Procedure

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[edit] Calculation of all the parameters on the table

Let the line we want to fit is y=mx+b,then

\chi^{2}={\sum(\frac{\Delta y_{i}}{\sigma_{i}})}^{2}

\chi^{2}=\sum\frac{1}{\sigma_{i}^{2}} {(y_{i}-mx_{i}-b)}^{2}

To find the value of m and b which will yield minimum value of χ2, We took the derivative of χ2 and equaled it to zero.The equations We got by doing so are

\sum\frac{y_{i}}{\sigma_{i}^{2}}=b\sum\frac{1}{\sigma_{i}^{2}}+m\sum\frac{x_{i}}{\sigma_{i}^{2}}

\sum\frac{x_{i}y_{i}}{\sigma_{i}^{2}}=b\sum\frac{x_{i}}{\sigma_{i}^{2}}+m\sum\frac{x_{i}^{2}}{\sigma_{i}^{2}}

Solving these equations,

Offsetb=\frac{1}{\Delta}(\sum\frac{x_{i}^{2}}{\sigma_{i}^{2}}\sum\frac{y_{i}}{\sigma_{i}^{2}}-\sum\frac{x_{i}}{\sigma_{i}^{2}}\sum\frac{x_{i}y_{i}}{\sigma_{i}^{2}})

Slopem=\frac{1}{\Delta}(\sum\frac{1}{\sigma_{i}^{2}}\sum\frac{x_{i}y_{i}}{\sigma_{i}^{2}}-\sum\frac{x_{i}}{\sigma_{i}^{2}}\sum\frac{y_{i}}{\sigma_{i}^{2}})

\Delta=\sum\frac{1}{\sigma_{i}^{2}}\sum\frac{x_{i}^{2}}{\sigma_{i}^{2}}-{(\sum\frac{x_{i}}{\sigma_{i}^{2}})}^{2}

We have 81 data points (fluxes in different bands).We took fluxes through 1B1 as xis and yis as 1A1,1A2,1B2,2A1,2A2,2B1 and 2B2 respectively for each fit.


\sigma_{i}^{2}= {(\sigma_{xi}/1.645)}^{2}+{(\sigma_{yi}/1.645)}^{2}

Substituting the values of m and b, we can calculate the minimum value of \chi^{2}=\sum\frac{1}{\sigma_{i}^{2}} {(y_{i}-mx_{i}-b)}^{2}

Hence the Reduced \chi_{\nu}^{2}=\frac{\chi^{2}}{80}

where number of degrees of freedom=80, one less than the number of observations.

For zero-offset, we took the line y=mx and proceeded in same way as above.

Slope for zero-offset m=\frac{\sum\frac{x_{i}y_{i}}{\sigma_{i}^{2}}}{\sum\frac{x_{i}^{2}}{\sigma_{i}^{2}}}

\chi^{2}=\sum\frac{1}{\sigma_{i}^{2}} {(y_{i}-mx_{i})}^{2}

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