CHAPTER 3 - MEASUREMENT ACCURACY

2. Definitions of Terms Related to Accuracy

**Precision **is the ability to produce the same value within given
accuracy bounds when successive readings of a specific quantity are measured.
Precision represents the maximum departure of all readings from the mean
value of the readings. Thus, a measurement cannot be more accurate than
the inherent precision of the combined primary and secondary precision.
**Error** is the deviation of a measurement, observation, or calculation
from the truth. The deviation can be small and inherent in the structure
and functioning of the system and be within the bounds or limits specified.
Lack of care and mistakes during fabrication, installation, and use can
often cause large errors well outside expected performance bounds. Since
the true value is seldom known, some investigators prefer to use the term
**Uncertainty**. Uncertainty describes the possible error or range of
error which may exist. Investigators often classify errors and uncertainties
into spurious, systematic, and random types.

**Spurious errors** are commonly caused by accident, resulting in
false data. Misreading and intermittent mechanical malfunction can cause
discharge readings well outside of expected random statistical distribution
about the mean. A hurried operator might incorrectly estimate discharge.
Spurious errors can be minimized by good supervision, maintenance, inspection,
and training. Experienced, well-trained operators are more likely to recognize
readings that are significantly out of the expected range of deviation.
Unexpected spiral flow and blockages of flow in the approach or in the
device itself can cause spurious errors. Repeating measurements does not
provide any information on spurious error unless repetitions occur before
and after the introduction of the error. On a statistical basis, spurious
errors confound evaluation of accuracy performance.

**Systematic errors **are errors that persist and cannot be considered
entirely random. Systematic errors are caused by deviations from standard
device dimensions. Systematic errors cannot be detected by repeated measurements.
They usually cause persistent error on one side of the true value. For
example, error in determining the crest elevation for setting staff or
recorder chart gage zeros relative to actual elevation of a weir crest
causes systematic error. The error for this case can be corrected when
discovered by adjusting to accurate dimensional measurements. Worn, broken,
and defective flowmeter parts, such as a permanently deformed, over-stretched
spring, can cause systematic errors. This kind of systematic error is corrected
by maintenance or replacement of parts or the entire meter. Fabrication
error comes from dimensional deviation of fabrication or construction allowed
because of limited ability to exactly reproduce important standard dimensions
that govern pressure or heads in measuring devices. Allowable tolerances
produce small systematic errors which should be specified.

Calibration equations can have systematic errors, depending on the quality of their derivation and selection of form. Equation errors are introduced by selection of equation forms that usually only approximate calibration data. These errors can be reduced by finding better equations or by using more than one equation to cover specific ranges of measurement. In some cases, tables and plotted curves are the only way to present calibration data.

**Random errors** are caused by such things as the estimating required
between the smallest division on a head measurement device and water surface
waves at a head measuring device. Loose linkages between parts of flowmeters
provide room for random movement of parts relative to each other, causing
subsequent random output errors. Repeating readings decreases average random
error by a factor of the square root of the number of readings.

**Total error** of a measurement is the result of systematic and
random errors caused by component parts and factors related to the entire
system. Sometimes, error limits of all component factors are well known.
In this case, total limits of simpler systems can be determined by computation
(Bos et al., 1991). In more complicated cases, different investigators
may not agree on how to combine the limits. In this case, only a thorough
calibration of the entire system as a unit will resolve the difference.
In any case, it is better to do error analysis with data where entire system
parts are operating simultaneously and compare discharge measurement against
an adequate discharge comparison standard.

**Calibration **is the process used to check or adjust the output
of a measuring device in convenient units of gradations. During calibration,
manufacturers also determine robustness of equation forms and coefficients
and collect sufficient data to statistically define accuracy performance
limits. In the case of long-throated flumes and weirs, calibration can
be done by computers using hydraulic theory. Users often do less rigorous
calibration of devices in the field to check and help correct for problems
of incorrect use and installation of devices or structural settlement.
A calibration is no better than the comparison standards used during calibration.

