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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, z :
n
z
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z
i
i 1
n
1
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The ‘average’ deviation between a single measurement and z is the standard deviation, s.
n
 z  z 
2
i
s
i 1
n 1
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:
smean 
s
n
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 = z + smean
Our level of confidence that the true value of z is between + smean depends on the number
of observations used to calculate z : 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 z , and 95% sure that the
true value of z lies within + 2smean of z .
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 =
desys 2  deran 2
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
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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 = (z/x) dx + (z/y) dy + ...
The maximum uncertainty is:
dz = z/x dx + z/y dy + ...
which for our example is:
dzmax = 8 (0.5) + 4(0.5) = 6
Note that the average of 5.75 and 6.25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 = [(z/x) dx)2 + ((z/y) 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
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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  sig. figs.
346,000 = 3.46 * 105  sig. figs.
0.0042 = 4.2 * 10-3  sig. figs.
0.004200 = 4.200 * 10-3  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
= 123
= 123.5
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if 100 is known to be + 50
if 100 is known to be + 5
if 100 is known to be + 0.5
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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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