1.3 Processing and analysis of experimental data
The ultimate objective measurements and experimental investigations is to find the most probable value by the measurement object or parameter investigated and, if possible, to establish mathematical relationships that best describe the patterns of change in these factors studied experimental conditions. While the graphics built directly on protocol records observations and the results of initial treatment, sometimes reflecting subjectivity testers, their experience and traces of errors of measurement, which reduces the likelihood of conclusions and generalizations.
To assess the accuracy and overall suitability of experimental data using known methods of the theory of random measurement errors and other provisions of mathematical statistics. Mathematical analysis of observations is absolutely necessary when between the studied parameters no apparent functional connection, but can probability, the so–called correlation, carried only average because of the complex nature of the mutual influence of factors that can not be reason to consider a pilot study.
It is known that nepodolanist random measurement errors in the principle exclude possibility of any true value of the measured physical quantity A. In other words, as an object of measurement, and always for us an unknown quantity x. Therefore, in practice forced treatment results are not true, and the probable value of the measured value.
Probable value of the measured (quantity x) is defined as the arithmetic mean value of all performed equally accurate measurements n some one of the same numerical values of physical quantities. Thus, the most probable value of the measured physical quantity:
where
—
the sum of the values obtained in the measurement.
Accuracy and reliability was assessed by measuring the relative error by comparing the absolute value of the error of measurement of the measured value, given confidence probability.
Absolute error Δx representing generally the difference between the desired values and the results of its measurement really can not exhaustively characterize the accuracy of measurements, since they depend on the numerical value of the measured value. Since the true value of the measurand generally remains unknown, for practical calculations take into account its probable value:
At the same time take into account that the possibility of receiving parcels of plus and minus sign and the same amount of random errors, therefore close to zero.
Indicators measuring accuracy are rms values, and average probable error.
Rms measurement error σ individual:
where
the sum of squared deviations of the
results of individual measurements from their average value.
Note that the relative value of the mean square error expressed as a percentage, called the coefficient of variation:
ω=(
)100
1.4 Errors of indirect measurements
Located on condition that the numerical value defined physical quantity is a function or one or a number of independent variables, ie
y = f (x1, x2,…, xi,……,xn),
where хi – independent variable to be measured to determine the results obtained by the numerical value of the function in.
In somewhat simplified interpretation, considering the magnitude of the absolute error Δxi direct measurement of the independent variables xi are very small compared with the measured values can be considered
where
–
differentials of the independent variables and their functions; dy –
change function for infinitesimal changes
independent variables, ie, the
absolute error of indirect measurement.
Errors in determining the function of one independent variable are then as follows. Let the unknown quantity in determined from direct measurements of size s, and in = f (x) and hence,
where
= f
(
).
Expanding the right–hand side of the penultimate equation in a Taylor series and neglecting members containing Δh2 and higher degree Δх, we get:
or
The relative error:
ie
where the sign d after differentiation should be replaced by Δ.