Accuracy and Precision
Accuracy and precision may be the most misused words in the test
and measurement lexicon
An awful lot of engineers and scientists seem to have a very muddy
idea of what the words accuracy and precision mean. The same is
true of the folks who write specification sheets, catalog descriptions
and other product literature describing data acquisition offerings.
Both of these words have very precise meanings, which you should
keep in mind in order to use them accurately. (Sorry about that
last sentence. I just couldnt resist it.)
A measuring instrument can be extremely precise, yet give you totally
bogus readings. An instrument can be extremely accurate, yet give
readings so imprecise that they are useless. Between you and me,
Id much rather have accurate though imprecise readings. At
least then you know theres a problem!
So, before I go into accuracy and precision specifications for
data acquisition equipment, I want to clear the air about what those
two words really mean.
Folks who know a thing or two about data acquisition boards will
often tell you that a 12-bit ADC doesnt necessarily give you
Of course, thats like saying that a red car wont necessarily
go 120 mph (or 192 kph, either). Being red and being fast are two
different things, just as being precise and being accurate are two
Lets, for the moment, forget accuracy and concentrate on
precision. Precision is a property of the measuring instrument and
Right now, Im looking at an amazingly ancient draftsmans
triangular scale. At least, thats what I think these things
were called back when draftsmen had any use for them. This particular
scale is so old that its made of hardwood. Its visibly
warped. It was made back when the metric system was "somethin
them furriners talk about." The only two uses I have for the
thing now are to keep papers from blowing off my desk when I have
the windows open, and as an object to help explain the concept of
If you pick this thing up, the odds are two to one that you wont
see anything that immediately makes sense. Turning it over once
or twice, however, will get you to a reasonably familiar scale:
12 inches divided into sixteenths of an inch. The same scale we
saw on the "foot rulers" we learned to use in grade school.
(My kids learned both metric and American Standard measurements.
I dont know what theyre teaching these days.) If you
measure the length of something with this "foot ruler"
scale, youll be able to report the results to the nearest
sixteenth of an inch. The basic precision of this instrument is
± 1/32 inch.
The other scales on this triangular scalethe ones that dont
look like anything that makes immediate sensedivide distance
in other ways that made it easy for the green-eye-shade draftsmen
of yore to convert measurements in scale drawings. These have all
sorts of different precisions. For example, there is one little
bit there that has 3/16 of an inch divided up into 12 subdivisions.
Thus, each small subdivision is 3/192 (or one sixty-fourth) of an
inch long. That gives a precision of ± 1/128 inch!
By now you should have the idea that an instruments precision
tells you the smallest difference that you can reliably read. For
a digital instrument that uses a binary representation, its
the least significant bit. For a decimal display, its the
least significant digit. You can either define the precision by
giving the size of that smallest difference or (what is actually
more meaningful) plus or minus half of the smallest difference.
Accuracy, on the other hand, tells you how well measurements made
using your instrument match up with some standard instrument. Its
a calibration issue.
Im a private pilot and as a private pilot I use standard
air navigational charts. For short trips, I use what are called
Sectional Charts, which have a standard scale of 8 statute miles
per inch. To measure how far it is from point A to point B (and
therefore how long it will take and how much fuel Ill use),
I use the standard sectional-chart plotter. I also have a little
pocket-sized plotter that I got as a promotional freebie.
The scales printed on the small plotter, however, are 2.5% shorter
than they should be. I was a little unhappy about that until I realized
that on a one-hour flight at 120 miles per hour, Id be off
by 3 miles, which is smaller than the Denver, Colorado airport.
If you cant see three miles when flying, you are figuratively,
legally and actually "in the soup." That 2.5% error is
not too important.
Anyway, assuming that the standard plotter is, as we used to say
around the accelerator lab, "dead nuts" (meaning perfectly
accurate), the little freebie plotter gives measurement results
that are too high by one part in four hundred. Its accuracy
Accuracy, obviously, means the same thing whether the instrument
is digital, analog orwhat else is there?
Another term that goes along with accuracy and precision is repeatability.
Whereas precision arises from the measuring instruments construction
details, and accuracy arises from comparing the instruments
results to those of a standard, repeatability expresses how well
its measurements compare with each other.
If I measure something now and then measure it again later, how
close can I expect the results to be? How well do the measurement
results now compare to measuring the same thing tomorrow, a week
from now, a month or a year from tomorrow, or just 30 seconds from
I ran across a perfect example while buying plants for the new
cactus garden were putting in the front yard. The fellow running
the nursery heard that I was into measurements and just had to tell
me this story.
Many moons ago, hed been a foreman in a sawmill. Being smarter
than the average bear, hed figured out that he could improve
operations if he started processing each log by cutting it off to
a standard length. The number floating around in my memory associated
with this length is 27 feet.
So, he set up the system and explained what to do to everyone involved,
then stepped back to watch it work.
