Page |1

Lecture 2
Errors in Measurement

 Types and Sources of Errors

 Error Propagation
 Effect of Averaging Results
 Method of Least Squares
 Calculations of Error
 Assignment

………………………………………………………………………………………………………………..

Errors in Measurements
 It is never possible to measure the true value of a dimension,
there is always some error. So, measurement is merely an
estimated value of dimension.
 The error in the measurement is the difference between the
measured value and the true value of measured dimensions.
Error in Measurement = Measured value-True value
 The error in measurement may be expressed or evaluated either
as an absolute error or as a relative error.

Absolute Error

True absolute error: It is the algebraic difference between the result of

measurement and the conventional true value of the quantity measured.

True Absolute Error = | Measured Value – Conventional True Value |

Apparent absolute error: If the series of measurement are made then the
algebraic difference between one of the result of measurement and the
arithmetical mean is known as apparent absolute error.

ApparentAbsolute Error = | MeasuredValue – ArithmeticMean Value |

Relative Error
 Page |2

It is the quotient of the absolute error and the value of comparison (may be true
value or the arithmetic mean of a series of measurements) used for calculation of

the absolute error.

It is an error with respect to the actual value.
Relative Error = | 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑉𝑎𝑙𝑢𝑒 − 𝑇𝑟𝑢𝑒/𝑀𝑒𝑎𝑛 𝑉𝑎𝑙𝑢𝑒 |
| 𝑇𝑟𝑢𝑒/𝑀𝑒𝑎𝑛 𝑉𝑎𝑙𝑢𝑒 |

Uncertainty
Uncertainty of measurement means doubt about the validity of the result of a
measurement. A measure of range of measurements from the average value. Also
called standard deviation.
Uncertainty characterize the dispersion of the measured quantity values.
Average Deviation: Estimation of Uncertainty by Repeated Measurements
The statistical method for finding a value with its uncertainty is to repeat the
measurement several times, find the average value, and find the either the
average deviation or the standard deviation.
Standard Deviation

The statistical measure of uncertainty. Average deviation

Factors Affecting the Amount of Error

 The accuracy and design of the measuring instruments
 The skill of the operator
 Method of measurement
 Page |3

 Temperaturevariations
 Elastic deformation of the part or instrument etc.

Sources of Errors
i. Calibration Error
Each measures instrument should be calibrated with a standard one at
certain time interval (may be once in a year or once in every 6 months.) If the
above procedure is not followed the instrument may give erroneous result, it
is called calibration errors.

ii. Environmental Error

These errors are due to surrounding pressure, temperature and humidity.
Internationally agreed standard value of temperature pressure are:
Temperature = 20 degree celcius
Pressure = 760 mm of Hg+ 10 mm of Hg vapour pressure
If the ambient condition varies from the above standard values the measured
value will be erroneous.

iii. Contact Pressure/Stylus Pressure

Errors are also introduced due to pressure exerted at stylus. It is more
prominent in case of soft work piece. Ideally the stylus should touch the top
surface of the w/p, due to stylus pressure both deformation and deflection of
w/p take place. This type of errors are also induced when the force applied on
the anvils of micrometer varies. To minimize this error: Skill inspector Fit with
ratchet mechanism, the ratchet slips when the applied pressure exceeds the
minimum required operating pressure.

iv. Error due to Supports

The elastic deformation/deflection of a long measuring bar due to position of
support cause error in measurement. So, Sir G.B. Airy found out the position
of supports to give minimum error. Two support conditions are shown in
figures.
 Page |4

Fig iii fig iv.

v. Error due to Alignment

 Abbe’s alignment principle: It states that “ the axis or line of measurement
should coincide with the axis of measuring instrument or line of the
measuring scale.”
 If while measuring the length of a work-piece the measuring scale is
inclined to the true line of the dimension being measured there will be an
error in the measurement.
 The length recorded will be more than the true length. This error is called
“Cosine error”. In many case the angle 𝜃 is very small and the error will be
negligible.

Cosine Error
Both sine and cosine error
If D = True diameter
L= apparent length
d = micrometer anvil diameter
Then, D = L cosθ – d sinθ
Error = L – D = L – (Lcosθ – d sinθ) = L(1 – cosθ) + d sinθ
 Page |5

The errors of above nature can be avoided by using gauges with spherical
ends.

vii. Error due to Dust

Presence of dust in the atmosphere change reading in the order of fraction of
micron. When high accuracy in measurement is required dust should be cleaned
by clean chamois cloth or by soft brush and gauges should never be touched by
moist fingers.
viii. Error due to Vibrations
The instrument anvil will not give consistent and repetitive reading if it is
subjected to vibration. So the measurement should be taken away from the
source of vibration, or use rubber padding for damping.

ix. Error due to Location

If the datum surface is not perfectly flat or if any foreign matter such as dirt, chip
etc are present between the datum and w/p, error occurs in measurement.

x. Error due to Poor Contact The measured dimension will be greater than the
actual dimension due to poor contact as shown in figure.
 Page |6

xi. Error due to Wear in Gauges

The anvil of micrometer is subjected to wear due to repeated use and lead to
error in measurement. The lack of parallelism due to wear of anvil can be checked
by optical flat.

