## Accuracy, Precision and Sensitivity

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On: 05 Jun, 2017

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Accuracy is the quality that characterizes the capacity of a...

Accuracy is the quality that characterizes the capacity of a measuring instrument for giving results close to the true value of the measured quantity.

The “true,” “exact,” or “ideal” value is obtained when measurements are made using an ideal method. A measurement method is considered to be ideal when experts agree that its results are sufficiently accurate for the intended application of the measurement data.

Sensor accuracy is determined by means of the static calibration process. It consists of keeping constant all sensor inputs, except that to be studied. This input is changed very slowly, thus taking successive constant values along the measurement range. The successive sensor output results are then recorded. Their plot against input values forms the calibration curve. Obviously each value of the input quantity must be known. Measurement standards are such known quantities. Their values should be at least ten times more accurate than that of the sensor being calibrated.

Any discrepancy between the true value for the measured quantity and the instrument reading is called an *error. *The difference between instrument reading and the true value is called *absolute error. *Sometimes it is given as a percentage of the maximal value that can be measured with the instrument (end-of-scale value) or with respect to the difference between the maximal and the minimal measurable values, that is, the measurement range. Therefore we have

Absolute error = Result — True value

The common practice, however, is to specify the error as a quotient between the absolute error and the true value for the measured quantity. This quotient is called the *relative error. *Relative error usually consists of two parts: one given as a percentage of the reading and another that is constant. The constant part can be expressed as a percentage of the end-of- scale value (or a threshold), a number of counts in digital instruments, or a combination of these. Then

Relative error = Absolute error / True value

Some sensors have a relative error specified only as a percentage of the full scale. If the measurement range includes small values, the full-scale specification implies that for them the measurement error is very large. Some sensors have a relative error specified as a percentage of the reading. If the measurement range includes small values, the percent-of-reading specification implies unbelievably low errors for small quantities.

To ease the comparison of several sensors with respect to their accuracy, use is made of the Accuracy Class concept. All sensors belonging to the same class have the same measurement error when the applied input does not exceed their nominal range and work under some specified measurement conditions. That error value is called the *index of class. *It is defined as the percent measurement error, referred to a conventional value that is the measurement range or the end-of-scale value. For example, a displacement sensor belonging to an accuracy class 0.2 and for which end-of-scale displacement is 10 mm, in the specified reference conditions has an error lower than 20 m when measuring any displacement inside its measuring range.

The measured value and its inaccuracy must be expressed with compatible numerical values. That is, the numerical result of the measurement must not have more figures than those that can be considered reliable by taking into account the uncertainty of the result. For example, when measuring ambient temperature, a result expressed as 20°C **± **1°C is in correct form, but the expressions 20°C **± **0.1°C, 20.5°C **± **1°C, and 20.5°C **± **10% are all incorrect because the measured value and the error do not have the same uncertainty.

Care must be taken also when converting units to avoid false gains of accuracy. For example, a 19.0-inch length (1 inch = *2.54 *cm) should not be directly expressed as 482.6 mm, since the original figure suggests an uncertainty of tenths of an inch while the converted figure indicates an uncertainty of tenths of a millimeter. That is, the original result indicates that the length is between 485 mm and 480 mm while the converted result would suggest that it is between 482.5 mm and 482.7 mm.

*Precision *is the quality that characterizes the capability of a measuring instrument of giving the same reading when repetitively measuring the same quantity under the same prescribed conditions (environmental, operator, etc.) without regard for the coincidence or discrepancy between the result and the true value. Precision implies an agreement between successive readi ngs and a high number of significant figures in the result. Therefore it is a necessary but not sufficient condition for accuracy. Figure *1.5 *shows different possible situations.

The *repeatability *is the closeness of agreement between successive results obtained with thesame method under the same conditions and in a short time interval. Quantitatively the repeatability is the minimum value that exceeds, with a specified probability, the absolute value of the differe nce between two successive readings obtained under the specified condit ions. If not stated, it is assumed that the probability level is 95%.

The *reproducibility *is also related to the degree of coincidence between successive readings when the same quantity is measured with a given method, but in this case with a long-term set of measurements or with measurements carried out by different people or performed with different instruments or in different laboratories. Quantitatively the reproducibility is the minimal value that exceeds, with a given probability, the absolute value of the difference between two single measurement results obtained under the above-mentioned conditions. If not stated, it is assumed that the probability level is *95%. *In sensors, when there is an output change with time, it is sometimes said that there are instabilities and that the sensor drifts. In particular, for some sensors zero and scale factor drifts are specified. The

*zero drift*describes output variations when the input is zero.

*Scale factor drift*describes sensit ivity changes.

The

*sensitivity*or scale factor is the slope of the calibration curve, whether it is constant or not along the measurement range. For a sensor in which output y is related to the input x by the equation y =

*f(x),*the sensitivity

**S(Xa),**at point

**Xa,**is

**S(Xa)**(1.2) =

*dx/*

*dy*

*It is desirable in sensors to have a high and, if possible, constant sensitivi ty. For a sensor with response*

y

*kx*+

*b*

the sensitivity is S

*k*for the entire range of values for x where it applies. For a sensor with response

y =

*kx2*+

*b*

the sensitivity is S =

*2kx,*and changes from one point to another over the measurement range.

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