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Probability Density Function

- Terms
- Uncertainty
- PDFs
  - Gaussian
  - Uniform
  - Triangular
- Procedure
- Combined
- Example
- References

A probability density function (pdf) is a way of describing the data that has been collected from a measurement or multiple measurements. Probability density is simply the probability of a variable existing between two values that bound an interval. The area under the pdf is always 1 or 100%. There are a large number of probability density functions (pdf) that are useful in a variety of applications. However, when making physical measurements, there are three continuous pdfs that are used most often and these will be the focus here. A Gaussian pdf is used for Type A evaluations of uncertainty, which are those involving a set of repeated readings of the measurand with some scatter in the reading. Multiple measurements should be made whenever possible, but when it A uniform or triangular pdf will be used for Type B evaluation of uncertainty, which are those associated with reading a scale or with the measurement instrument calibration will rely on knowledge about the measurement process. The table below summaries the pdf that are used most often when making physical measurements.

Standard Uncertainty and Expanded Uncertainty

All contributing uncertainties should be expressed at the same confidence level, by converting them into standard uncertainties. A standard uncertainty is usually shown by the symbol u (small u), or u(y) (the standard uncertainty in y). The standard uncertainty is a parameter characterizing the range of values within which the true value of the measurand can be said to lie with a specified level of confidence. The standard uncertainty characterizes the uncertainty of an mean (not the spread of readings). One standard uncertainty equates to a mathematical function called the second moment, which describes the width of the probability density function (It can be thought of as "plus or minus one standard deviation"). For example when the standard uncertainty for a uniform pdf is calculated, this indicates that there is a 68% probability that the true value of the measurand will fall within the calculated range.

The coverage probability (or confidence level) in a result is increased by multiplying the standard uncertainty (u) by a constant, called the coverage factor (k) which results in the expanded uncertainty (U). In other words U = ku. For the extended uncertainty range of 2u the probability of the true value falling within the uncertainty range increases to 95%. At 3u, there is a 99% probability that the true value of the measurand will fall within this extended uncertainty range. It should be noted that Type A evaluations other coverage factors may be more appropriate. For example, a coverage factor larger than two may be used with the evaluation is based a small number of repeated measurements. The references should be consulted to learn about methods to determine other coverage factors.

Click on the pdf type to learn more about the use of these functions.

Evaluation Type

PDF Type

Usually Used When the Measurement is:

Standard Uncertainty (u)

Expanded Uncertainty and Probability that the Measurand Lies Within the Uncertainty Range

Type A

Multiple values

Gaussian for bell-shaped pdf Bell-Curve A set of repeated and scattered readings Gaussian Equation
(Std. deviation of the mean)
             1u = 68%
             2u = 95%
             3u = 99%

Type B


Uniform, flat, or rectangular pdf Square A single digital reading Uniform Equation              1u = 58%
        1.65u = 95%
        1.73u = 100%

Type B


Triangular pdf Triangle A single analog reading Triangle Equation              1u = 65%
        1.81u = 95%
        2.45u = 100%

Absolute, Relative and Percent

When a measurement is expressed as 25.3 ± 0.05 cm, the uncertainty is being given in absolute terms. Sometimes it is useful to express the uncertainty in terms of relative or percent uncertainty. Calculating the relative uncertainty is particularly important when calculating the combined standard uncertainty, which is covered on a subsequent page. For now, simply know that all the uncertainties must be in the same units before they can be combined. Therefore, how is the case handled when one measurement has an uncertainty in one unit (say cm3), and another measurement has its uncertainty expressed in a second unit (say grams). In order to combine uncertainties with different units, the units must be eliminated. This is done by calculating the relative uncertainty, which is simply the ratio of the standard uncertainty u(m ) over best approximation of m (ex. 1.8grams / 23.4grams = 0.043). Since the uncertainty is now unitless, it can be combined with other unitless uncertainties to arrive at a combined uncertainty.

The uncertainty is sometimes stated as the percentage uncertainty, which is simply the relative uncertainty quoted as a percentage. The example measurement given in the previous paragraph would be stated something like, "23.4g with a percentage uncertainty of 4.3% .”