# Evaluation of the Variability of Vibration Measurement Results in Rolling Bearing Quality Control
## 4 Results and Discussion  
The following commonly used statistical parameters were calculated for a population of data specific to each system: arithmetic mean, mean square deviation for a single measurement, spread and final result of a single measurement with a probability of 0.95. Arithmetic mean and mean square deviation are the most common indicators of the position and variability of a population of measurement variables. The arithmetic mean is an estimator of the expected value of the distribution of a characteristic in the population. It can be used to estimate the expected mean outcome of a random experiment when repeated many times. The mean square deviation is an estimator of the standard deviation. The standard deviation is a measure of the dispersion of the results around the expected value. The spread characterises very well the area of variability of the attribute under study, taking into account only the extreme results (minimum and maximum). The final result of a single measurement is included in the article to make it easier to visualise the result obtained and its uncertainty associated with the statistical analysis of a series of measurement results [24; 25; 26]. Furthermore, the spread asymmetry expressed with the skewness parameter and the spread flattening expressed with kurtosis were tested. The statistic \(\chi^{2}\) was calculated and compared with the critical value selected from the spread \(\chi^{2}\) tables to check the obtained spreads' conformity with the normal spread. The obtained spreads' parameters are presented in Tables 2-4, while the result histograms divided by frequency bands are presented in Figures 3-5. The available statistical data (aside from determining individual features of the measurement series' spread for each system) also allow for conducting comparative tests between the spreads' most important parameters, such as the arithmetic mean or variance. Two additional tests were also conducted in the analysis: a comparative test of two mean values from two populations and a comparative test of variances from two populations with normal spreads. The results of the comparative test of mean values of all combinations of the three tested systems, conducted for three frequency bands, \begin{table}
\begin{tabular}{l c c c c} \hline
**Subassembly** & **System 1** & **System 2** & **System 3** \\ \hline \multirow{2}{*}{Spindle} & & hydrodynamic & air bearing spindle & hydrodynamic \\  & & pneumatic, on a & pneumatic and & spindle \\ \multirow{2}{*}{Clamp} & \multirow{2}{*}{type} & swivel arm with an & mechanical, & pneumatic, \\  & & offset joint, & constantly coaxial & constantly coaxial \\ \multirow{4}{*}{Sensor} & \multirow{2}{*}{} & contact with the & three-point & three-point & circumferential \\  & bearing & & & applied during the \\ \multirow{2}{*}{Sensor} & \multirow{2}{*}{} & stationary & stationary & measurement \\ \multirow{2}{*}{Signal processing} & \multirow{2}{*}{} & analogue and digital & analogue & analogue and digital \\ \hline \end{tabular}
\end{table} 
Table 1: Specification of the tested systems’ most important subassemblies. Figure 2: General view of the test stand. (**a**) test measurement system 1, (**b**) test measurement system 2, (**c**) test measurement system 3.  
are presented in Table 5. The statistic's critical value was chosen based on the adopted probability \(p\) = 0.95. The results of the comparative test of mean values of all combinations of the three tested systems are presented in Table 6. The statistic's critical value was chosen based on the adopted probability \(p\) = 0.95 and the degrees of freedom related to the size of the measurements. Due to the fact that the calculated statistics \(\chi^{2}\) (which are decisive for treating a spread as normal) were based in the critical area for all systems in the low frequency band, this band was omitted during the presentation of the variance comparison results.