Sources of Errors (Metrology)

1.14.
In any measurement, there is always a degree of uncertainty resulting from measurement
error, i.e. all measurements are inaccurate to some extent Measurement error is the difference
between the indicated and actual values of the measurand. The error could be expressed either as
an absolute error or on a relative scale, most commonly as a percentage of full scale. It is important
to examine fully the errors in measurement systems that cause these uncertainties, the meaning
and interpretations of these errors and methods of reducing or circumventing of errors. Each
component of the measuring system has sources of errors that can contribute to measurement error.
Instrument or indication errors may be caused by defects in manufacture of adjustment of
an instrument, imperfections in design, etc.
The error of measurement is the combined effect of component errors due to various causes.
There may be errors due to method of location, environmental errors, errors due to the properties
of object of measurement, viz. form deviation, surface roughness, rigidity, change in size due to
ageing etc., observation errors.
The total error of measurement includes indication errors, errors of gauge blocks or setting
standards, temperature change errors, and errors caused by the measuring force of the instrument.
During measurement several types of errors may arise such as static errors, instrument
loading errors or dynamic errors, and these errors can be broadly classified into two categories viz.
controllable errors and random errors.
Static Errors. These result from the physical nature of the various components of the
measuring system as that system responds to a fixed measurand input. Static errors result from
the intrinsic imperfections or limitations in the hardware and apparatus compared to ideal
instruments. The environmental effect and other external influences on the properties of the
apparatus also contribute to static erros. Other sources of static errors could be inexactness in the
calibration of the system, displaying the output of the measuring system in a way that requires
subjective interpretation by an observer. From above it could be concluded that static errors stem
from three basic sources : reading error, characteristic error and environmental error. In the
measurement of length of a surface table with a rule, these errors will be encountered when aligning
the ends of the rule and surface table, and when estimating the length of the table. The static error
divided by the measurement range (difference between the upper and lower limits of measurement)
gives the measurement precision. Reading error describes such factors as parallax, interpolation,
optical resolution (readability or output resolution). Reading errors apply exclusively to the readout
device and have no direct relationship with other types of errors within the measuring system.
Attempts have been made to reduce or eliminate the reading errors by relatively simple techniques.
Where there is possibility of error due to parallax, the use of mirror behind the readout pointer or
indicator virtually eliminates occurrence of this type of error. Interpolation error can be tackled by
increasing the optical resolution by using a magnifier over the scale in the vicinity of the pointer.
The use of digital readout devices is increasing tremendously for display purposes as it eliminates
most of the subjective reading errors usually made by the observer. However, there exists a
possibility of plus or minus one count error in digital readout devices also and its value can be
effectively reduced by arranging full range to correspond to huge number of pulses so that one pulse
has very negligible value. Digital counting devices are capable of counting each and every pulse,
however short may be the duration, but it is only during start and at stop that one pulse is likely
to be missed which can lead to error.
Environmental errors result from effect of surrounding temperature, pressure and humidity
on measuring system. It can be reduced by controlling the atmosphere according to estipulated
requirements. External influences like magnetic or electric fields, nuclear radiation, vibration or
shock, periodic or random motion etc., also lead to errors. It is important to note that these factors
affect both the measuring system and measurand, and usually the effects of these factors on each
component are independent. Thus the environmental errors of each component of the measuring
system make a separate contribution to the static errors. Due to this reason, the number of
environmental variables and external influences that could affect the measurement should be
minimised and where it is not possible to do so then their effect should be computed and taken into
account.
Characteristic error is defined as the deviation of the output of the measuring system under
constant environmental conditions from the theoretically predicted performance, or from nominal
performance specifications. If the theoretical output is a straight line, then linearity, hysteresis,
repeatability, and resolution errors are part of the characteristic error. Linearity errors, hysteresis
and repeatability errors are present to some degree in each component of a measuring system. Other
characteristic errors include gain errors and zero offset, often collectively called calibration errors.
Similar characteristic errors in each component of the measuring system tend to be additive.
Thus, system linearity is usually the sum of the errors in individual components ; and as such the
study of combination and accumulation of errors is very important and will be discussed later.
It has been found that the static erros introduced by the components of the measuring system
are the cause of major concern. However, the loading errors and dynamic errors which are generally
encountered in process measurements and not in the field of Metrology, will also be discussed in
brief here to complete the subject.
Loading errors result from the change in the measurand itself when it is being measured,
i.e. after the measuring system or instrument is connected for measurement. Instrument loading
error is thus the difference between the value of the measurand before and after the measurement
system is measured. One example of such an error could be the deformation of soft component under
contact pressure of measuring instrument. The effects of instrument loading are unavoidable and
must be determined specifically for each measurement and measurand. Such loading erros are often
the single greatest uncertainty in a physical measurement. Therefore, measuring system should
be selected such that its sensing element will minimise instrument loading error in the particular
measurement involved. In a steady state measurement, the cumulative effect of static errors and
instrument loading errors determines the accuracy of the measurement.
Dynamic error is caused by time variations in the measurand and results from the inability
of a measuring system to respond faithfully to a time-varying measurand. Usually the dynamic
response is limited by inertia, damping, friction or other physical constraints in the sensing or
readout or display system. Dynamic error is characterised by the frequency and phase response
(Bode criterion) of the system for the cyclic or periodic variations in the measurand input. For
random or transient inputs, the dynamic error is described by the time constant of response time.
In both the cases , it is essential that dynamic characteristics of the measuring system be known
before putting the system to measure time varying inputs.
It is thus seen that different errors entering into any observation arise due to a variety of
reasons. Many times it may not be possible to identify the source of errors. Therefore it is more
fruitful to classify errors according to the effects they produce rather than on the basis of sources
which produce them.
For statistical study and the study of accumulation of errors, errors are categorised as
controllable errors and random errors.

