Poisson errors describe the intrinsic statistical uncertainty of discrete event counts when events occur independently at a constant average rate, as modeled by the Poisson distribution.
Poisson counting model and the meaning of poisson errors
A count \(N\) collected over a fixed exposure (time, area, volume, mass, or integrated luminosity) is often modeled as \(N \sim \text{Poisson}(\lambda)\), where \(\lambda\) is the expected number of events for that exposure. The defining property is equality of mean and variance: \[ \mathbb{E}[N] = \lambda,\qquad \mathrm{Var}(N) = \lambda. \] Poisson errors refer to the standard deviation implied by this variance.
Core rule (counts): \(\sigma_N = \sqrt{\lambda}\). With an observed count \(N\), the plug-in estimate is \(\sigma_N \approx \sqrt{N}\).
Relative Poisson error: \[ \frac{\sigma_N}{N} \approx \frac{1}{\sqrt{N}}\quad (N>0), \] showing that precision improves slowly; increasing the exposure by a factor of 4 typically halves the relative counting uncertainty.
Visualization of poisson errors in counts and relative uncertainty
Poisson errors for rates and derived quantities
Many measurements are functions of one or more independent Poisson counts. Variances add for independent random variables, so poisson errors propagate naturally by quadrature.
Rate from a single count. For exposure time \(T\) (or any known exposure factor), the estimated rate is \(r = N/T\). With \(\sigma_N \approx \sqrt{N}\), \[ \sigma_r \approx \frac{\sqrt{N}}{T}. \]
Sum of independent counts. If \(N_1,\dots,N_m\) are independent Poisson counts, then \(N_{\text{tot}}=\sum_i N_i\) has \[ \sigma_{N_{\text{tot}}}^2 \approx \sum_{i=1}^m \sigma_{N_i}^2 \approx \sum_{i=1}^m N_i, \qquad \sigma_{N_{\text{tot}}} \approx \sqrt{\sum_i N_i}. \]
Background subtraction. For a signal region count \(S\) and a background region count \(B\) scaled by a known factor \(\alpha\), \[ N_{\text{net}} = S - \alpha B, \qquad \sigma_{N_{\text{net}}} \approx \sqrt{\sigma_S^2 + (\alpha\,\sigma_B)^2} \approx \sqrt{S + \alpha^2 B}. \] The net count can be negative from statistical fluctuation; the variance formula remains a variance statement under the model.
General propagation (delta method). For a smooth function \(y=f(N)\) of a single Poisson count, \[ \sigma_y \approx \left|\frac{dy}{dN}\right|\sqrt{N}, \] and for several independent counts \(N_i\), \[ \sigma_y^2 \approx \sum_i \left(\frac{\partial f}{\partial N_i}\right)^2 \sigma_{N_i}^2 \approx \sum_i \left(\frac{\partial f}{\partial N_i}\right)^2 N_i. \]
Small counts and asymmetric confidence intervals
The approximation \(\sigma_N \approx \sqrt{N}\) is most reliable when counts are moderate to large. For small counts, poisson errors are inherently asymmetric for the underlying mean \(\lambda\), and exact confidence intervals are often preferred.
A standard exact interval for the Poisson mean \(\lambda\) uses chi-square quantiles. For confidence level \(1-\alpha\), \[ \lambda_{\text{low}} = \begin{cases} 0, & N=0,\\[4pt] \tfrac12\,\chi^2_{2N,\,\alpha/2}, & N\ge 1, \end{cases} \qquad \lambda_{\text{high}} = \tfrac12\,\chi^2_{2(N+1),\,1-\alpha/2}, \] where \(\chi^2_{\nu,\,p}\) denotes the \(p\)-quantile of a chi-square distribution with \(\nu\) degrees of freedom.
The symmetric “\(N \pm \sqrt{N}\)” interval is a normal-approximation heuristic. The chi-square interval above remains well-defined at \(N=0\) and captures asymmetry when counts are small.
| Observed count \(N\) | Normal-style heuristic for \(\lambda\) (about 1σ) | Exact 68.27% CI for \(\lambda\) (chi-square) |
|---|---|---|
| 0 | 0.000 to 0.000 | 0.000 to 1.841 |
| 1 | 0.000 to 2.000 | 0.173 to 3.300 |
| 2 | 0.586 to 3.414 | 0.708 to 4.638 |
| 5 | 2.764 to 7.236 | 2.840 to 8.382 |
| 10 | 6.838 to 13.162 | 6.891 to 14.267 |
| 25 | 20.000 to 30.000 | 20.034 to 31.067 |
The \(N=0\) row illustrates a common practical issue: poisson errors cannot be represented by a symmetric “±√N” band around zero, while the upper confidence bound remains positive.
Interpretation and common pitfalls
Poisson errors represent random counting uncertainty under an independence-and-constant-rate model; systematic effects (changing rate over time, detector dead time, missed events, misclassification) add additional uncertainty that does not follow \(\sqrt{N}\).
Overdispersion (observed variability larger than Poisson) can arise from clustering, heterogeneity, or time-varying rates; in such cases poisson errors underestimate uncertainty and alternative models (for example, a negative binomial model) or empirical variance estimation becomes appropriate.
Weighted counts (non-integer event weights) are not Poisson counts; a common approximation for a weighted sum \(W=\sum w_i\) is \(\sigma_W \approx \sqrt{\sum w_i^2}\), which differs from \(\sqrt{W}\) when weights vary.