They cut a bunch of logs to standard length on the first day. The
next morning, he went out to measure their lengths as a quality
control check. They were all too long.
So, he chewed people out and sent them back through the saw to
make them right. After lunch, he checked the logs recut that morning.
Now, they were too short!
After banging his head on a tree for a while, he figured out what
was wrong with his system: he was using a steel tape measure. The
thermal expansion coefficient of the steel tape was much larger
than that of the logs, and the logs retained heat better than the
tape as well. Thus, the tape was expanding and contracting as the
ambient temperature changed, but the logs werent.
Hed explained his system and gotten the work organized during
the morning of the first day, so all the logs had been measured
and cut during the heat of that afternoon. The next morning, when
it was cool, hed checked the work with that same steel tape
measure. The tape, being relatively cooler, was relatively shorter.
Thus, it erroneously told him that the logs were too long.
They recut the logs during the morning to match the measurements
using the cool tape. When he went back to recheck in the afternoon
with a warm tape, the measurements said the logs were too short.
The difference had never bothered anyone because they werent
trying to precisely measure the log lengths. What they previously
most cared about was the cross-sectional dimensions of the boards,
which were just a few inches. The thermal variation had not been
measureable with the equipment they used, so they didnt know
It only showed up when they started measuring long distances reasonably
accurately, and then rechecking their measurements later. Then,
they started seeing variations on the order of an inch from measurement
Repeatability is only a problem if you want to repeat your measurementsand
have them come out the same each time.
In the example, the measurements repeatability was limited
by thermally induced variations. There are lots of other possible
physical phenomena that can limit measurement repeatability as well.
Mechanical flexure of the measuring apparatus is the second most
common cause of distance-measurement repeatability problems.
For data acquisition systems, the repeatability is limited by a
large number of mechanisms, from thermal effects to piezoelectric
effects. A detailed discussion of these problems and what to do
about them is well beyond the scope of this book. If you really
want to pursue the subject, get a copy of Low Level Measurements
Handbook, from Keithley Instruments by visiting their website at
Ideally, you would like an instruments repeatability to be
better than its accuracy, and its accuracy to be better than its
precision. That way, you can feel confident that the number you
get truly represents the property of the object youre measuring
to within the obvious limit of the measurement. Whenever we make
a measurement, we tacitly assume this situation exists by reporting
all the digits measured.
Figure 3.1 shows how accuracy, precision and repeatability relate
to a set of measurements.
Click image to see full size
Figure 3.1: Precision, accuracy and repeatability relate to the
probability of getting various measurement outcomes.
You might look at the diagram and say: "Oh, the diagram shows
that the instrument reports values at 0.1 unit increments. That
must mean its a digital instrument."
You might say that, but youd be wrong if you did. Remember
that the instruments precision means that you can report values
only to "the nearest such-and-such value." This measuring
instrument can be read to the nearest 0.1 unit, just as an analog
watch can (usually) only be read to the nearest minute. The instrument
shown in the figure could be analog or digital with no affect on
The next thing to realize is that this is a really stinko instrument.
Its repeatability isnt much better than its accuracy, which
is much worse than its precision. This thing needs a digit knocked
off its readings and a quick trip to the nearest cal lab. Of course,
if it werent such a piece of junk, we wouldnt have much
to talk about, would we?
In laying out Figure 1, I chose to denote the repeatability by
reporting the half-width of the probability distribution. That is,
Ive indicated the repeatability by showing the distance between
the points where the probability curve crosses 50%. I could have
used the RMS deviation, the points between which 95% of the measurements
fall, or any of several other ways of reporting repeatability. Any
of those definitions is valid and all of them are used by various
people. The important thing is that when someone quotes a repeatability
for an instrument, you have to ask how theyre defining it.
Accuracy is the result of convolving a whole lot of different effects.
As such, it doesnt really scale with anything reliably. Its
just a mish-mash! This instrument is clearly disgustingly inaccurate.
The average of measurements is 18.6% higher than the true value.
Accuracy specifications, however, can be stated in almost any way
the manufacturer feels confident in stating them. An accuracy specification
is a promise by the manufacturer that under specific circumstances,
which include being within a range of environmental conditions,
proper calibration and nobody mucking about in the units guts,
what the unit reads when compared to a NIST (National Institute
of Standards and Technology)-traceable transfer standard will differ
from the standard value by no more than the specified amount.
A promise is a promise, but you have to look carefully at what
really is being promised. What one manufacturer promises may be
quite different from what another manufacturer promises for a similar
product. Read the fine print.
Accuracy and Precision in Data Acquisition
Accuracy, precision and repeatability figure into data-acquisition-equipment
specifications in a particular way because of the digital-sampling
nature of data acquisition boards. Figure 3.2 illustrates the processing
of a voltage-input signal that increases linearly with time.
Click image to see full size
Figure 3.2: An ideal analog-to-digital convertor produces a stair-step
readout when presented with a steadily rising voltage input.
