How to Report the Uncertainty of Results

Kelly Kissock

“Nobody believes an analysis except the analyst. Everybody believes an experiment except the experimenter.”

“Uncertainty analysis is more valuable before an experiment than afterward.”

Precision, Accuracy, Random and Systematic Error

“The accuracy of an instrument indicates the deviation of the reading from a known input... The precision of an instrument indicates its ability to reproduce a certain reading with a given accuracy. As an example of the distinction between precision and accuracy, consider the measurement of a known voltage of 100 volts with a certain meter. Five readings are taken, and the indicated values are 104, 103, 105, 103, and 105 volts. For these values it is seen that the instrument could not be depended on for an accuracy of better than 5 percent (5 volts), while a precision of +1 percent is indicated since the maximum deviation from the mean reading of 104 volts is only 1 volt.” (Holman, pg. 7.)

The distinction between precision and accuracy is similar to the distinction between random and systematic error. Systematic error indicates a lack of accuracy, and is generally dealt with by properly calibrating instruments before measurements are taken. Random error is an indication of the precision with which measurements can be made.

Traditional uncertainty analysis deals mainly with random error (i.e. the precision of measurements and calculations) because in most experimental situations the “true” value of a measurement is not known, and hence the “accuracy” of a measurement can not be determined.

Explicit Uncertainty

The clearest way to indicate the level of uncertainty with which a number is known is to explicitly state it next to the number, i.e.:

2 + 0.1

or algebraically:

z + dz

In experimental situations, the best estimate of z is the mean value of a series of observations, :

The ‘average’ deviation between a single measurement and is the standard deviation, s.

Most measurements which are not subject to systematic error can be described by a “normal” distribution. The normal distribution looks like the familar bell curve, with most observations occuring near the mean. If the distribution of measurements follows a “normal” distribution, then 68% of the measurments will be within the interval + 1s, and 95% of the measurments will be within the interval + 2s.

We are generally more interested in how certain we are about the mean value (which is our best indication of the true value of the measurement) than about any single measurement. The uncertainty with which we know the mean value is given by:

Thus, if we want to explicitly state our best estimate of a series of measurements and the uncertainty of our estmate, we usually say:

z + dz = + smean

Our level of confidence that the true value of z is between + smean depends on the number of observations used to calculate : the more observations, the more confident we are about our estimate of z. The level of confidence is quantified in a t-table (see for example, Box et. al.). For a large number of independent observations, we can say that we are 68% sure that the true value of z lies within + 1 smean of , and 95% sure that the true value of z lies within + 2smean of .

From this it can be seen that the reported uncertainty, dz, is usually a statement of the repeatability (or precision) of a measurement. We usually assume that systematic error, which is a measure of accuracy, is negligble. If one can identify a systematic error, then the total error (or uncertainty) and is given by:

de =

Propagation of Error: Reporting Uncertainty Explicitly

If the uncertainties of intermediate values are known and explicitly given, then the uncertainty of a combination of the intermediate values can be determined on a case by case basis. For example, if x = 4 + 0.5, y = 8 + 0.5, and z = xy, then the smallest and largest that z could be is:

zmin = 3.5 * 7.5 = 26.25

zmax = 4.5 * 8.5 = 38.25

and z could be reported as:

z = 32 (-5.75 + 6.25)

More generally, if z = z(x, y, ...), then

dz = (dz/dx) dx + (dz/dy) dy + ...

The maximum uncertainty is:

dz = ½dz/dx ½ dx + ½dz/dy ½ dy + ...

which for our example is:

dzmax = 8 (0.5) + 4(0.5) = 6

Note that the average of ½-5.75½ and ½6.25½ is 6.

In most cases, however, the maximum uncertainty overstates the true uncertainty because of the improbability that both measurements will be of maximum extent in the same direction at the same time. This improbability is greatest when the errors (dx, dy, ...) are random and independent. In this case we can add the errors in quadrature (Holman, pg. 38; Taylor pg. 73) as:

dz = [((dz/dx) dx)2 + ((dz/dy) dy)2 + ...]1/2

In our example, the uncertainty of z would be:

dz = [(8 (0.5))2 + (4(0.5))2]1/2 = 4.47

and, if the component uncertainties are independent, we should report our result as:

z = 32 + 4.47

.

Reporting Uncertainty Implicitly with Significant Figures

Most of the time, the uncertainty with which a number is known is not explicitly stated and the general procedure is to estimate its uncertainty from the number of significant figures with which the number is reported. An easy way to determine how many significant figures a number has is to write it in scientific notation, noting that trailing zeros to the left of a decimal point are merely placeholders and do not count as signficant figures, while trailing zeros to the right of a decimal point indicate increased precision and count as significant figures.

346 = 3.46 * 102 Þ 3 sig. figs.

346,000 = 3.46 * 105 Þ 3 sig. figs.

0.0042 = 4.2 * 10-3 Þ 2 sig. figs.

0.004200 = 4.200 * 10-3 Þ 4 sig. figs.

It is assumed that numbers are known to the precision of the least significant figure. Thus, reporting a result as 346 implies that the true value is between 345.5 and 346.5. And, according to this rule, a reported result of 12,000 implies that we know the true result to be between 11,500 and 12,500.

However, it is best to use some judgment when determining the precision of numbers which end with zero or a string of zeros. For example, based on the preceding rule, “50” has one significant figure and is known to be between 45 and 55. But 50 may also be known to be between 49.5 and 50.5, in which case it should carry two significant figures and not one. Because its precision is not explicitly stated, the reader should use his/her own judgment in assigning its precision.

Finally, a special case exists when a number is known exactly, as in “2 cars”. In these cases, there is no uncertainty associated with the number and it does not propagate any error.

Propagation of Error: Reporting Uncertainty Implicitly with Significant Figures

The general rule for propagation of implicit uncertainty is: carry lots of significant figures in intermediate calculations, but report the final result using one more significant figure than the least precise number.

Example 1:

4.513 * 8 = 36

Example 2:

3.142 kg/person * 2 persons = 6.284 kg

Because “2” is known exactly, it does not effect the precision of the result.

Example 3:

1.2346 * 100 = 120 if 100 is known to be + 50

= 123 if 100 is known to be + 5

= 123.5 if 100 is known to be + 0.5

References

Box, G., Hunter, W.G., Hunter, J.S., “Statistics for Experimenters”, John Wiley and Sons, 1978.

Holman, J.P., “Experimental Methods for Engineers”, McGraw-Hill Book Company, Second Edition, 1971.

Taylor, J. R., “An Introduction to Error Analysis”, University Science Books, 1982.

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