# Monitoring and Improving Aircraft Maintenance Processes Using IT Systems
## 5 Assessment of the Usefulness of the Collected Data
### Examination of Probability Distributions of Maintenance Tasks  
The identification of the density function of the distribution of a random variable is one of the main issues in the application of statistical methods. Traditional identification methods rely on assuming a predetermined type of distribution of a random variable and then specifying the parameters that define that distribution. The proprietary probability distribution identification algorithm is presented below (Figure 6). Figure 6: An algorithm for identifying the distribution of a maintenance task. The probability values for individual elements in the sample were estimated using the Median Rank method. The calculation involves solving a binomial distribution equation [7; 45]. This method calculates the probability of the ii-th event in a sample of nn items at a 50% confidence level according to the following equation (the calculation of the equation presented above requires the use of numerical methods):  
\[0.5=\sum_{k=i}^{n}\binom{n}{k}Z^{k}(1-Z)^{n-k} \tag{11}\]  
The estimation of distribution parameters was carried out based on the least squares method. The method of least squares fits a straight line to a set of data points so as to minimize the sum of the squares of the points' deviations from the straight line. For the Y-axis linear regression (RRY) method, the sum of the squares of the vertical deviations is minimized; in the case of the RRX method, the sum of the squares of the horizontal deviations is minimized [46; 47]. The choice between the RRY and RRX method depends on the direction of expected deviations. Since in the analysis of the duration of technical tasks, the precisely known values are the measured durations on the X-axis and the Y-axis Rank Regression method that will be used in further analysis. Suppose a set of data pairs \((x_{1},\ y_{1}),\ (x_{2},\ y_{2}),\ \ldots,\ (x_{n},\ y_{n})\) was determined, plotted, and all values of \(x\) were known exactly. The straight line best describing the considered data is defined by the equation \(y=\hat{a}+\hat{b}x\). The values of \(\hat{a}\) and \(\hat{b}\) are estimated as follows:  
\[\hat{a}=\frac{\sum_{i=1}^{n}y_{i}}{n}-\hat{b}_{i}\frac{\sum_{i=1}^{n}x_{i}}{n} =\overline{y}-\hat{b}\overline{x} \tag{12}\]  
\[\hat{b}=\frac{\sum_{i=1}^{n}x_{i}y_{i}-\frac{\sum_{i=1}^{n}x_{i}\sum_{i=1}^{n} y_{i}}{n}}{\sum_{i=1}^{n}x_{i}^{2}-\frac{\left(\sum_{i=1}^{n}x_{i}\right)^{2}}{n}} \tag{13}\]  
In order to estimate the parameters of the distributions for the tested maintenance tasks, formulas for the coordinates and parameters for the four considered probability distributions were determined and summarized in Table 3.  
where  
* \(F(t_{i})\)--probability value estimated using the Median Rank method;
* \(\mu,\ \sigma,\ \gamma\)--are the parameters of the scale, shape, and position of the probability distribution, respectively;
* \(\Phi\)--the distribution function of the standard normal distribution, N(0, 1);
* \(\Phi^{-1}\)--the inverse distribution function of the standard normal distribution, N(0, 1). The coefficient of determination \(R^{2}\) informs us about what part of the variability (variance) of the explained variable in the sample coincides with the correlations with the variables included in the model [48]. It is, therefore, a measure of the extent to which the model fits the sample. The coefficient of determination takes values in the range (0, 1); the