## 3 Methods and Data
### Assumptions for Quantitative Analysis and Probabilistic Approach  
The collected and processed data are supposed to serve for calculations, resulting in a series of discrete values representing actual efficiencies of construction works. The calculated efficiencies should then be ordered according to the types of work. The discrete empirical data should be processed to obtain an interval series (or so-called frequency distribution) which, in turn, allows for fitting a continuous distribution of probability. The range of empirical data must be divided into intervals, preferably of equal width. The next step is fitting continuous distribution to the interval series. One may consider different types of continuous distribution, such as beta, Kumaraswamy, metalog, quantile-parametrized, trapezoidal, triangular, or others (continuous distributions are widely discussed in statistical literature, e.g., [31; 32; 33]).  
In this study, the triangular distribution was chosen due to its simplicity and ease of application. Moreover, the triangular distribution is expected to be understandable for practising construction engineers.  
To provide examples of how the triangular distribution is used in engineering, some applications are mentioned in this paper: analysing and examining cost estimates for electrical services contracts in government clinics in Hong Kong [34], deriving the occurrence probabilities of major construction project risk factors for proactive scheduling [35], estimating the duration of horizontal directional drilling works [36], assessing the duration of construction works, such as finishing plastering [37], estimating the duration of construction tasks for project evaluation and review techniques (PETT) [38], and analysing construction cost risk for practical application in the bidding process [39].  
The triangular distribution is a continuous probability distribution named after its triangle-shaped density function. Some issues with its application are discussed in [40]. This distribution is characterised by three values: the minimum, peak, and maximum (referred to as \(a\), \(m\), and \(b\), respectively), and it is a simple and suitable model for skewed distributions. By defining finite lower and upper limits, unwanted extreme values can be avoided when the distribution is applied. The general form of the density function for the triangular distribution is provided below.  
\[f(x)=\begin{cases}\begin{aligned} & 0,\ x<a\\ &\frac{2(x-a)}{(b-a)(m-a)},\ \ x<a;\ m\end{aligned}\\ &\frac{b-a}{b-a},\ \ x=m\\ &\frac{2(b-x)}{(b-a)(m-a)},\ \ x\in(m;b>\\ & 0,\ x>b\end{aligned}\end{cases} \tag{1}\]Following this, the expected value \(E(X)\), median \(Me\), standard deviation \(\sigma\), and mode \(D\) can be computed, respectively. The equations for these calculations are given below.  
\[E(X)=\frac{a+m+b}{3} \tag{2}\]  
\[Me=\begin{cases}a+\frac{\sqrt{(b-a)(m-a)}}{\sqrt{2}},&m\geq\frac{b-a}{2}\\ b-\frac{\sqrt{(b-a)(b-m)}}{\sqrt{2}}\;,&m\leq\frac{b-a}{2}\end{cases} \tag{3}\]  
\[\sigma=\sqrt{\frac{a^{2}+m^{2}+b^{2}-ab-am-mb}{18}} \tag{4}\]  
\[D=m \tag{5}\]  
On the basis of the fitted distribution, it is possible to assess the probability of obtaining an efficiency that is not less than a certain value, which can be denoted as \(x_{c}\) and referred to as critical. The formal notation for this is given below:  
\[P(X>x_{c})=\int_{x_{c}}^{a}f(x)dx \tag{6}\]  
Assuming the distribution allows for assessing a certain value of efficiency for which a probability is assumed. Furthermore, it is possible to compute work duration, work output, or gang size with the assumed probability. Examples of applications for formwork assembly works based on real-life data are presented in Section 4 of this paper.