Error Propagation

A problem that we encounter when combining data from different images is how the error propagates with our new combined data. When we calculated the variance for our plot we needed to take the difference between two identical images so that we can isolate the noise. Since we combined two different data sets, we are essentially combining the errors of the data sets as well.

$$\bar{x}_c = \bar{x}_a - \bar{x}_b$$

$$s_c^2 = s_a^2 + s_b^2$$

So in essence we calculated $s_c^2$ when we took the variance of our subtracted image. However, we do not want to find the variance of the subtracted image, we want the variance of our original image without the variance due to just the variations in the brightness of the image itself. But since the two original images are identical, we can say that the variances of the original images are identical, so we get the relation

$$s_a^2 = s_b^2$$

so,

$$s_c^2 = 2s_a^2 \;\; \rightarrow \;\; s_a^2 = \frac{1}{2}s_c^2.$$