# Comfort Distance--A Single-Number Quantity Describing Spatial Attenuation in Open-Plan Offices
## 2 Materials and Methods  
We utilized the measurement data of \(D_{2,\rm S}\) and \(L_{\rm P,A,S,4m}\) of Keranen and Hongisto [17], which represents well the range of values where the SNQs of ISO 3382-3 could usually lie. They reported altogether 26 measurements in acoustically different open-plan offices (Table 1, Figure 2). Each measurement corresponds to a single path in one direction.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Office** & \(L\) & \(L_{\rm P,A,B}\) & \(L_{\rm P,A,S,4m}\) & \(D_{\rm 2,\rm S}\) & \(r_{\rm D}\) \\
**ID** & **[m]** & **[dB]** & **[dB]** & **[dB]** & **[m]** \\ \hline
1 & 16 & 39 & 53.8 & 4.0 & 14.2 \\
2 & 27 & 45 & 57.2 & 4.2 & 18.5 \\
3 & 16 & 42 & 52.5 & 4.6 & 9.5 \\
4 & 60 & 41 & 49.4 & 5.7 & 5.6 \\
5 & 18 & 35 & 50.9 & 6.0 & 15.4 \\
6 & 36 & 44 & 52.6 & 6.2 & 5.4 \\
7 & 19 & 31 & 47.5 & 6.3 & 13.8 \\
8 & 19 & 39 & 52.4 & 6.4 & 10.3 \\
9 & 42 & 40 & 54.4 & 6.7 & 15.3 \\
10 & 23 & 39 & 43.4 & 9.0 & 5.5 \\
11 & 34 & 35 & 48.3 & 9.2 & 9.9 \\
12 & 32 & 37 & 49.4 & 9.4 & 9.3 \\
13 & 36 & 31 & 46.5 & 11.4 & 9.5 \\
14 & 35 & 31 & 47.1 & 11.5 & 6.2 \\
15 & 70 & 31 & 49.0 & 11.7 & 8.1 \\
16 & 27 & 33 & 49.9 & 12.4 & 10.0 \\ A & 18 & 34 & 47.4 & 4.9 & 16.2 \\ B & 33 & 32 & 49.1 & 6.0 & 15.3 \\ C & 69 & 29 & 44.0 & 6.4 & 11.4 \\ D & 17 & 38 & 50.4 & 6.4 & 11.9 \\ E & 23 & 34 & 47.9 & 7.8 & 8.8 \\ F & 16 & 35 & 51.5 & 8.2 & 11.1 \\ G & 36 & 32 & 50.3 & 9.3 & 14.0 \\ H & 28 & 38 & 50.3 & 9.4 & 6.0 \\ I & 30 & 38 & 53.9 & 9.0 & 9.7 \\ J & 33 & 39 & 49.3 & 11.6 & 9.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The data of the 26 offices of Ref. [17] used in our study. \(L\) is the length of the office in the direction of the measurement path. The other quantities were defined in Section 1. The notation (both numbers and letters) is adopted from Ref. [17].  
We determined the comfort distance for the 26 offices of Table 1 using 21 different values for the comfort criterion level, \(L_{\text{P,A,C}}\). The values ranged from 30 to 50 dB in 1-dB steps. The calculation was made using Equation (2). The method is depicted in Figure 3 for office ID 1. This way, each office was assigned by 21 different comfort distances. Simple statistics (mean, minimum, maximum 68% confidence intervals) were determined at every comfort criterion level for the distribution of comfort distances over the 26 offices.  
It was justified to presume in general that comfort distance should not be larger than the length of the office. Therefore, we calculated for every \(L_{\text{P,A,C}}\) value the probability \(P\) that the comfort distance \(r_{\text{C}}\) was larger than the room length within the sample, by  
\[P=\frac{N_{0}}{N} \tag{3}\]  
where \(N_{0}\) is the number of offices (out of 26 offices in question) fulfilling the adverse criterion \(r_{\text{C}}>L\) and \(N\) is the total number of offices (26). The room length \(L\) of each office is given in Table 1. The desirable situation is \(P=0\). It indicates a high probability that the comfort distance is shorter than a room in most offices beyond the sample of Table 1 since the sample of Table 1 represents a broad range of acoustically different offices.  
Figure 3: Example of the comfort distances, \(r_{\text{C}}\), obtained for office ID 1 for different comfort criterion levels, \(L_{\text{P,A,C}}\), from 30 to 50 dB.  
Figure 2: Measured A-weighted SPL of speech, \(L_{\text{P,A,S}}\), as a function of distance, \(r\), from the omnidirectional loudspeaker, for the 26 open-plan offices of Table 1. The number of measurement positions per office ranged from 4 to 13. The required minimum number of positions is four.  
Our second purpose involved a comparison between previous studies. Some important previous studies are described in Table 2.