You can tell by the scales that the data acquisition card was set
up so that input voltages ranging from zero to 100 V read out in
engineering units from 0 to 100 units. Being analog, the input voltage
takes on a continuum of values. Being digital, the output takes
on only specific discrete values. In this case, Ive arranged
ten discrete values from 0 to 90. That is, this setup has a resolution
of one decimal digit. For a data acquisition card, the well-defined
term resolution serves as proxy for the generally poorly understood
term precision. In either case, we are talking about the smallest
difference the data acquisition card can reliably report. In this
case, thats one tenth of full scale or 10 units.
Figure 3.3 makes this point a little more clearly by comparing
low resolution and "high" resolution digitization. (By
the way, the correct term is "digitization." The word
"digitalization," which I sometimes see used is a medical
term that refers to treatment with the drug digitalis, which is
something even the most mercenary of health-care professionals will
not do to analog measurements.)
Click image to see full size
Figure 3.3: The analog-to-digital convertors resolution determines
the measurement precision.
The low-resolution version has only five levels, which makes it,
by anybodys lights, truly coarse, crude and low resolution.
The "high" resolution version is high only by comparison
with the low-resolution version. It has 20 digital levels. In an
era when 12-bit DAQ boards (which give 4,096 levels) are readily
available, a 20-level digitization (comparable to something absurd
like 4.325 bits) is really poor. It is, however, about the best
I can illustrate and still have the graph make its point.
So, measurement precision translates pretty clearly to ADC resolution.
What about accuracy?
Accuracy collects the effects of a whole bunch of error sources.
Figure 3.4 shows three obvious ones and how they relate to DAQ-card
operation. These errors have their counterparts in all measurement
Click image to see full size
Figure 3.4: Accuracy specifications encompass a number of error
For example, the freebie sectional plotter I described earlier
suffers from a good case of gain error. That is, the measured value
is equal to the true value times a constant other than one. In the
case of the plotter, the constant was 1.025. In the case of the
Gain Error graph, the constant is 0.80. (I know, because thats
what I put in the simulation to get the chart.)
Linearity error means that the transfer function between the input
and the output is a power law with quadratic or higher terms. In
other words, it aint straight. A step from one ADC output
level to the next means something different at one end of the scale
than it does at the other.
Gain error and linearity error are fairly consistent errors that
can be backed out of the data during post analysis if you have some
way of knowing the transfer function. In fact, the entire field
of calibration boils down to determining the gain and linearity
errors, then compensating for them to bring the measurement system
within its specifications. When you read in some specifications
that an instruments accuracy is plus-or-minus such-and-such,
it means that each instrument of that model has been calibrated
to produce readings within that range.
Your particular unit is going to have a unique transfer characteristic
within the range. Transfer characteristics will vary from unit to
unit, but will be stable for any given unit. In other words, when
you use your unit to measure 3.257 Volts, it will be off by some
amount. There are two things you can be sure of: the amount it is
off will be the same every time you use it to measure 3.257 Volts
and that amount will be within the stated accuracy for that model.
What I said in the last paragraph is true, provided the unit has
been calibrated per specification and is operating within its environmental
(temperature and humidity) specifications. Calibration intervals
arise from the fact that each units transfer characteristic
changes slowly over time in a predictable way. The cal lab knows
how fast the unit will drift and schedules a recalibration at intervals
that let them catch the unit before its readings drift out of tolerance.
If you use an out-of-calibration instrument of any kind, you simply
dont know just how bogus the readings are.
The third error source illustrated in Figure 3.4 is noise. Unlike
gain and linearity errors, noise is not repeatable. Therefore noise
is more of a repeatability issue than an accuracy issue.
I created the noisy graph by adding in plus or minus 0-10 units
from a random number generator before digitizing the simulated signal.
Every time I ran the simulation, I got an entirely different chart.
It works the same way in real life.
This kind of noise can get into acquired data in two ways: your
DAQ-board circuitry can inject it, or it can float in from outside.
Noise from data acquisition boards generally arises from thermal
noise in the front ends of instrumentation amplifiers. Generally,
you can figure that electromagnetic junk getting into your system
through the sensor or through the hookup leads will swamp what your
DAQ board puts in. Obviously, the DAQ-board manufacturer has no
control over what garbage you let in through the front end, so it
has nothing to do with quoted catalog specifications.
What you look for is a board whose noise specifications are considerably
better than what you can hope to achieve in your systems front
end. The boards noise gives a lower limit beyond which you
cannot expect to go. The manufacturer cannot give you any guarantee
of ever achieving it in your real system.
Precision (in the guise of digitizer resolution), accuracy (as
specified by gain error and linearity error) and repeatability (in
the form of instrumental noise level) apply to all instruments,
not just data acquisition products. They also appear in some similar
form whether the instrument is digital or analog.
Chapter 1 | Chapter
2 | Chapter 3 | Chapter 4