Averaged Histogram

Averaging Raw Histograms

Given raw histogram counts from independent experiments, we can combine them into one histogram with error bars as follows. Let \( x_k^{(i)} \) be the raw histogram counts in bin \( k\) of experiment \( i\) of \( n\). Then the full histogram counts \( y_k \), with corresponding error bars \( \Delta_k \), is

\begin{align*} y_k &= \frac1n \sum_i x_k^{(i)} \ ; \\ \Delta_k &= \sqrt{\frac1{n-1}\left(\langle (x_k^{(i)})^2 \rangle - \langle x_k^{(i)} \rangle^2\right)}\ , \end{align*}

where \( \langle\cdot\rangle = \frac1n \sum_i \cdot \) is the average over the different experiments.

Averaging Histograms Which Already Have Error Bars

Let \( x_k^{(i)} \) raw histogram counts, and suppose that we already have error bars \( \delta_k^{(i)} \) on these counts (e.g., from binning analysis).

The combined histogram \( y_k \), with final corresponding error bars \( \Delta_k \), is

\begin{align*} y_k = \frac1n \sum_i x_k^{(i)} \ ; \\ \Delta_k = \frac1n\,\sqrt{\sum_i \left(\delta_k^{(i)}\right)^2} \ . \end{align*}

(propagation of errors in error analysis in physics: \( \Delta f = \sqrt{ \left(\frac{\partial f}{\partial x}\right)^2 \Delta x^2 + \cdots } \) ).