There will be uncertainty around the results of any measurement and, as such, data on the quantity and resolution of such measurements should always be provided. For anyone who relies upon these measurements in their work, this data is indispensable, as it is required to assess the reliability of the measurements. When this data is missing the measurements cannot be compared to references mentioned in regulations and norms. This means that estimating and presenting the uncertainty is obligatory for a measuring process.
The error and the analysis of errors are well established in metrology and yet the notion of uncertainty as a digitally expressed feature is a relatively new approach.
Corrections are made as the calculation of all known and anticipated errors is completed. Uncertainty remains around the method for obtaining the result and whether this result correctly represents the measurand. To apply to any measurement and any data processed in measurements, the ideal method of estimating and expressing the uncertainty of measurement should be as universal as possible.
Uncertainty is verifiably bound to the measurement results. A range within the measurement limits, covering the distribution of the calculated data, is used to express the level of uncertainty. The relationship between the uncertainty and the results has received particular attention in recent years, both in standard laboratory and industrial conditions.
According to the International Dictionary of Basic and General Terms of Metrology, the uncertainty of measurement is a parameter bound to the measuring result, which is characterized by dispersion of measurand, and an attribute of the measurand.
An example of a parameter involving uncertainty is standard deviation. Standard deviation from a series of measurements is also uncertainty. And thus lies a division of uncertainty by the source of parameters. There are two types of uncertainty here, type A and type B, both of which will now be discussed in more detail.
Uncertainty Type A
The calculation of standard uncertainty is described in method A through an analysis of the statistical series of observations. In this situation, the standard uncertainty is the standard deviation. A large number of measurements and repetitions are required for this method.
Method A is only applied in situations where it is possible to carry out a series of equal measurements in equal measuring conditions. Such is the case when assessing the repeatability of an electronic balance, where a series of approximately 10 measurements are performed in accordance with European Union regulation PN-EN 45501:1992. The measurements need to all be performed with the same standard mass, by the same operator, in the shortest possible period and within a stable environmental condition. It is possible to estimate the standard deviation for these measurements by:
n – quantity of repetitions (measurements)
xi – result i of the measurement,
x – the average value of all measurements for n repetitions, calculated according to below relation:
Both formulae are well known within mathematics and are commonly used for measurement analysis. The standard distribution is employed for uncertainty type A and is expressed graphically as a Gaussian curve. Such curves can be set experimentally for extremely large quantities of measurements (for example, n = 400).
For training new personnel and to help in understanding the notion, this makes for a very good example. A series of aspects are usually involved in uncertainty. Statistic distribution of results in a series of measurements can be used to determine some of these, and they are estimated as standard deviation as detailed above. While also described by standard deviation, the other components of uncertainty are defined according to an assumed distribution of probability. This distribution is based on experiments or other data.
Uncertainty Type B
A scientific analysis of all accessible information on the changeability of the initial value is used to determine uncertainty type B. Such data is based on previously performed measurements, the experience of the operator and the characteristics of measured materials and measuring devices.
Furthermore, data from the manufacturer’s product specification, uncertainty reference data, handbook and manual content, and all accessible publications can all be utilized by method B. Data taken from the calibration certificates of measuring devices and standard masses are also very important.
It is possible to determine the constituents of uncertainty type B through the application of the previously mentioned electronic balance. These constituents are:
- Reading unit d
- Repeatability, determined by standard deviation set by an operator or during calibration process
- Balance indication error, specified in the calibration certificate
- Uncertainty while determining an indication error
Many other parameters could be focused on but the accuracy of a measurement will dictate whether these have any influence on the uncertainty value. The most common distribution is rectangular-shaped when the uncertainty is estimated through method B.
The initial data should be divided by √3 to estimate the value of uncertainty. The uncertainty can be determined by dividing the reading unit by 2√3 in cases where it is possible for the upper and lower limit of the initial value to be set. By dividing the enhanced uncertainty (specified on the calibration certificate) by the enhanced ratio k, (also specified on the calibration certificate) it is possible to estimate the uncertainty of determining the indication error. Further definitions, namely complex uncertainty and extended uncertainty, have resulted from the above calculations.
Complex uncertainty is a connection of uncertainty types A and B. In some cases, however, complete uncertainty analysis is based solely on type B.
Related to the initial rate, the sensitivity ratio is a partial derivative describing how the estimator of the initial quantity changes in relation to changes in the values of the initial estimators. This parameter is characterized by the relationship:
ci – sensitivity ratio
xi – initial rate estimator
Xi – initial rate value
The participation in the complex uncertainty can be expressed through the ratio:
ui(y) = ci • u(xi)
ui(y) – participation in standard complex uncertainty
ci – sensitivity error
u(xi) – standard uncertainty
The range of values surrounding the measuring result is termed as extended uncertainty. This covers a large part of the distribution of the values. The letter u denotes uncertainty and the expression of extended uncertainty is denoted with a capital letter U, in accordance with the Guide to Expression of Uncertainty in Measurements. The chart below presents a graphical presentation of measurement uncertainty:
x – measurement result
xP – measurand
As result of value measurement xP, value x has been obtained x. The graph above demonstrates that the result of the measurement is not equal to measurand. It is only possible to discuss the range in which the measurand is positioned. The range can have a bigger or smaller scope dependent on the accuracy of a measuring process and the uncertainty related to it. The measuring device, environmental conditions, the operator, and the proper analysis of measuring uncertainty all influence the scope size.
