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# Measurement Uncertainty (The Probabilistic Approach)

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The overall purpose of performing measurements in science is to increase knowledge and document this knowledge about some physical quantity. The measurand has a true value but, except for straightforward counting situations, the true value can never be fully determined. When a quantity is measured, the outcome depends on the measuring system, the measurement procedure, the skill of the operator, the environment, and other effects. For example, when the length of a steel bar is measured with a scale, a measurement of 10.2cm may be recorded. However, when the same bar is measured using a digital caliper, it may be determined that the bar is 10.23cm long. More sophisticated measurement equipment will increase the precision of the measurement and, if the equipment is well calibrated, move the measured value closer to the true value of the length. Theoretically, it is possible to keep using smaller and smaller divisions to make the measurement, so it is not possible to perfectly determine the true value of a measurand. Even with no other factors affecting the measurement processes, the scale will limit the measurement and the final result will always be an interval that cannot be reduced to a point.

Therefore, the measurement process shouldn't be thought of as an exercise to determine the true value of the measurand. Rather it should be viewed as an attempt to estimate the true value and characterizes the range of values within which the true value is asserted to lie. The range of values that is believed to encompass the true value with a specified level of confidence is called the uncertainty. The size of the uncertainty is affected by a number of factors, which include: the interval of the scale, environmental conditions affecting the measurement, equipment calibration, point-to-point variation, and others.

A measurement that does not report the likely range of uncertainty communicates limited information because it does not indicate how well the result represents the value of the measurand. Thus, the result of any physical measurement has two essential components: (1) A numerical value (in a specified system of units) giving the best approximation of the true value, and (2) the degree of uncertainty associated with this estimated value. Whenever possible the systematic error should be removed from the total uncertainty so that only random error remains. This can be done by collecting a set of measurements on a reference standard with an accepted value that is traceable to a standards organization such as the National Institute of Standards and Technology and statistically analyzing the data set to determine the bias between the measurement results and the accepted value.

There are two methods to describe uncertainty: The use of significant digits to imply uncertainty and explicitly describing the uncertainty. When using significant digits to imply uncertainty, the last digit is considered uncertain. For example, a result reported as 1.23 implies a minimum uncertainty of ± 0.005 and a possible range of 1.225 to 1.235. However, there are problems with using significant digits, so it is better to explicitly describe the uncertainty. An example of explicitly describing the uncertainty is expressing measurement results as 25.3 ± 0.05 cm.  This result is communicating that the measurement is believed to be closest to 25.3cm but it could be as small as 25.25cm or as large as 25.35cm.

Uncertainty Analysis Using a Probabilistic Approach
One simple method that is often used to report the uncertainty of a simple measurement is to calculate the half-value of the interval. For example, when measuring the length of an object using a scale graduated in millimeters, the uncertainty can be reported as 0.5 mm.  In this case where the same value is easily obtained (barring an operator mistake) it is generally not necessary to make multiple measurements. However, the uncertainty arrived at with this method does not relate well to the standard deviation that is obtained using statistical methods of evaluating uncertainty. This is a problem especially when combining uncertainties where all contributing uncertainties should be expressed at the same confidence level, by converting them into standard uncertainties. A standard uncertainty is a margin whose size can be thought
of as ‘plus or minus one standard deviation’. The standard uncertainty tells us about the uncertainty of an average (not just about the spread of values). A standard uncertainty is usually shown by the symbol u (small u), or u(y) (the standard uncertainty in y).To eliminate this disconnect, the International Standards Organization (ISO) recommends that metrology use a probabilistic approach to evaluating measurement uncertainty. This approach does not use the true value in the analysis because the true value of a measurand is never known. The probabilistic approach uses mathematical functions called probability density functions to model the information available about a particular measurement. For most people the probabilistic approach is a easier to understand and apply than other approaches.

There are two basic methods of evaluating uncertainties, which are:

Type A evaluation - evaluation of uncertainty by the statistical analysis of series of observations.

Type B evaluation - evaluation of uncertainty by means other than the statistical analysis of series of observations.

Type A evaluations of standard uncertainty may be based on any valid statistical method for treating data. Examples are:

• calculating the standard deviation of the mean of a series of independent observations.
• using the method of least squares to fit a curve to data in order to estimate the parameters of the curve and their standard deviations.
• carrying out an analysis of variance (ANOVA) in order to identify and quantify random effects in certain kinds of measurements.

Note: Type A evaluations are only possible when there is scatter in the repeated measurements. Sometimes repeated measurements are identical. This usually means that the measuring apparatus was not sensitive enough to display scatter in the readings. When repeated measurements are identical, then the data should be treated as a single reading and a Type B evaluations of uncertainty should be conducted.

Type B evaluations of standard uncertainty are usually based on scientific judgment using all of the relevant information available, which may include:

• data from a single digital or analog reading.
• previous measurement experience with, or general knowledge of, the behavior and property of relevant materials and instruments.
• manufacturer's specifications.
• data provided in calibration and other reports.
• uncertainties assigned to reference data taken from handbooks.

Both methods of uncertainty analysis make use of probability density functions which are covered on the following pages.

Last updated on November 24, 2010