## 2 Methodology
### Measurement Uncertainty  
#### 2.5.1 Measurement Uncertainty According to the First Averaging Method  
For each frequency band from the measurement, we obtain a decay curve. Since we determine the reverberation time from each decay curve (for a given frequency band), we obtain \(N\) reverberation time results determined as the product of sound source arrangements by the number of measurement points and the number of measurement repetitions at each point. Thus, we have a finite set of measured values \(\mathrm{T_{1},\ T_{2},\ \ldots,\ T_{N}}\) from which we find the best approximation for T and \(\sigma\) (standard deviation). As the best approximation, we adopt the numbers that provide the greatest probability of obtaining \(\mathrm{T_{1},\ T_{2},\ \ldots,\ T_{N}}\), i.e., the numbers that provide the greatest value of the probability given by the formula:  
\[P_{T,\sigma}(\mathrm{T_{1},\ \ldots,\ T_{N}})=\frac{1}{\sigma^{N}}e^{-\sum \left(T_{i}-T\right)^{2}/2\sigma^{2}} \tag{4}\]  
Obviously, it can be easily evidenced that the best approximation of the true value T is the average of \(N\) measurements (trivial proof) expressed by the formula:  
\[\overline{\mathrm{T}}=\frac{\sum\mathrm{T_{i}}}{N} \tag{5}\]  
By calculating the derivative of the function (4) with respect to \(\sigma\) and equating it to zero, we obtain the value of \(\sigma\) which ensures the maximum probability of Equation (4) and is the best approximation of \(\sigma\). \[\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\mathrm{T_{i}-T})^{2}} \tag{6}\]  
Since the true value T is unknown, in practice this value is replaced by the best approximation described by Equation (5), and the following is obtained:  
\[\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\mathrm{T_{i}-T}\right)^{2}} \tag{7}\]  
Naturally, it is often argued that the best approximation of \(\sigma\) is not described by Equation (7) but by its multiplication by \(\sqrt{\frac{N}{N-1}}\), which ultimately yields the estimate:  
\[\sigma_{\mathrm{T}}=u(T)=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}\left(\mathrm{T_{i} -T}\right)^{2}} \tag{8}\]  
The following is assumed in all mathematical formulas:  
\(N\)--number of measurements;  
\(\mathrm{T_{i}}\)--i-th measurement of reverberation time;  
\(\overline{\mathrm{T}}\)--average reverberation time;  
\(\sigma\)--sample standard deviation. Therefore, it can be concluded that the standard deviation expressed by the Formula (8) characterizes the average uncertainty of the measurement results. 2.5.2. Measurement Uncertainty According to the Second Averaging Method  
The uncertainty of measurement for the averaging method presented in this way is determined as the uncertainty expressed by the following formula:  
\[u_{c}=\sqrt{\left(\frac{\partial T(\Delta L,\ \Delta\tau)}{\partial\Delta L} \right)^{2}u^{2}(L)+\left(\frac{\partial T(\Delta L,\ \Delta\tau)}{\partial\Delta\tau} \right)^{2}u^{2}(\tau)} \tag{9}\]  
Using the estimated model (3), we determine \(\Delta L\), which depends only on the direction coefficient \(b_{1}\). Thus, we can adopt the estimated standard error for this coefficient as the estimation uncertainty \(u(L)\). And to estimate the uncertainty \(u(\tau)\), which is the average of the averages from all reverberation curves, we use the following relation:  
\[u(\tau)=\frac{\sigma_{\tau}}{\sqrt{N}} \tag{10}\]  
where \(\sigma_{\tau}\)--standard deviation of sound decay time measurements for a 30 dB drop; \(N\)--number of measurements of the decay curve. 2.5.3. Proof of Formula (10)  
It is generally known that the best approximation of measurement results is the average, which was demonstrated on the example of average reverberation time described by Formula (5). Thus, the calculated quantity described by Formula (11),  
\[\overline{\tau}=\frac{\tau_{1}+\ldots\ +\tau_{N}}{N} \tag{11}\]  
is a function of the measured quantities \(\tau_{1},\ldots,\tau_{N}\). Thus, we can easily find the distribution of the variable \(\overline{\tau}\) using the error transfer rule.