Metrological quality of the measuring devices

The metrological quality of a measuring device is the whole of the characteristics which will make that a measuring device will take measurements with quality corresponding to waiting of the user.

The quality of an apparatus is defined by the following characteristics:

*L' wide

*la sensitivity

*la precision

** the accuracy

** fidelity

Extent of measuring

It is the field of variation possible of the size to be measured. It is defined by a minimal value and a maximum value.

Example:

A Voltmètre will have an effective range ranging between 1 Volt and 10 volts.

Sensitivity

The sensitivity of an apparatus is the smallest variation of measurement which it can detect.

General case

The sensitivity is a parameter expressing the variation of the output signal of a measuring device according to the variation of the entry signal.

An apparatus is all the more sensitive as a small variation of the size G to be measured will cause a larger change of the measuring device.

Foot-note: if the value of entry is of comparable nature that the value of exit, the sensitivity is called profit .

The sensitivity in the vicinity of a value given of the size G to measure is expressed in the following way:

$S=\backslash frac$ : $I$: Indication given by the test

$G$: Quantity of size to measure

It is generally considered that it is about the slope of the curve of graduation on an interval: average sensitivity.

One can write then:

$\backslash \; bar\; S=\; \backslash \; frac\; \{\backslash \; Delta\; I\}\; \{\backslash \; Delta\; G\}$

Example:

The level of a Volucompteur is of 1 cm and the value of level is of 5 Cl.

$S=\; \backslash \; frac\; \{10\; mm\}\; \{5\; Cl\}\; =\; \backslash \; frac\; \{2\; mm\}\; \{Cl\}$

Case of a linear apparatus

The formula has direction only so on this interval of measurement the apparatus is linear. I.e. if the exit is proportional to the entry.

$S=\backslash frac=\backslash operatorname\{Arctan\}\; \backslash \; \backslash varphi$

In practice, one carries out a calibration curve where the linearity is approximate . To determine the line one can use the Method of least squares.

Sensitivity and resolution

Not to confuse the sensitivity with the resolution (or to be able of resolution).

The resolution is the smallest difference between two values, such as the apparatus gives a different measurement of it

Example:

A voltmeter indicates 100 volts. A variation of 0,1 volt makes move the needle of the voltmeter whereas a variation of 0,05 volt does not make move the needle. The resolution of the voltmeter is of 0.1 volt and the sensitivity of 0,05.

Precision

An measuring instrument is all the more precise as the results of measurement which it indicates coincide with the true value (by theoretical definition) which one seeks to measure.

The precision is easier to define by the error of precision. It is expressed in unit of size (absolute error) or expressed as a percentage (relative error).

Apart from the operating conditions, the precision of an apparatus is primarily related to two types of characteristics: accuracy and fidelity. Apparatuses is precise if it is at the same time right and faithful.

The precision of measuring device can also be sullied by external causes: operational error, error caused by the actuating quantities (temperature, pressure etc), error of reference or calibration, error of Hysteresis, error of smoothness etc

Accuracy

The error of accuracy is the total error resulting from all the causes for each result of measurement separately taken. It is thus the aptitude of the apparatus to give results which are not sullied with error.

In the case of multiple measurements it is the difference between the average result and the true value.

$J=\; \backslash \; bar\; v\; -\; V$

$\backslash \; bar\; v$: Arithmetic mean of a great number of measurements

$V$: Value true (or conventionally true)

If one carries out a representation in two dimensions by considering the true value as the origin one can regard the error of accuracy as the Barycentre of the whole of measurements.

$\backslash begin\{matrix\}$

& \ bar D & = & \ sqrt \ \ & & = & \ sqrt \end{matrix}

Fidelity

Definition: Fidelity is the aptitude of a measuring device to give measurements free from accidental errors.

Fidelity defines the dispersion of the results. If only one measurement is taken, fidelity represents the probability that it is representative of the average result. This last would have been obtained by carrying out an infinity of measurements.

Foot-note: the result average itself being sullied with the error of accuracy.

If one carries out a set of measures of a size G, one obtains a maximum value (V_max) and a value minimum (V_min). The limiting errors of fidelity are then:

$F\_\; \{max\}\; =+\; \backslash \; frac\; \{V\_\; \{max\}\; -\; V\_\; \{min\}\}\; \{2\}$

$F\_\; \{min\}\; =\; \backslash \; frac\; \{V\_\; \{max\}\; -\; V\_\; \{min\}\}\; \{2\}$

Example:

The measurements repeated using a voltmeter give

$U\_\; \{max\}\; =\; 100,2\; \{\backslash \; rm\; V\}$

$U\_\; \{min\}\; =\; 99,7\; \{\backslash \; rm\; V\}$

$F\; =\; \backslash \; pm\; \backslash \; frac\; \{100.2-99,7\}\; \{2\}\; =\; \backslash \; pm\; \backslash \; frac\; \{0,5\}\; \{2\}\; =\; \backslash \; pm\; 0.25\; \{\backslash \; rm\; V\}$

Conclusion

One can symbolically represent fidelity, the accuracy and the precision in the following way:

See also

• Error (metrology)

• Calculation of uncertainty
• Criteria of dispersion
• Metrology

Standards

International standards (ISO)

• ISO 5725-1: Exactitude (accuracy and fidelity) of the results and methods of measurement. Part 1: general principles and definitions.

• ISO 5725-2: Exactitude (accuracy and fidelity) of the results and methods of measurement. Part 2: basic method for the determination of the repeatability and the reproducibility of a method of standardized measurement.
• ISO 5725-3: Exactitude (accuracy and fidelity) of the results and methods of measurement. Part 3: intermediate measurements of the fidelity of a method of standardized measurement.
• ISO 5725-4: Exactitude (accuracy and fidelity) of the results and methods of measurement. Part 4: basic methods for the determination of the accuracy of a method of standardized measurement.
• ISO 5725-5: Exactitude (accuracy and fidelity) of the results and methods of measurement - Part 5: alternative methods for the determination of fidelity of a method of standardized measurement
• ISO 5725-6: Exactitude (accuracy and fidelity) of the results and methods of measurement. Part 6: use in practice of the values of exactitude.
• ISO/TR 22971 : Exactitude (accuracy and fidelity) of the results and methods of measurement - practical Guiding lines for the use of the ISO 5725-2: 1994 for the design, the implementation and the statistical analysis of the results of repeatability and interlaboratory reproducibility

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