CE207L
Measurement Error
A major part of engineering work is making and reporting measurements. When communicating
technical information such as numerical quantities, it is important to report numbers so that the
reader can know that the results are representative of the precision achieved by the original
measurement. To do otherwise may give your reader an incorrect idea about how well your
answer represents reality. Since making completely error-free measurements is an elusive goal,
you must understand and be able to correctly communicate numerical results so that your
audience will understand or at least be given information about the accuracy of your work.
Measurement terms (ASTM D 2777-96)
accuracy - a measure of the degree of conformity of a single test result generated by a
specific procedure to the assumed or accepted true value and includes both precision and
bias.
bias – the persistent positive or negative deviation of the average value of a test method
from the assumed or accepted true value.
precision – the degree of agreement of repeated measurements of the same property,
expressed in terms of dispersion of test results about the arithmetical mean result
obtained by repetitive testing of a homogeneous sample under specified conditions. The
precision of a test method is expressed quantitatively as the standard deviation computed
from the results of a series of controlled determinations.
Measurement Error
The difference between the true value of a measured object or parameter and the observed value
of what is measured is called error. Sources of error include
1. instrument errors, such as manufacturing or calibration defects
2. user problems typically related to eyesight limitations including lighting, physical
problems, etc., or user inexperience, mistakes etc.
3. inability to make a careful measurement due to environmental conditions, physical
obstructions, changes in the object being measured, etc.
4. errors made in writing the original measurement value or in subsequent operations.
There are two general categories of error – systematic (bias) error and random (precision) error.
An illustration of each is given in Figure 1.
1. Systematic or bias error occurs when a measurement or group of measurements is
consistently offset by some amount higher or lower than the true value. In many cases
this type of error can be discovered and accounted for to make a correction to the original
readings. Systematic errors tend to accumulate proportionately to the number of
measurements made and used in subsequent calculations (Moffitt 1998).
2. Random or precision error occurs when the magnitude and sign of the error cannot be
predicted. Although random errors may not be easily corrected, the effects of the random
errors may be reduced by taking the average of several readings or by using other
mathematical techniques that cause a part of the errors to cancel each other. Random
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CE207L - Measurement Error
errors tend to grow proportionately to the square root of the number of measurements
(Moffitt 1998).
As shown in Figure 1, the true value of a measured quantity can be estimated by adding the bias
error to the average measured reading. For example, if the measured length of an object averaged
over 7 readings is 12.56 mm and the known bias error is 1.02 mm, then the true length would be
12.56 mm + 1.02 mm = 13.58 mm.
Figure 1 - Illustration of measurement bias and precision errors.
measured value
true value
bias error
average value
precision error
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7
measurement readings
The goal of making any measurement is to get a true (perfect) indication of what the actual value
is. In order to do this, the measurement technique or equipment must be precise and unbiased.
To be able to do this means that an accurate measurement can be made. About the only way a
measurement may be made with zero error is by chance. If an error is made due to an obvious or
known mistake, such as by malfunctioning equipment or personnel, it should be noted as such
and omitted from related calculations and conclusions.
Making a measurement
To make an accurate measurement the appropriate instrument must be selected for the task.
Table 1 summarizes a variety of measurement task and equipment used in our labs to perform
those measurements. Several points need to be considered before the most appropriate
instrument choice can be made.
1. The type of measurement – length, weight/mass, time, pressure (differential or gage),
temperature, strain, electrical potential in volts, electrical current in amperes, etc.
2. Must the instrument make physical/electrical contact with the object, and if so, how? Are
there any obstructions to overcome?
3. What other conditions or factors might complicate the measurement task – accessibility,
time of test, time duration of test, vibration, personnel requirements/training/availability,
etc.?
4. How accurate must the measurement be? Will one test suffice or must there be multiple
test samples taken?
