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used foil strain gauges with a grid length of 3 mm and with gauge factor, e.g., 2.03, this represents an uncertainty of, at most, 0.0203. This may seem like a relatively high value, but its actual effect on the torque calculation is fully eliminated by calibration of measuring points. Then, regarding the achieved accuracy of calibration processes and full temperature compensation of full bridges used, the gauge factors of individual strain gauges are also considered absolutely exact for our purpose. Thus, finally, the recorded data in mV/V are converted to torques in \(Nm\) without the above errors, and therefore, the uncertainty of required mechanical efficiency of the tested gearbox results just from the character of the recordings.  
The following data processing procedure is based on the assumption of a normal distribution of the measured data.If the measured data approach the normal distribution, but there are also less significant side spectra of the measured data frequencies, it is necessary to accept the assumption of using the main peak, which carries essential information to determine efficiency and further consider these data as normally distributed. Figure 10 shows histograms of the measured data, specifically Mode 7, which is demonstrated in Figure 11. Mode 7 is the designation of one mode of the total measured data shown in Section 2.5 in Figure 9. It can be seen in Figure 10 that the data can be considered normal, as they are in accordance with the normality test.
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The recorded data are cropped for each operating point. The output time series \(\{x_{i}\}\) and the input time series \(\{y_{i}\}\), where \(i=1\ldots n\) are used to calculate the efficiency \(\mu\) for each operating point. The efficiency is traditionally expressed as the ratio between the output \(\bar{x}\) and the input \(\bar{y}\):  
\[\frac{\bar{x}}{\bar{y}}, \tag{5}\]  
where \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i};g=\frac{1}{n}\sum_{i=1}^{n}y_{i}\). Figure 9 illustrates that the time series for each operating point repeats infinitely with the same periodicity, and therefore, the time series is considered stationary. In such a case, covariance between any two points of the time series only depends on the distance between these points. Covariance \(\left\{c_{j}\right\}\) of the time series \(\left\{x_{i}\right\}\) a \(\left\{x_{i+j}\right\}\) is estimated as  
\[c_{j}=\frac{1}{n-j}\sum_{i=1}^{n-j}(x_{i}-\bar{x})\left(x_{i+j}-\bar{x}\right), \tag{6}\]  
where \(c_{0}\) is equal to variance of \(\left\{x_{i}\right\}\), Refs. [20,21]. Variance of \(\bar{x}\) is calculated from the covariance matrix \(\left\{c_{\left|i-j\right|}\right\}\), where \(i,j=1\ldots n\) according to the variance sum law  
\[\mathrm{var}\bar{x}=\frac{2}{n^{2}}\sum_{j=1}^{n}(n-j)c_{j}+\frac{c_{0}}{n}. \tag{7}\]  
Var\(\bar{y}\) is calculated likewise [20,21].  
As the returned variances are relatively small compare to the means, the variance formula for the ratio between output and input data in Equation (5), i.e., efficiency, can be stated using the Taylor series expansion as  
\[\mathrm{var}\frac{\bar{x}}{\bar{y}}=\frac{1}{\bar{y}^{2}}\mathrm{var}\bar{x}- 2\frac{\bar{x}}{\bar{y}^{3}}\mathrm{cov}\bar{x},\bar{y}+\frac{\bar{x}^{2}}{ \bar{y}^{4}}\mathrm{var}\bar{y}. \tag{8}\]  
As the point synchronization is not ensured between the output and input series, covariance between \(\{x_{i}\}\) and \(\{y_{i}\}\) is neglected, and the standard deviation on establishing the efficiency is equal to  
\[\frac{\bar{x}}{\bar{y}}\sqrt{\frac{1}{\bar{x}^{2}}\mathrm{var}\bar{x}+\frac{1 }{\bar{y}^{2}}\mathrm{var}\bar{y}} \tag{9}\]  
In this way, the uncertainty is determined respecting the calculation principle of the required efficiency from torques identified by their direct measurement.