Uncertainties of Type B

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Community Portal → Data Analysis → Uncertainties of Type B

1 Evaluating Type B Uncertainties
2 50/50 Bound
3 Bounds Only
4 References

Although we cannot quantify errors, we can quantify uncertainties: an uncertainty is a range of values that is likely to contain the (unknown) exact value. Following the ANSI guide,^[1] we divide uncertainties into two kinds: type A and type B. Type B uncertainties are those in which a distribution is assumed, based on experience, other experiments, reference works, etc. Hence, we do not use a statistical analysis to obtain these uncertainties. We caution that type B uncertainties are not necessarily confined to systematic errors because uncertainties are independent of errors.

Evaluating Type B Uncertainties

Table 1. Ten measured values for flow rate of water through a pipe.
Run #	Flow Rate	Run #	Flow Rate
1	5.5	6	5.6
2	5.85	7	5.75
3	5.55	8	5.65
4	5.8	9	5.4
5	5.9	10	5.7

To estimate uncertainties that cannot be extracted from repeated measurements, we appeal to experience with the equipment and with physical properties, to judgement about factors that cause variations in measured values, to manufacturer’s specifications, to calibration standards, etc. To illustrate, consider the measurements of flow rate for water flowing through a pipe, as in Table 1. For such measurements, we identify three sources of uncertainty that we treat as Type B.^[2]

The flow meter was calibrated by the manufacturer at 21°C with an uncertainty of ±2% of scale reading. For a mean flow rate of 5.67 gpm, this gives an uncertainty of ±0.113 gpm.
Our measurements were not done at 21°C, so we include uncertainties caused by variations in fluid temperature. We estimate the temperature might vary by as much as ±2°C; but from NIST tables for water [12], even a ±5°C change at 20°C causes only 0.1% change in the molar volume of water. This gives a variation of ±0.006 gpm in the flow. This will probably be negligible compared to other uncertainties.
The scale resolution on the meter is 0.1 gpm. We believe we can read the meter to within half of this, so we estimate that meter readings may be uncertain up to ±0.025 gpm.

This example is typical: Type B uncertainties are often composed of several independent contributions, $u B i$ . The total $u B$ is computed from the individual components by,^[3]^[4]

(1)

$u_B \ = \ \sqrt{ \sum_i^N u_{Bi}^2}$

In our flow-rate problem we identified three Type B uncertainties: one of 0.113 gpm, one of 0.006 gpm, and one of 0.025 gpm. Hence,

(2)

$u_B \ = \ \sqrt{ (0.113)^2 + (0.006)^2 + (0.025)^2} \ = \ 0.12\, \mbox{gpm}$

50/50 Bound

Often we are not confident that we can identify all contributions to Type B uncertainties. But if, based on our knowledge and experience, we can bound the expected value, then we can still assign a value to $u B$ .^[3] For example, assume we are certain that the (unknown) exact value for $y$ has a 50/50 chance of lying between a lower bound $y l o$ and an upper bound $y h i$ . Further, we believe values for $y$ are normally distributed on this range. Then let $a$ be the half-width of the interval,

(3)

$a \ = \ \tfrac{1}{2} (y_{hi} - y_{lo})$

and take the Type B uncertainty to be

(4)

$u_B \ = \ 1.48a$

because the range [– $a$ /1.48, $a$ /1.48] captures 50% of a normal distribution. Obviously, this procedure can be generalized to bounds having other probabilities of occurring. For example, if the interval 2 $a$ has a 2 out of 3 chance of bounding the desired value, then the factor in (4) becomes 1 rather than 1.48.

Bounds Only

In other cases we may only be able to bound the value. That is, we are certain the (unknown) exact value for $y$ lies between $y l o$ and $y h i$ , but we do not know the distribution. Then we can only assume a uniform distribution: every value in the range is equally likely to occur. In this case, take the Type B uncertainty to be the standard deviation for the midpoint of the range [ $y l o$ , $y h i$ ]; that is, use

(5)

$u_B \ = \ \frac{a \sqrt{3}}{3}$

where $a$ is the half-width, as in (3).

Once we have estimates for both type A and type B uncertainties, then we must combine them to obtain the total. That procedure is describe on the page entitled Combined Uncertainty.

References

↑ U. S. Guide to the Expression of Uncertainty in Measurement, ANSI/NCSL Z540-2-1997, American National Standards Institute, NCSL International, Boulder, CO, 1997.
↑ J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.
↑ ^3.0 ^3.1
↑ J. R. Taylor, An Introduction to Error Analysis, University Science Books, Mill Valley, CA, 1982.ISBN 0-935702-10-5