Naive Limit Combination

When combining limits from two different measurements, the naive combination – which assumes the measurements are completely independent is given by

\[ \sigma_c = \sqrt{\frac{1}{\frac{1}{\sigma_1^2} + \frac{1}{\sigma_2^2}}} = \frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2+\sigma_2^2}} \]

If \(\sigma_1=\sigma_2\), \[ \sigma_c = \frac{\sigma^2}{\sqrt{2\sigma}} = \frac{\sigma}{\sqrt{2}} \]