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# Measurement Uncertainty Using a Uniform Probability Density Function

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A continuous, uniform probability density function (pdf) is used when the interval of possible values is known, but the measurement provides little additional information. A continuous, uniform pdf (which is also known as a rectangular or flat pdf) is a density function that is constant, making it the simplest kind of density function. Evaluation of the uncertainty associated with a digital readout uses a uniform probability density function. Since this measurement provides a single value, a Type B uncertainty evaluation is used, which is based on scientific judgment using all of the relevant information available. So, what scientific judgment is available with a digital readout?

Using scientific judgment, it is known that the best approximation of the true value is clearly the value shown by the digital readout. Additionally, it is known that the value displayed was likely rounded up or down from a slightly different value. Is the value closer the the displayed value or closer to the upper or lower limits of the interval of possible values? Since it is impossible to tell where the true values is within the interval of possible values, the probability is uniform throughput the interval.

Consider the case where a digital scale was used to determine the mass of an object. A single digital reading of 76.34grams was produced. The probability density function is shown in the image below. If the scale is calibrated correctly, the reading of 76.34grams would be produced if the mass of the object was anywhere between 76.335grams and 76.345grams (due to rounding of these values). There is no way of knowing where the true value exists within this range, so the probability function (solid line) is the same for all values. The total probability is represented by the area under the graph and this must equal 1 or 100%.

From the table on the previous page (see portion of table copied below), it is given that the standard uncertainty (u) for a uniform probability density function, is represented by the following equation where (a) is the width of the interval of possible values.

After doing the math, it is determined that the standard uncertainty for this measurement is ± 0.00289grams, which was rounded to ± 0.003grams. The uncertainty was rounded to three decimal places, which is one more than the original reading. The results of the measurement would then be reported as 76.34 ± 0.003grams with a confidence level of 58% using a uniform pdf. The confidence level of the measurand falling within the uncertainty range can be increased to 95% by multiplying 0.00289 by 1.65 to arrive at 0.00477. Therefore, it would also be correct to express the measurement as 76.43 ± 0.005grams with 95% confidence using a uniform pdf.

Note on Significant Digits and Precision:
The uncertainty should generally be quoted giving two significant figures. The best approximation of the measurand should then be rounded to the same decimal place as the second digit of the uncertainty. However, if using two significant digits causes the uncertain to exceed the precision of the best approximation value by more than one decimal place, then the uncertainty should be rounded up (like in the example above). It should also be noted that when rounding, the uncertainty is always rounded up (never down) to produce a conservatively larger interval.

 Evaluation Type PDF Type Usually Used When the Measurement is: Standard Uncertainty (u) Confidence that the Measurand Lies Within the Uncertainty Range: Type B Single value Uniform, flat, or rectangular pdf A single digital reading 1u = 58% 1.65u = 95% 1.73u = 100%

Procedure for Using a Uniform PDF

• Estimate lower and upper limits a- and a+ for the value of the input quantity in question such that the probability that the value lies in the interval a - and a + is, for all practical purposes, 100 %.
• Provided that there is no contradictory information, treat the quantity as if it is equally probable for its value to lie anywhere within the interval a - to a +; i.e. model using a uniform probability distribution.
• The best estimate of the value of the quantity is then (a+ + a-)/2
• The standard uncertainty is then ( a + - a -) divided by the quantity of two times the square root of three
• Quote the uncertainty giving two significant figures and then round off the best approximation of the measurand to the same digit as the second digit in you uncertainty.

More About the Uniform Probability Density Function

The standard uncertainty equation and the confidence levels used with the uniform distribution, are fairly easy to determine. The equation is based on a mathematical moment. A moment is a quantitative measure of the shape of a set of points and the "second moment” characterizes a particular aspect of the width of a set of points in one dimension. Other moments describe other aspects of a distribution such as how the distribution is skewed from its mean, or peaked. The calculation of uncertainty is based on the second moment. When the second moment for a uniform distribution is calculated, it is found to have the value a2/12, where a is the overall width of the distribution. The square root of this quantity is taken to find the standard uncertainty which results in the value a /√12 or equivalently a /2 √3. This value is approximately 0.289 of the width of the interval a.

The figure below shows a uniform pdf. By definition the width of the pdf is indicative of the interval of possible values. Since it is known that the true value of the measurand fall somewhere between the limits of the interval, the area under the pdf represents a probability of 100%. The area of the region shaded light blue indicates the probability that the true values fall within plus or minus one standard deviation. The area of the light blue shaded rectangle is 2(0.289a)(1/a) = 0.58. Hence, for the rectangular distribution the probability that the measurand lies within plus or minus one standard uncertainty of the best approximation is 58%.

Last updated on November 24, 2010