(a) Systematic or Controllable Errors

. Systematic error is just a euphemism for ex-
perimental mistakes. These are controllable in both their magnitude and sense. These can be
determined and reduced, if attempts are made to analyse them. However, they can not be revealed
by repeated observations. These errors either have a constant value or a value changing according
to a definite law. These can be due to:
1. Calibration Errors. The actual length of standards such as slip gauges and engraved scales
will vary from nominal value by small amount. Sometimes the instrument inertia, hysteresis effect
do not let the instrument translate with complete fidelity. Often signal transmission errors such as
drop in voltage along the wires between the transducer and the electric meter occur. For high order
accuracy these variations have positive significance and to minimise such variations calibration
curves must be used.
2. Ambient Conditions. Variations in the ambient conditions from internationally agreed
standard value of 20°C, barometric pressure 760 mm of mercury, and 10 mm of mercury vapour
pressure, can give rise to errors in the measured size of the component. Temperature is by far the
most significant of these ambient conditions and due correction is needed to obtain error free results.
3. Stylus Pressure. Error induced due to stylus pressure is also appreciable. Whenever any
component is measured under a definite stylus pressure both the deformation of the workpiece
surface and deflection of the workpiece shape will occur.
4. Avoidable Errors. These errors include the errors due to parallax and the effect of
misalignment of the workpiece centre. Instrument location errors such as placing a thermometer
in sunlight when attempting to measure air temperature also belong to this category.
5. Experimental arrangement being different from that assumed in theory.
6. Incorrect theory i.e., the presence of effects not taken into account.

(b) Random Errors.

These occur randomly and the specific cases of such errors cannot be
determined, but likely sources of this type of errors are small variations in the position of setting
standard and workpiece, slight displacement of lever joints in the measuring joints in measuring
instrument, transient fluctuation in the friction in the measuring instrument, and operator errors
in reading scale and pointer type displays or in reading engraved scale positions.