An extension ratio k is a numerically expressed ratio used as a multiplier of standard complex uncertainty, determined to set extended uncertainty.
The following relationship is used to express the extended uncertainty:
U = k • u(x)
U – extended uncertainty
k – extension ratio
u(x) – complex uncertainty
The extension ratio k is described as k = 2 where the distribution of measurand is characterized by a standard (normal) distribution (Gaussian) and the standard uncertainty is related to the initial value estimator. This assignment of extended measurement uncertainty refers to trust level, which is approximately equal to 95%. This is correct for the vast majority of calibration processes.
As such, a decision has been taken by international organizations that laboratories performing calibration that are accredited as EAL members should specify the extended uncertainty U, obtained by multiplication of standard uncertainty u(y), initial value estimator y by extension ratio k = 2.
It should be noted that the uncertainty of measurement is an effect of random errors that occurs in the measuring process. The International Dictionary of Basic and General Terms of Metrology describes the error of measurement as a difference between measuring result and a real measured value.
In line with this definition, the below errors can be determined:
- Relative error is a ratio between the measurement error and real value of measurand
- Random error is a difference between measurement value and an average from an infinite number of measurements of the same measurand, performed in repeatable conditions.
- Systematic error is expressed as a difference between the averages from the infinite quantity of measurements of the same measurand. The notion of systematic error refers to the correction process, which is a value added to the sole measuring result. This correction compensates the systematic error and is easily presented as a measuring error with reverse value.
An example of uncertainty of measurement calculation is given below for a 5 g sample measured on an electronic balance with readability of 0,01 mg.
In accordance with the regulations on estimating uncertainty of measurement, the very first step is to determine the measurement equation, which should include all elements that may influence the measurement. Here, this equation is:
m = m0 + δ m1 + δ m2 + δ m3 + δ m4
m – measured mass
m0 – weighed mass
δ m1 – repeatability of balance indications
δ m2 – balance reading unit
δ m3 – balance indication error
δ m4 – uncertainty on determining indication error.
Focus should be placed on determining the equation for the uncertainty of measurement, which can be used to calculate uncertainty for each element of the equation:
u2(m) = u2(δ m1)+u2(δ m2)+u2(δ m3)+u2(δ m4)
ci = 1
Here, the sensitivity ratio equals 1 for each element of the equation. Determining the calculation of uncertainty for initial value of each element is the next step:
- Weighed mass – m0: the mass of the sample is placed on the weighing pan and is displayed as 5000 mg (all below masses are presented as mg for simplification)
- Repeatability of balance indications – δm1: standard deviation equaling s = 0,02 mg is determined, based on several series of measurements
- Balance reading unit – δ m2: reading unit δ of the applied balance equals 0,01 mg, and therefore the uncertainty referring to the resolution of the measuring device should equal:
- Balance indication error – δ m3: the calibration certificate of the balance gives indication error of + 0,01 mg for 5 g, with uncertainty of measurement equal U = 0,02 mg and extension ratio k = 2. The following equation can be used to calculate uncertainty:
All of the above results then need to be collected and an uncertainty budget formulated. It is possible to observe which of the elements influence the uncertainty the most through chart 1. The uncertainty value is determined as a radical sum of squares of all uncertainty elements.
Chart 1. An instance of uncertainty budget. Source: Radwag Balances & Scales
||Contribution into complex uncertainty
The following step is to calculate extended uncertainty U. In the example above, it has been assumed that the extension ratio k = 2, which corresponds to a trust level of approximately 95%. If the relationship describing extended uncertainty is applied, then the value of extended uncertainty can be calculated through the equation:
U = k • uc(m) = 2 • 0,023 mg = 0,046 mg
The indication of the balance with a 5 g load on its weighing pan is the final result of the measurement. This equals:
m = (5000,00 ± 0,05) mg
and so, the measurand is within the thresholds from 4999,95 mg to 5000,05 mg.
The lack of accurate data on the quantity of the measured value was evident in the uncertainty of a measuring result. An infinite quantity of information is required for accurate knowledge of the quantity of measured value. However, in practice, this is unobtainable.
The source of uncertainty is the phenomena affecting the uncertainty, and, consequently, the fact that the measurement result cannot be expressed through a single value. Several possible sources of uncertainty can be identified in practice, including:
- Incomplete definition of a measurand
- Imperfect realization of the definition of the measurand
- Imprecise sampling, i.e. The measured sample is not representative of the defined measured quantity
- Incomplete knowledge of the environmental impact on the measurement procedure, or imperfect measurement of the parameters that characterize these conditions
- Subjective errors in reading the indications of analog instruments
- Inaccurate values assigned to standard masses and reference materials
- Inaccurate values of constant and other parameters, obtained from external resources, and applied in data processing procedures
- Simplified approximations and assumptions used in methods and measuring procedures
- Dispersion of values obtained in the process of observations repeated in seemingly equal conditions
A thorough and correct analysis of the whole measuring process is crucial for the successful estimation of uncertainty of measurement. As such, it is very important that an appropriate estimation of measurement uncertainty is carried out, as the components of the uncertainty are not always influential in the process itself.
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