## 3 Statistical Methods
### Goodness-of-Fit Test
#### 3.2.1 Normal Distribution  
The normal distribution is a probability function that describes how the measured experimental values are distributed; it is characterized by a symmetric distribution where most observations cluster around the central peak, with values tapering off sharply as they move away from the mean [35]. Accordingly, the important parameters in this distribution are the mean and standard deviation. The mean and standard deviation for the normal distribution are calculated as follows [24]:  
\[\mu\approx x_{m}=\overline{x}=\sum_{i=1}^{n}h_{i}x_{i}=\frac{1}{n}\sum_{i=1}^{ n}x_{i} \tag{5}\]  
\[\sigma^{2}\approx s^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-x_{m})^{2} \tag{6}\]  
where \(x_{i}\) is discrete measured values, \(n\) is a limited number, \(h_{i}\) is the rate of occurrence defined as \(1/n\), \(x_{m}\) is the arithmetic mean value (or expectation value), and \(\mu\) is the empirical estimate of the expectation value. Additionally, \(s\) denotes the empirical standard deviation (root mean square deviation, r.m.s.d.), and \(\sigma\) is the standard deviation. These parameters are used when analyzing the experimental data (e.g., AC-BDS). In addition, the probability density function (pdf) and cumulative distribution function (cdf) can be obtained using these values. The two probability distribution functions are expressed as follows [24]:  
\[D_{N}(x)=\frac{1}{\sigma\sqrt{2\pi}}\text{exp}\left(-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right)\text{ for }-\infty<x<\infty \tag{7}\]\[F_{N}(x)=\int_{-\infty}^{x}D_{N}(x)dx \tag{8}\]  
where \(D_{N}(x)\) is the probability density function of normal distribution and \(F_{N}(x)\) is its cumulative distribution function. For a detailed explanation of the normal distribution, please refer to [24].  
Meanwhile, there are important indicators to consider in a normal distribution. These include skewness, which measures the asymmetry of the data distribution relative to the center, and kurtosis, which characterizes how peaked the shape of the data set is. A skewness value closer to 0 indicates that the distribution is more symmetric. For a data point that shows a perfectly symmetric bell-shaped normal distribution, the skewness would be 0, whereas it may have positive skewness or negative skewness otherwise. The skewness, \(S_{k}\), could be calculated as  
\[S_{k}=\frac{\sum{(x_{i}-\mu)^{3}}\cdot p_{i}}{\sigma^{3}} \tag{9}\]  
where \(x_{i}\) represents the measurement value (here, AC-BDS), \(\mu\) is the mean value, \(\sigma\) is the variance, and \(p_{i}\) is the probability of occurrence of the measured value. In addition, the kurtosis provides information on how concentrated the data are around the center. Kurtosis, \(K_{u}\), can be expressed as  
\[K_{u}=\frac{\sum{(x_{i}-\mu)^{4}}\cdot p_{i}}{\sigma^{4}} \tag{10}\]  
and \(K_{u}=3\) implies that the distribution has the same kurtosis as the normal distribution, representing a mesokurtic distribution. While a \(K_{u}\) value being greater to 3 means that the distribution has heavier tails and more outliers, representing a leptokurtic distribution, and \(K_{u}\) being smaller than 3 indicates that the distribution has thinner tails and fewer outliers, representing a platykurtic distribution. It is worth noting that both \(S_{k}\) and \(K_{u}\) values are commonly used during partial discharge (PD) pattern analysis for the classification and identification of defects in electrical insulation [36; 37].