This chapter presents statistical commands and its applications to various problems of the mechanics and tribology (M&T). Descriptive statistics, data statistics tool, specialized statistical graphs, probability distributions, and hypothesis tests are discussed. The solutions of various applied problems are given. In particular, surface roughness indices are calculated by the measured data using the descriptive statistics command; the histogram generated by a runout data are matched with the theoretical distribution; capability plot generation is shown by the data for the piston ring gaps; friction torques for two different oil additives are compared using a hypothesis test.
TopIntroduction
Statistical calculations are widely used in M&T sciences and engineering to analyze and present data obtained in scientific or manufacturing processes, in particular to describe the mechanical or tribological characteristics of materials, surface roughnesses, mechanical part tolerances, as well as to quality assurance and fabrication process control, and in many other areas.
In this chapter, the basic MATLAB® toolbox commands for descriptive statistics and the Statistics and Machine Learning ToolboxTM commands for statistical graphs, probability distributions and hypotheses tests are highlighted. All discussed commands are provided with problem-oriented examples. In the final subsection the following application are presented:
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Surface roughness indices by the descriptive statistics commands;
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Histogram and theoretical pdf curve for a runout data;
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Process capability calculations for piston ring gaps;
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Comparing the friction torque data for the two oil additives;
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Sample size evaluation for desired precision of the diameter of a machine part.
Top11.1. Descriptive Statistics
A summary description of the obtained data, called descriptive statistics, occupies an important place in science and technology in general and, in particular, in the areas of M&T. MATLAB® has significant set of commands and specialized means for this purpose. Below we describe the commands and the tool “Data Statistics”, designed for statistical processing of samples or the entire population of data.
11.1.1. About Commands for Quantifying Data
To calculate descriptive statistics the basic commands max, mean, median, min, mode, std and var are used, most of these commands were presented in the Chapter 2 (Table 4). The simplest forms of new two commands are
mode(a)
and
var(a)
These commands compute respectively the most frequently value and variance (dispersion 
of the N values); a – is a vector or matrix, in the latter case these command return vector of values calculated for each of the column in a.
Problem: In a laboratory of engine parts, the bore (inner diameter) of a worn cylinder was tested. The data is 60.1, 60.19, 60.25, 60.62, 60.47, 60.35, and 60.83 mm. Write program named ExDescrStat that calculate and outputs all of the above descriptive statistics indices and determine outliers (if any) - values that are more than three standard deviations than the average. In case of absence of the outliers, the program should type ‘Data does not have outliers’. Use the fprintf command to output names and resulting values of the descriptive statistics.
The ExDescrStat program that solves this problem are.
d=[60.1 60.19 60.25 60.62 60.47 60.35 60.83];
Aver=mean(d);
Minim=min(d);
Maxim=max(d);
Mod=mode(d);
StDev=std(d);
Rng=range(d);
varia=var(d);
outliers=d(abs(d-Aver)>3*StDev);
if outliers>0
outliers
else
fprintf('\n Data does not have outliers\n')
end
DescrStat=[Aver Minim Maxim Mod StDev Rng varia]';
fprintf('\n Mean Minimum Maximum Mode StDev Range Variance\n')
fprintf('%7.2f %7.2f %7.2f %7.2f %5.3f %5.3f %7.4f\n',DescrStat)After starting, the program outputs the following results:
>> ExDescrStat
Data does not have outliers
Mean Minimum Maximum Mode StDev Range Variance
60.40 60.10 60.83 60.10 0.257 0.730 0.0662
In case the outliers were found among the data, this program outputs the found outlier values.