30 July, 2017

Uncertainty Estimation (III): Resolution, Drift, Temperature & Measured Samples

6. Examples of uncertainty calculation: resolution, drift, influence of temperature on measurements



1st example: Resolution and Rounding


Evaluate the contribution to the uncertainty due to the resolution of a measuring instrument. For example the resolution of a millesimal micrometer (r = 0.001 mm) in the measurement of a standard gauge block

 For both analog and digital equipment, the use of an instrument with an 'r' resolution assumes that the uncertainty of the measured values is between ± r/2 with equal probability throughout this range; In this case we have a rectangular distribution with a half-amplitude-magnitude of 0,5 µm.

 Therefore, its contribution to uncertainty is assessed as:

Both digital and analogue resolution, as well as rounding, give rise to an uncertainty contribution that can be analyzed by taking into account a rectangular distribution with a r/2 limit value

2nd example: Drift

Calibrated standard blocks are used for the calibration of a micrometer. Assuming that the blocks are within the calibration period and is allowed a drift of ± d = ± 30 nm between calibrations, evaluate the contribution to the uncertainty due to the drift of the block value since the last calibration.

 A triangular distribution can be considered for the drift, although a rectangular distribution could be taken. Taking into account that between calibrations does not exceed the value ± e, you get:

3rd example: Calibration Certificate

The gauge blocks used to calibrate one external micrometer are calibrated by comparison and have a certified uncertainty of 60 nm with a coverage factor k=2.

 The assessment of the typical uncertainty associated with a certified value is obtained from the certificate itself knowing the coverage factor employed. The result in this case is:

4th example: Reference Standard

In order to calibrate an external micrometer some set of certified by comparison gauge-blocks are used, its certificate shows an uncertainty of 60 nm with a coverage factor k=2. Assuming that the gauge-blocks are within the calibration period and is allowed a drift of ± 30 nm between calibrations; Evaluate the uncertainty associated with the use of standard blocks as a reference in the micrometer calibration.

 It is evaluated by the uncertainty propagation law as the positive square root of the quadratic combination of the uncertainty due to the calibration certificate and the one due to the derivation of the value blocks in time, ie:

5th example: Temperature Influence

In the above-mentioned case, we assume that the gauge blocks are made of steel, with an expansion coefficient of α = (11,5 ± 1) × 10-6 °C-1 and the measurement is performed under laboratory conditions in which we allow a temperature variation during the measurement process of ± 0,025 °C. In order to perform the measurement we use temperature sensors whose uncertainty is 0,02 °C and their resolution is 0,001 °C.; Evaluate the uncertainty associated with temperature

 In the mentioned case we can assume a mathematical model:





In the mathematical model the corrections are not specified, since what is sought is to evaluate the uncertainty associated with temperature.

 The coefficients of sensitivity are evaluated by calculating the partial derivatives of the mathematical model with respect to the coefficient of expansion (α) and with respect to the variation of temperature(Δt).

 With this we obtain:

And the contribution to the uncertainty associated with the length:

In this case, we must take into account two different uncertainty contributions:

 1.: One due to the uncertainty associated with the expansion coefficient of the gauge blocks used and,

 2.: Two due to the uncertainty associated with the non-exact knowledge of the temperature difference with respect to 20°C reference temperature at the time of calibration.

 1. u(α) : Uncertainty due to the coefficient of expansion

 To evaluate this uncertainty we will assume a rectangular distribution around the mean value of the expansion coefficient whose half-range is 1 × 10-6, so the associated uncertainty is obtained as:

2. u(Ѳ), Uncertainty due to non-exact knowledge of the temperature difference with the reference temperature (20 °C) at the calibration time

 To evaluate the value of the temperature of the blocks use temperature sensors Pt-100 with resolution 0.001 °C and allows a variation of temperature during the measurement of ± 0,025 °C.

