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The previously discussed rectangular distribution is a reasonable default model in the absence of any other information. But, if it is known that value of the quantity in question is more likely near the mean of the interval and less likely near the limits, a triangular distribution is a better model. A triangular probability density function is often used when making measurements using analog devices. Consider the measurement of the rectangular block using a steel rule as shown in the image. A general rule when making measurements with an analog device is to read to the nearest division mark. If more precision is needed, a device with finer divisions should be used. However, different organizations have different rules when it comes to estimating between division marks. In fact, when measuring with a metric rule marked in millimeters it is usually acceptable to estimate to the nearest 0.5mm without magnification and to the nearest 0.1mm with magnification. (For more information see Note 1 at bottom of page.) It appears that the best approximation of the length is 1.60cm. However, it may be a little more or a little less. To estimate the uncertainty of the measurement, the interval of possible values must be determined. To accomplish this, it is usually easiest to identify the values above and below the best approximation where there is likely zero probability that the true value of the measurand exist. For example, for this test piece, the upper and lower limits of the probability density function could be estimated to be 1.55cm and 1.65cm. Since we know that the probability of the true value is highest at the center and tapers down to zero at the limits, the probability density function would look like the one shown below. 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 ± 0.0204cm, which was rounded to ± 0.02cm. The results of the measurement would then be reported as 1.60 ± 0.02cm with a confidence level of 65% using a triangular pdf. The confidence level of the measurand falling within the uncertainty range can be increased to 95% by multiplying 0.00204 by 1.81 to arrive at 0.037. Therefore, it would be correct to express the measurement as 1.60 ± 0.004cm with 95% confidence using a triangular 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 to produce a conservatively larger interval.
Using the Triangular pdf When Both End Measurements Are UncertainWhen using a metric rule as shown in the image to the right, there is uncertainty in reading the rule on both ends. In this situation, it can be said with certainty that the left end of the test piece is between 0.95cm and 1.05cm and the right end of the test piece is between 1.55 and 1.65cm. The uncertainty would be calculated in the same manner as above for the readings at both ends of the rule. This will result in twice the uncertainty when the two uncertainty values are combined. Procedure for Using a Triangular pdf
More About the Triangular Probability Density FunctionThe 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 triangular distribution is calculated, it is found to have the value a^{2}/24, 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 /√24 or equivalently a /2 √6. This value is approximately 0.204 of the width of the interval a. The figure below shows a triangular 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 + or  one standard deviation. The area of the light blue shaded region, which can be is 2((0.204a)(1.184/a)+1/2(0.204a)(0.816/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 65%.
Note 1: We recommend estimating to 0.5mm when using a metric rule marked in millimeters because a study found that it is difficult to accurately determine measurements closer than 1/64th of an inch. This converts to approximately 0.4mm. With magnification, it may be possible to estimate to 0.1mm. Ref: Kennedy, Clifford; Hoffman, Edward; and Bond, Steven; Inspection and Gaging: Sixth Edition; Industrial Press, Inc.; New York, NY; 1987; pp. 167168. Last updated on November 24, 2010

