Variance and Uncertainty Range

Normal Distribution

Type B uncertainty is also of the nature of standard deviation of the estimated value of xp. For the estimation of standard deviation from the given range of uncertainty (U) the range U is to be divided or multiplied by a certain factor, whose value will depend upon the confidence level at which the result was stated. For example if the result is stated at a confidence level of 95%, then the range U is to be divided by 1.96 for infinite number of degree of freedom or by the “Student’s ‘t’ factor” for the given degrees of freedom.

In older literature, uncertainty figures are given at the confidence level of 50%; in that case one has to multiply it by a factor of 1.48 to obtain the standard deviation.

Rectangular Distribution Uncertainty

If a result has been indicated with a range of aC and at and it is assumed that it is equally likely for the estimated value to lie anywhere within the given range, i.e. the result has a rectangular probability distribution, then corresponding variance u2.xp/
will be given as

giving standard deviation u.xp/as

Here it is assumed that

thus giving

Uncertainty Triangular Distribution

In many cases, it is more realistic to expect that the chance of reported value lying near the bounds is less than that lying near the
midpoints of the range. The probability of occurrence of the result at the extreme boundary points is zero and increases linearly and becomes a maximum at the midpoint of the range. That is the reported value under consideration follows a triangular distribution.
Then u.xp/ is given as

Trapezoidal Distribution

It is more reasonable to assume that the result under considerationhas a maximum probability within a range of ˙aˇ about the midpoint of the range and decreases linearly to zero at its ends. That is the input quantity has a symmetric trapezoidal probability distribution having equal sloping sides with a base of width of 2a and top of width 2aˇ. Here ˇ is a proper fraction and may take any value between 0 and 1. When ˇ is 0, the probability distribution becomes triangular distribution and it becomes rectangular distribution for ˇ D 1. The standard deviation u.xp/, in this case, is given as

Rectangular distribution should be used only when no data are available. Otherwise, logically it will be better to use triangular distribution. Firstly this is similar to normal distribution and secondly more logical. While stating the range of uncertainty, the measurements are carried out, which follow normal distribution.
When extended uncertainty is stated, a multiplying factor to standard deviation is given. So when using the uncertainty range to calculate back the standard deviation, one should assume that the reported result is following normal distribution unless contrary is stated or otherwise evident. Type B evaluation of uncertainty should be carried out keeping in view of the hierarchy of standards and laboratories. In practice, for example mass measurement, National prototype kilograms are calibrated by International Bureau of Weights and measure. The calibration certificate contains, besides other data, the mass value with semi-range of uncertainty normally equal to 2 times the standard deviation. So for a national metrology laboratory (NPL in the case of India), the standard deviation should be semi-range divided by 2. It is not justified to assume that reported result is following the rectangular distribution and obtaining standard deviation by dividing semi-range by the square root of 3.
All other laboratories should follow a similar method for Type B evaluation of uncertainty. It is emphasized that this method should be used only for applying the mass value of standard and uncertainty associated with the mass value.

In some cases, standards and measuring instruments are calibrated, but no specific value of the standard input versus scale observations with the corresponding uncertainty or correction to the specific points of the scale of the instrument is given. The calibration only ensures that the instrument will perform within certain specified limits. In that case, rectangular distributions or its modified versions may be used for Type B evaluation of uncertainty.

The proper use of available information for Type B evaluation of standard uncertainty of measurement needs greater insight based on experience and general knowledge. It is the skill that can be learned with practical experience and deep study of the mathematical statistics. A well-based Type B evaluation of standard uncertainty can be as reliable as Type A evaluation of standard uncertainty. Type B evaluation of uncertainty assumes greater importance in those cases where direct observed measurement data are small for Type A evaluation of uncertainty. There are many valid reasons for not able to take larger number of observations.

When only single value is known of the quantity Xp, for example a single measured value, a resultant value of the previous measurement, a reference value from the literature or a correction value, this value will be used as xp. The standard uncertainty u.xp/ associated with xp is adopted where it is given. Otherwise it has to be calculated from unequivocal uncertainty data. If data of this kind are not available, the uncertainty has to be evaluated on the basis of the experience taken as it may have been stated (often in terms of an interval corresponding to expanded uncertainty).

When probability distribution can be assumed for the quantity Xp, based on theory or experience, then the appropriate expectation or expected value (mean value) and the standard deviation of this distribution have to be taken as the estimate of xp and the associated standard uncertainty u.xp/, respectively.

Source : measurement uncertainty