 We will consider the following sources of uncertainty:

2.1 u(ΔѲ) Temperature difference ΔѲ, during the measurement
Assuming a rectangular distribution we obtain:

2.2 u(Ѳr) Temperature sensor resolution Pt100 The associated uncertainty is:

2.3 u(Ѳc) Termometer Calibration

 As specified in the calibration certificate its expanded uncertainty is U (Ѳcal)= 0,02 °C for k = 2. Therefore:

2.4 u(Ѳd) Termometer Drift

 We can estimate a drift between calibrations of ± 0.01 ° C. In this case we will assume a rectangular distribution, more conservative than a triangular

Taking into account all the contributions associated with the unknown gage-block temperature value (47 to 50), we obtain:

By using formula (45) we obtain:


7. Steps to follow to estimate the result of a measurement

In a schematic way and following the recommendations of the GUM the steps to follow to estimate the uncertainty of measurement are the following:

 1) Mathematically express the relationship between the measurand Y and the input quantities Xi on which Y depends in the form Y = f(Xi). The function f must contain all magnitudes, including corrections.

 2) Determine xi, estimated value of the input variable Xi, either from the statistical analysis of a series of observations, or by other methods.

 3) Evaluate the typical uncertainty u(xi) of each input estimate xi. The evaluation may be type A or type B.

 4) Evaluate the covariates associated with all input estimates that are correlated.

 5) Calculate the measurement result; That is, to estimate y of the measurand Y, from the functional relation f using for the input quantities Xi the estimations xi obtained in step 2.

 6) Determine the combined standard uncertainty uc(y) of the measurement result y based on the typical uncertainties and covariance associated with the input estimates.

 7) If it is necessary to give an expanded uncertainty U, whose purpose is to provide an interval [y − U, y + U] in which one can expect to find most of the distribution of values that could reasonably be attributed to the measurand Y, Multiply the combined standard uncertainty uc(y) by a coverage factor k, usually 2 or 3, to obtain U = k uc(y).
Select k considering the confidence level required for the interval [ if k = 2 ( 95%) ].

 8) Document the measurement result and, together with its combined standard uncertainty uc(y), or its expanded uncertainty U.
When the result of a measurement is accompanied by the expanded uncertainty U = k uc(y), we must:

 a) fully describe the manner in which the measurand Y has been defined;

 b) indicate the result of the measurement in the form Y = y ± U, and give the units of y, and of U;

 c) include the relative expanded uncertainty U/⎜y⎜, ⎜y⎜≠0, when applicable;

 d) give the value of k used to obtain U [or, to facilitate to the user the result, provide both the value of k and that of uc(y)];

 e) give the approximate confidence level associated with the interval y ± U, and indicate how it has been determined;


8. Conclusions


 All measurement process is aimed at obtaining information about the measurand in order to evaluate its conformity with specifications, make comparisons or make other decisions. In any case, the outcome of the measure is as important as the quality of the measure. The quality of a measure is quantified by measuring the uncertainty of that measurement.

 The result of any measurement and should be documented together with its combined standard uncertainty uc(y), or its expanded uncertainty U, indicating the coverage factor or confidence level associated with the interval and ± U

 The evaluation of uncertainty is not a single mathematical task, but thanks to the Guide for the expression of measurement uncertainty can be analyzed according to general rules. 

 This guide has facilitated the comparison of interlaboratory results since it has been widely extended and in this way a common language is used.

 In this article we have tried to give a brief description, accompanied by examples, of the steps to follow in the determination of uncertainties following the GUM.

 In a first step the physical model of the measurement must be represented by means of a mathematical model and it is necessary to identify each of the input quantities on which it depends, as well as their relationships, if they exist.

 Subsequently, uncertainties are evaluated from an objective, statistical, and subjective point of view, ie taking into account all aspects that influence the outcome of a measure, such as factors inherent in the instrument, environment conditions, etc.

 The uncertainty propagation law or other methods yields the combined standard uncertainty associated with the final estimate of the measurand. Finally, this uncertainty is amplified by a coverage factor to obtain an expanded uncertainty so that the confidence level of the interval  y± U  is greater.

Useful References:

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