Measurement of a radioactive decay

The number $A$ of beta particles per second (activity) correspond to the decay rate of the radioactive atoms $A = - \dot N = \lambda N_0 \exp(-\lambda t)$ The radiation is homogeneously distributed over the whole sphere with area $S_0 = 4 \pi r^2$ . The Geiger counter detects only the particles inside the area $S = \pi d^2/4$ , so that the mean counting rate is $\dot n = \frac{S}{S_0} A = \frac{d^2}{16 r^2} \lambda N_0 \exp(-\lambda t)$ The total number of counts, $\Delta n$ , for a time intervall $\Delta t = 1\,\text{s}$ can be approximated by $\Delta n \approx \dot n \Delta t$ provided that $\Delta t \ll 1/\lambda$ applies. For the logarithmic number $\log(\Delta n)$ results in a linear time dependence $\begin{aligned} \log(\Delta n) &= \log\left(\frac{d^2}{16 r^2} \lambda N_0 \Delta t \right) - \lambda t \\ &= A + B \cdot t \\ \Rightarrow \quad N_0 &= -\frac{16 r^2}{d^2 \Delta t} \frac{\exp(A)}{B} \end{aligned}$ So, if you present the results as a semi-logarithmic plot, you can use points to draw a regression line:

In this case, the regression line correspond to the parameters $\begin{aligned} A &\approx 6.52 \pm 0.14 \\ B &\approx -0.0718 \pm 0.0093 \,\frac{1}{\text{s}}\\ \Rightarrow \quad N_0 &\approx (0.9 \pm 0.2) \cdot 10^{9} \end{aligned}$