Characteristics of random errors

The various characteristics of random errors are:
— These are due to large number of unpredictable and fluctuating causes that can not be
controlled by the experimenter. Hence they are sometimes positive and sometimes
negative and of variable magnitude. Accordingly they get revealed by repeated observa-
tions.
— These are caused by friction and play in the instrument’s linkages, estimation of reading
by judging fractional part of a scale division, by errors in positioning the measured object,
etc.
— These are variable in magnitude and sign and are introduced by the very process of
observation itself.
— The frequency of the occurrence of random errors depends on the occurrenceprobability
for different values of random errors.
— Random errors show up as various indication values within the specified limits of error
in a series of measurements of a given dimension.
— The probability of occurrence is equal for positive and negative errors of the same
absolute value since random errors follow normal frequency distribution.
— Random errors of larger absolute value are rather than those of smaller values.
— The arithmetic mean of random errors in a given series of measurements approaches
zero as the number of measurements increases.
— For each method of measurement, random errors do not exceed a certain definite value.
Errors exceeding this value are regarded as gross errors (errors which greatly distort
the results and need to be ignored).
— The most reliable value of the size being sought in a series of measurements is the
arithmetic mean of the results obtained.
— The main characteristic of random errors, which is used to determine the maximum
measuring error, is the standard deviation.
— The maximum error for a given method of measurement is determined as three times
the standard deviation.
— The maximum error determines the spread of possible random error values.
— The standard deviation and the maximum error determine the accuracy of a single
measurement in a given series.
From the above, it is clear that systematic errors are those which are repeated consistently
with repetition of the experiment, whereas Random Errors are those which are accidental and
whose magnitude and sign cannot be predicted from knowledge of measuring system and conditions
of measurement.

Spurious errors and Dixon test.

Errors due to operator mistakes or malfunction of
instrument are called spurious errors and need to be ignored in the statistical analysis. Statistical
outlier or Dixon test is applied to discard spurious readings. All good observations follow normal
distribution and spurious reading will fall outside the normal distribution. In Dixon test, all
observations are arranged in ascending order if spurious reading is suspected for high value, or
descending order if spurious reading is suspected for low value.

Statistical Treatment of Errors.

Random and systematic errors are evaluated and studied
by statistical procedures which make it possible to state from a limited group of data the most
probable value of a quantity, the probable uncertainty of a single observation, and the probable
limits of uncertainty of the best value that can be derived from the data. It may be noted that the
object of the statistical methods ; based on laws of chance which operate only on random errors and
not on systematic errors ; is to achieve consistency (precision) of value and not their accuracy
(approach to the truth).
It is also important to note that in quality control of a product we must consider variations
in the repeat measurement of a single part as well the variations in the single measurements of a
large number of’so-called’ identical parts. The first is largely due to error in the instrument whereas
in the second there is also a contribution caused by variations as a result of the manufacturing
process. The first is the study of errors (dealt here) and the second is the subject of statistical quality
control (dealt in chapter 18).
Let us first understand some terms used in statistical analysis as under :
Population of Measurement. An infinite number of independent measurements carried
out for determination of a certain quantity constitute a population.
Sample of Measurements. In practice, only a finite number of measurements are carried
out for determination of a certain quantity which constitute a sample.
Sample Mean. If x\, x2, x3,… xn be n measurements then sample mean 3c is

Sample Standard Deviation. Sample standard deviation V is defined as

Population Mean. The limiting value of sample mean as number of measurements tends
to infinity is called population mean

Population Standard Deviation. The limiting value of sample standard deviation as
number of measurements tends to infinity is called population standard deviation

Estimate of Population Standard Deviation. An estimate S of the population standard
deviation is obtained from sample standard deviation as

Random Uncertainty (Ur).

where t is Student’s't’ factor and S/Vre is the standard error of the mean, assuming that measure-
ments follow the Gaussian (Normal) distribution.