**Comparison standards** for water measurement are systems or devices
capable of measuring discharge to within limits at least equal to the desired
limits for the device being calibrated. Outside of the functioning capability
of the primary and secondary elements, the quality of the comparison standard
governs the quality of calibration.

**Discrepancy** is simply the difference of two measurements of the
same quantity. Even if measured in two different ways, discrepancy does
not indicate error with any confidence unless the accuracy capability of
one of the measurement techniques is fully known and can be considered
a working standard or better. Statistical **deviation** is the difference
or departure of a set of measured values from the arithmetic mean.

**Standard Deviation Estimate** is the measure of dispersion of a
set of data in its distribution about the mean of the set. Arithmetically,
it is the square root of the mean of the square of deviations, but sometimes
it is called the root mean square deviation. In equation form, the estimate
of standard deviation is:

(3-1)

where:

*S* = the estimate of standard deviation

*X _{Avg} *= the mean of a set of values

*X _{Ind} *= each individual value from the set

*N* = the number of values in a set

= summation

The variable *X* can be replaced with data related to water measurement
such as discharge coefficients, measuring heads, and forms of differences
of discharge.

The sample number, *N,* is used to calculate the mean of all the
individual deviations, and (*N* - 1) is used to calculate the estimate
of standard deviation. This is done because when you know the mean of the
set of *N* values and any subset of (*N* - 1) values, the one
excluded value can be calculated. Using (*N*-1) in the calculation
is important for a small number of readings.

For the sample size that is large enough, and if the mean of the individual
deviations is close to zero and the maximum deviation is less than __+__3*S*,
the sample distribution can be considered normally distributed. With normal
distribution, it is expected that any additional measured value would be
within __+__3*S* with a 99.7 percent chance, __+__2*S*
with a 95.4 percent chance, and __+__*S* with a 68.3 percent chance.

Measurement device specifications often state accuracy capability as
plus or minus some percentage of discharge, meaning without actually stating,
__+__2*S*, two times the standard deviation of discharge comparisons
from a calibration. However, the user should expect an infrequent deviation
of __+__3*S*.

Error in water measurement is commonly expressed in percent of comparison standard discharge as follows:

(3-2)

where:

*Q _{Ind}* = indicated discharge from device output

*Q _{Cs}* = comparison standard discharge concurrently measured
in a much more precise way

*E%QCS*= error in percent
comparison standard discharge

Comparison standard discharge is sometimes called actual discharge, but it is an ideal value that can only be approached by using a much more precise and accurate method than the device being checked.

Water providers might encounter other terms used by instrument and electronic manufacturers. Some of these terms will be described. However, no universal agreement exists for the definition of these terms. Therefore, water providers and users should not hesitate to ask manufacturers' salespeople exactly what they mean or how they define terms used in their performance and accuracy claims. Cooper (1978) is one of the many good references on electronic instrumentation.

Error in **percent full scale**, commonly used in electronics and
instrumentation specifications, is defined as:

(3-3)

where:

*Q _{Ind}* = indicated discharge

*Q _{Cs}* = comparison standard discharge concurrently measured

*Q _{FS}* = full scale or maximum discharge

*E _{%QFS
}*= error in percent full-scale discharge

To simply state that a meter is "3 percent accurate" is incomplete. Inspection of equations 3-2 and 3-3 shows that a percentage error statement requires an accompanying definition of terms used in both the numerator and denominator of the equations.

For example, a flowmeter having a full scale of 10 cubic feet per second
(ft^{3}/s) and a full scale accuracy of 1 percent would be accurate
to __+__0.1 ft^{3}/s for all discharges in the flowmeter measurement
range. Some manufacturers state accuracy as 1 percent of measured value.
In this case, the same example flowmeter would be accurate to within __+__0.1
ft^{3}/s at full scale; and correspondingly, a reading of 5 ft^{3}/s
would be accurate to within __+__0.05 ft^{3}/s for the same
flowmeter at that measurement.