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Zhang Yue
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Accuracy and precision&uncertainty
Introduction:
Accuracy and precision
In the fields of science, engineering, industry and statistics, accuracy is the degree of
closeness of a measured or calculated quantity to its actual (true) value. Accuracy is
closely related to precision, also called reproducibility or repeatability, the degree to
which further measurements or calculations show the same or similar results.[1]
Accuracy indicates proximity to the true value, precision to the repeatability or
reproducibility of the measurement
The results of calculations or a measurement can be accurate but not precise, precise
but not accurate, neither, or both. A measurement system or computational method is
called valid if it is both accurate and precise. The related terms are bias (non-random
or directed effects caused by a factor or factors unrelated by the independent variable)
and error (random variability), respectively.
uncertainty
Uncertainty is a term used in subtly different ways in a number of fields,
including philosophy, physics, statistics, economics, finance, insurance,
psychology, sociology, engineering, and information science. It applies
to predictions of future events, to physical measurements already made,
or to the unknown.
It is easy to see an uncertainty in the reading of a meter with a scale. Uncertainty can
also be seen on digital device. For example, the uncertainty of a digital milliammeter
may be quoted as ±1% ±digit. For a reading of 800mA, the uncertainty would be :
1% of 800 (i.e. ±8) ±1=
±9,
And the reading would be 800±9mA
Details:
Accuracy and precision
Accuracy versus precision; the target analogy
High accuracy, but low precision
High precision, but low accuracy
Accuracy is the degree of veracity while precision is the degree of
reproducibility.[citation needed] The analogy used here to explain the difference between
accuracy and precision is the target comparison. In this analogy, repeated
measurements are compared to arrows that are shot at a target. Accuracy describes the
closeness of arrows to the bullseye at the target center. Arrows that strike closer to the
bullseye are considered more accurate. The closer a system's measurements to the
accepted value, the more accurate the system is considered to be.
To continue the analogy, if a large number of arrows are shot, precision would be the
size of the arrow cluster. (When only one arrow is shot, precision is the size of the
cluster one would expect if this were repeated many times under the same conditions.)
When all arrows are grouped tightly together, the cluster is considered precise since
they all struck close to the same spot, if not necessarily near the bullseye. The
measurements are precise, though not necessarily accurate.
However, it is not possible to reliably achieve accuracy in individual measurements
without precision — if the arrows are not grouped close to one another, they cannot
all be close to the bullseye. (Their average position might be an accurate estimation of
the bullseye, but the individual arrows are inaccurate.) See also Circular error
probable for application of precision to the science of ballistics.
Ideally a measurement device is both accurate and precise, with measurements all
close to and tightly clustered around the known value.
uncertainty
There are two types of uncertainty, systematic and random
A systematic uncertainty (or error) will result in all readings being either too large or
to small. This uncertainty cannot be eliminated by taking an average of several values
Examples:



Non-zero reading on a meter (a zero error)
Incorrectly calibrated scale
Reaction time of experimenter
(associated with poor experimental technique)
Random uncertainty (or error) gives rise to a scatter of readings about the true value.
The uncertainty can be reduced by repeating readings and taking an average.
Examples:





Reading a scale
Taking a reading which changes with time
Counting oscillations
Measuring out a certain volume of liquid
Reading a scale from the wrong position (a parallax error)
Measurements
In metrology, physics, and engineering, the uncertainty or margin of error
of a measurement is stated by giving a range of values which are likely
to enclose the true value. This may be denoted by error bars on a graph,
or by the following notations:


measured value ± uncertainty
measured value(uncertainty)
The latter "concise notation" is used for example by IUPAC in stating the
atomic mass of elements. There, the uncertainty applies only to the least
significant figure of x. For instance, 1.00794(7) stands for 1.00794 ±
0.00007.
Often, the uncertainty of a measurement is found by repeating the
measurement enough times to get a good estimate of the standard deviation
of the values. Then, any single value has an uncertainty equal to the
standard deviation. However, if the values are averaged, then the mean
measurement value has a much smaller uncertainty, equal to the standard
error of the mean, which is the standard deviation divided by the square
root of the number of measurements.
When the uncertainty represents the standard error of the measurement,
then about 68.2% of the time, the true value of the measured quantity falls
within the stated uncertainty range. For example, it is likely that for
31.8% of the atomic mass values given on the list of elements by atomic
mass, the true value lies outside of the stated range. If the width of
the interval is doubled, then probably only 4.6% of the true values lie
outside the doubled interval, and if the width is tripled, probably only
0.3% lie outside. These values follow from the properties of the normal
distribution, and they apply only if the measurement process produces
normally distributed errors. In that case, the quoted standard errors are
easily converted to 68.3% ("one sigma"), 95.4% ("two sigma"), or 99.7%
("three sigma") confidence intervals.
In this context, uncertainty depends on both the accuracy and precision
of the measurement instrument. The least the accuracy and precision of
an instrument are, the larger the measurement uncertainty is. Notice that
precision is often determined as the standard deviation of the repeated
measures of a given value, namely using the same method described above
to assess measurement uncertainty. However, this method is correct only
when the instrument is accurate. When it is inaccurate, the uncertainty
is larger than the standard deviation of the repeated measures, and it
appears evident that the uncertainty does not depend only on instrumental
precision.
Numerical and solved examples:
1. The length of a pencil is measured with a 30cm rule. Suggest one
possible source of
(a) a systematic uncertainty
(b) a random uncertainty
In each case, suggest how the uncertainty may be reduced
Ans: (A) End of ruler damaged. Measure length from 10cm mark; (b) parallax
error. Place pencil on top of scale.
2. The mean diameter of the wire is found to be 0.50±0.02mm .Calculate
the percentage uncertainty in
(i) the diameter,
(ii) the area of cross-section of the wire
Ans: (i)uncertainty = o.o2/0.5 = 4%
(ii)Area = ∏r2
△ y/y = 2△x/x
△ y = 2 * 0.02/0.5 * 0.5 2
=0.02
Uncertainty = 0.02/0.5 2 = 8%
3. Two distances are measured using a meter rule. The distances are 975mm
and 964mm. Explain why, although the readings are precise, the result
when the distances are subtracted may not be accurate.
Ans: Each distance is measured to about 0.1% and may be thought to be
precise. Result is 11mm with uncertainty likely to be ±2mm i.e. 18%.
Probably inaccurate.
References
1. ^ John Robert Taylor (1999). An Introduction to Error Analysis: The Study of
Uncertainties in Physical Measurements. University Science Books.
pp. 128-129. ISBN 093570275X.
http://books.google.com/books?id=giFQcZub80oC&pg=PA128.
2. wikipedia
3. ^ Knight, F.H. (1921) Risk, Uncertainty, and Profit. Boston, MA: Hart,
Schaffner & Marx; Houghton Mifflin Company
4. ^ Douglas Hubbard "How to Measure Anything: Finding the Value of
Intangibles in Business", John Wiley & Sons, 2007
5. ^ Tannert C, Elvers HD, Jandrig B (2007). "The ethics of uncertainty.
In the light of possible dangers, research becomes a moral duty.". EMBO
Rep. 8 (10): 892–6. doi:10.1038/sj.embor.7401072. PMID 17906667.
6. ^ Douglas Hubbard "How to Measure Anything: Finding the Value of
Intangibles in Business", John Wiley & Sons, 2007
7. ^ Flyvbjerg, B., "From Nobel Prize to Project Management: Getting Risks
Right." Project Management Journal, vol. 37, no. 3, August 2006, pp. 5-15.