Systematic Uncertainty (US)

Contributions due to measuring instruments, operating conditions and inherent charac-
teristics of the instrument are taken into consideration.
Uncertainty reported in the certificates of calibration for measurement standards and
instruments normally follow Rectangular Distribution with semi-range a. Then variance may be
taken a a2/3. Systematic uncertainty Us – K as.
K is the value of student’t’ for n = <» at the desired confidence level and value of K = 0.675
for CL of 50%, 1.00 for CL of 68.3%, 1.96 for CL of 95%, 2.58 for CL of 99% and 3.0 for CL of 99.7%.

Overall Uncertainty U

There is no universal agreement for combining the systematic and random uncertainties.
One view is to add the two, another is to use the quadrature method while third is to report them
separately.

Principles of Least Squares.

Assessment of deviation of errors relative to some particular
datum may be done with the help of the principle of least squares. The principle states that the
most probable value of observed quantities is that which renders the sum of the squares of residual
errors a minimum.
Let a number of repeated readings on a component be represented by xlf x2, £3,……xn. It can
now be shown by least squares principle that the most probable value of the series of observed
results is the arithmetic mean 2 x/n as follows :
Let the most probable value be assumed to be x’. Then the deviation of any particular value
x from the most probable value x’ is Or – x’). From the least square principle 2(x – x’) 2 should be
minimum, i.e..

Error distribution. Virtually all instrument errors are random in nature. Exceptions to
these, such as non-linearity errors and other errors are called systematic errors.
Random errors have positive and negative values and their magnitudes are generally
distributed in accordance with the Gaussian Distribution—the familiar bell-shaped curve shown
in Fig. 1.6.

where P (x) ■ probability density, x ■ error value
P(x\ < x< x2) ■ probability that x (error value) lies within the interval x\, x2.
The curve (Fig. 1.6) and its mathematical
expression [equation (1.1)] represents the prob-
ability distribution of the random errors. Since an
error within the limits – oo to + oo is certain to occur,
the area under the curve is numerically 1; repre-
senting a probability of 1. The probability that the
error value lies betweenxi and”x2> P(x\ <x<x2) is
simply the area under the curve between these two
points [equation (1.2)]. On a frequency basis, the
area under the curve between error values xi and
x2 represents the percentage of all errors lying
between these two values.

Systematic errors are certain to occur and are, therefore, not treated statistically. If such
errors are present in a system of random errors, they are simply added directly to the statistically
combined random errors.

Errors Accumulation.

The total static error of a measurement system can be measured
in terms of root-mean-square (rms) of the component characteristic errors, if the following conditions
are fulfilled:
Component characteristic errors are independent and of the same order of magnitude, and
the distribution of errors is normal (Gaussian), i.e. we consider only the random errors ; and
wherever possible this latter condition should be verified by experimental analysis.
The total static error of a measuring system, therefore,

where LE = linearity errors of individual component; RE = reading errors,
CE – characteristic errors (other than linearity) and EE = environmental errors.

Variance of Error Distribution

The basic measure of the random error distribution is the variance (a2) which indicates the
spread or dispersion of the distribution function. Mathematically it is defined as :

If the variance is large, the error distribution curve (Fig. 1.6) is broad (curve B). Conversely,
if the variance is small, the error distribution curve is quite narrow and peaked (curve C).
The square root of the variance or the “root mean square error” is also called the standard
deviation (a). It can be used to evaluate the most probable value of the measurement and is also
quite useful in statistical analysis of Gaussian error distributions. For example, the mathematical
nature of the Gaussian error distribution function is such that 68% of the errors represented by the
distribution lie between the limits of ± la, 94% between ± 2a, and 99.7% between ± 3a. Most of the
instrument errors generally expressed are based on the 3a limits. Thus, allowing for an error three
times as great as the RMS static error gives a 99.7% probability that the measurement error is no
greater.