigma_i^2}(y_i-ax_i)} = 0 \\
\sum_{i=1}^{N}{\frac{x_i y_i}{\sigma_i^2}}
= a\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} \\
a= \sum_{i=1}^{N}{\frac{x_iy_i}{x_i^2}} \\
\sigma_a^2 = \sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}} S^2 \\
S^2
= \frac{1}{N-1}\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}} \\
\sigma_a^2 =
\frac{1}{N-1}\left(\sum_{i=1}^{N}{\frac{x_i^2}{\sigma_i^2}}\r ight)\left(\sum_{i=1}^{N}{\frac{(y_i-ax_i)^2}{\sigma_i^2}}\right) \\
r_n^2 = \left(n-\frac{1}{2}\right)\lambda R \\
a= \sum_{n=1}^{15}{\frac{r_n^2\left(n-\frac{1}{2}\right)}{(r_n^2)^2}} \\
\sigma_a^2
= \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2} {\sigma_r^2}}\right)\left(\sum_{i=1}^{N}{\frac{(r_ n^2-a\left(n-\frac{1}{2}\right))^2}{\sigma_r^2}}\right) \\
\Psi(x) = x^2 \\
\sigma_\Psi^2 = (\frac{\partial}{\partial x}(x^2)\Delta x)^2 = 4x^2(\Delta x)^2 \\
\sigma_{r_n^2} = 4r_n^2(\Delta r)^2 \\
\sigma_a^2 = \frac{1}{14}\left(\sum_{n=1}^{14}{\frac{(r_n^2)^2} {4r_n^2}}\right)/\left(\sum_{i=1}^{N}{\frac{(r_n^2-a\left(n-\frac{1}{2}\right))^2}{4r_n^2}}\right)
3. The attempt at a solution
The error on r comes from the fact it's a measurement and we square it. I think Delta R should be there somewhere even though it is constant...