## 4 FRA, Statistical Analysis
### Statistical Indicators and Results  
The statistical indicators can reveal the changes between two FRA data sets \(X(i)\) and \(Y(i)\) numerically. The CC has been recommended as a numerical indicator to measure the association of FRA signatures in previous studies [28]. The ASLE is one of the highly recommended statistical indicators to evaluate the absolute difference between two FRA data sets [28; 30; 31]. The use of SD has been proposed to calculate the variation among two FRA data sets. According to [26; 32], the MSE, also known as sum, squared error (SSE), has a high sensitivity to changes in the FRA signatures. The RMSE is similar to the SD indicator. Compared with the MSE indicator, the RMSE is more appropriate to quantify the difference between two FRA traces. The absolute difference (DABS) metric is similar to ASLE, except that it does not perform logarithmic data conversion [32; 33]. COVAR is used to measure the connection between two data sets while MM is very sensitive to the deviation of the response amplitude [32; 33; 34]. The reliance on one statistical indicator to diagnose FRA signatures is not satisfactory. For instance, the SD value is mainly controlled by the large data in the FRA set with less contribution from other data which affects its reliability. The CC cannot be used as a diagnostic indicator in some cases while the application of ASLE to diagnose FRA signatures still needs further research. Furthermore, the use of individual indicators does not reflect the actual fault level and an inconsistent trend of these indicators with the fault level has been reported in the literature [35; 36]. The simultaneous use of several statistical parameters improves the reliability of the fault diagnosis process based on FRA measurement. Therefore, it is recommended to use a set of statistical parameters in a complementary way to increase confidence in the final decision. The selected statistical indicators in this paper are given by questions (1 to 8) in which \(X\) and \(Y\) represent the two FRA data sets to be compared and \(N\) is the total number of points within each set. The calculated values of these statistical indicators for normal and faulty conditions of the tested transformer and IM with a comprehensive comparison are presented in Tables 7-9. The color coding refers to the threshold limits of the variation between the normal response and the fault responses. This method is used to evaluate and compare the indicators' ability for fault detection. The green color refers to the non-visible variation between the normal response and the fault response which can be detected by the indicator. The red color refers to the obvious variation between the normal and fault responses which can be noted in the indicator value. Thus, based on the notable indicator value, the variation can be detected and expressed by the red color. \begin{table}
\begin{tabular}{l l l} \hline \hline
**FRA Regions** & **Transformers** & **IMs** \\ \hline LF & 20 Hz–2 kHz & 20 Hz–1 kHz \\ MF & 2 kHz–20 kHz & 1 kHz–80 kHz \\ HF & 20 kHz–1 MHz & 80 kHz–1 MHz \\ SHF & \textgreater{}1 MHz & \textgreater{}1 MHz \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The common FRA regions limits for power transformers and the proposed regions for IMs. The statistical indicators calculated in Tables 7-9 are analyzed in Table 10. The analysis was based on the indicator sensitivity to a fault (variation between the normal and fault responses), and the difference between its sensitivity to a fault in the power transformer and the IM.