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: $\Delta f = \sqrt{ \left(\frac{\partial f}{\partial x}\right)^2 \Delta x^2 + \cdots }$ ).