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Table 1 - Measurement tasks and the typical instruments used in CE labs
Measurement Instrument
type
Typical capacities
Length
Scale(ruler)
typically up to 3 ft
(1m)
Surveyor’s tape
Mechanical micrometer
100 ft
0-1inch
Mechanical vernier caliper
Electronic micrometer
Electronic vernier caliper
Linear displacement transducer
Electronic scale
Electronic scale
0-6 inch, 0-12 inch
Weight/mass
Pressure
Temperature
Pressure transducer, gage
Pressure transducer, differential
Pressure transducer, differential
Mechanical thermometer
Digital thermometer
Time
Electronic timer, manual
operation
Electrical
Potential
0 to 2 inches
4100 x 0.01 g
8100 x 0.1 g
125 psi
0- 138 inch H2O
0- 15 inch H2O
0 – 300 VDC
Typical accuracy
of instrument and
readout system
+/- 1/32 inch to
0.5% of the full
scale
+/- 0.01 foot
+/-0.001 inch
+/-0.001 inch
+/-0.001 inch
+/-0.001 inch
+/- 0.001 inch
+/- 5 psi
+/- 1 inch H2O
+/- 0.2 inch H2O
+/- 1° C to 5° C
+/- 1° C
+0/-0.1 sec and
depends on manual
operation to
start/stop timer
+/- 0.1 % of full
scale
Using measuring instruments
Instrument precision generally means that a measurement device will allow the person making
the measurement to read the dimension in sufficiently finely divided increments and to be able to
observe the same measurement repeatedly. In order to use the device and report its output in
calculations or a report, the user must know something about the instruments accuracy. The
discussion below details some things to be considered for selecting and using certain
instruments. While specific devices are covered here, the general comcepts can be applied to
most measurement situations.
Linear measurements
A mechanical micrometer with a one-inch range may be used to measure anything from the
thickness of a sheet of paper to another object up to one inch thick. Since a sheet of paper may
have an average thickness of a few thousandths of an inch, it would be desirable to be able to
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make a measurement in 0.001-inch increments over the entire expected range. Provided the
micrometer is designed to make accurate measurements to the nearest 0.001 inch, any such
measurement could be reported to the nearest 0.001 inch. If you had a mechanical micrometer
with a vernier scale, or a digital version, that enabled readings to the nearest 0.0001 (1x10-4
inch) you could report your readings to the nearest 0.0001 inch.
On the other hand, consider a standard metal scale marked to one sixty-fourth of an inch as
shown in Figure 2. Even though one sixty-fourth inch is numerically equivalent to 0.015625
inch, you should not report all of those digits in a report for at least two reasons:
1. To do so would imply that you were able to read the measurement to the nearest 1x10-6
inch, which is not possible for the average person.
2. The level of error in a scale this finely divided is probably not much better than one-half
the smallest marked increment, or the width of the marks on the ruler, whichever is
greater.
Figure 2 - A typical steel scale marked in sixty-fourths of an inch.
What would be the appropriate number to report in this case? You should report to a level of
precision that you can defend if asked to verify your measurements. Assuming that your visual
ability allows you to determine which mark on the scale is closest to the reference on the object
you are measuring, then you should round to the nearest ½ mark. Since one sixty-fourth is about
0.0156 and one-half of that is 0.0078, you should round to the nearest 0.01 inch since that is
about the finest increment you are likely to resolve.
A linear displacement transducer (LDT) is an electrical device typically designed to work with a
signal conditioner/digital readout device and calibrated with the resolution of both the LDT and
readout devices in mind. In our labs, the LDTs have nearly infinite resolution but a stated
nonlinearity of 0.1 percent (1/1000) of the full scale value. What this means is that for a 1-inch
capacity LDT at any point within the 1-inch range the output from the device could be as much
as 1 inch x 1/1000 = 0.001 inch away from the true value. The digital display units we use have
accuracy (linearity) of 0.01 percent of full scale although the electrical output of the transducer
will normally stay well within the display units range. Since the linearity error of the LDT is
several times greater than that of the digital display it is presumed that the LDT error control the
final error level of the system. Therefore, we can say that the accuracy of the LDT system in our
labs will be plus or minus 0.001 inch.
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