Types of Errors
i.Controllable or Systematic Error
ii. Uncontrollable or Random Error

Controllable or Systematic Error

These type of errors are regularly repetitive in nature and are of similar form. If
properly analyzed, they can be determined and reduced effectively.
Systematic errors includes:
 Calibration errors,
 Ambient or atmospheric conditions,
 Stylus pressure error, etc.

Uncontrollable or Random Error

Random errors are accidental, non-consistent in nature and as they occur
randomly, they cannot be eliminated since no definite cause can be located.
The possible source of random errors are:
Small variations in the position of setting standard and work-piece.
light displacement of lever joints of measuring instruments.
Operator error in scale reading.
Fluctuations in the friction of measuring instrument etc.

SystematicError Random Error

These errors are repetitive in nature and These are non-consistent. The sources giving rise
are of constant and similar form. to such errors are random
 Page |7

These errors result from improper Such errors are inherent in the measuring system
conditions or procedures that are or measuring instruments
consistent in action.
Except personal errors, all other systematic Specific causes, magnitude and sense of these
errors can be reduced or eliminated. errors cannot be determined from the knowledge
of measuring system or condition.
If properly analyzed these can be These errors cannot be eliminated, but the results
determined and reduced or eliminated. obtained can be corrected.
These include calibration errors, variation These include errors caused due to variation in
in contact pressure, variation in position of setting standard and workpiece, errors
atmospheric conditions, parallax errors, due to displacement of lever joints of
mis-alignment errors etc. instruments, errors resulting from backlash,
friction etc.

During measurement several types of error may arise, these are:

1) Static errors which includes:
 Reading errors.
 Characteristic errors.
 Environment errors.
2) Instrument Loading errors:
3) Dynamic errors:

1)Static errors
These errors result from the physical nature of the various components of
measuring system. Static errors result from the intrinsic imperfections or
limitations in the hardware and apparatus compared to ideal instruments.
There are three basic sources of static errors.
 Reading Errors : These types of errors apply exclusively to instruments
These errors may be the result of parallax, readability, and interpolation.
 Parallax error creeps in when the line of sight is not perpendicular
to the measuring scale. The magnitude of parallax error increases
if the measuring scale is not made flush to the component.
 Interpolation error is the error resulting from the inexact
evaluation of the position of index with regards to two adjacent
graduation marks between which the index is located.
 Page |8

 Characteristic Errors : It is defined as the deviation of the output of the

measuring system from the theoretical predicted performance or from
nominal performance specifications.
linearity errors, repeatability, hysteresis and resolution errors are the
examples of characteristic errors.Calibration error is also included in
characteristic error.
 Environmental Errors : These errors results from the effect of
surrounding such as temperature, pressure, humidity etc. on measuring
system. Environmental errors of each component of the measuring
system make a separate contribution to the static error. It can be
reduced by controlling the atmosphere according to the specific
requirement.
2) Instrument loading errors
Instrument loading error is the difference between the value of the measurand
before and after the measuring system is connected/contacted for measurement.
Example: a soft or delicate component is subjected to deformation during
measurement due to the contactpressureof the instrument.

3) Dynamic error
Dynamic errors are 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.

……………………………………………………………………………………………………………………………
Propagation of Errors
Propagation of Errors The method of determining an uncertainty in a function of
the given independent variables each with an uncertainty is propagation of errors.
Page |9
P a g e | 10
P a g e | 11
P a g e | 12
 P a g e | 13

……………………………………………………………………………………………………………………………..
The Effect of Averaging Results
 The accuracy of determination is the amount by which it is estimated that a
measurement could deviate from its true value, but it is not necessarily true that it
does so.
 By chance the measurement could possibly be exactly right or it could deviate
from the true size by any fraction of the accuracy of determination. There is no
wayof knowing other than using a method of measurement with a better accuracy
of determination.
 The statistical parameters of arithmetic mean and standard deviation may be
used to assess random errors by taking a series of repeated measurements.
Thus, the complete measurements are repeated a great many times, then
frequency distribution is obtained by plotting a tally chart of these values.
P a g e | 14
 P a g e | 15

Graphical Methods
If an experiment is carried out to find the law relating two measured variables x and y, it
is usual to plot a graph of the readings and determine the law of graph by plotting a
mean line, i.e. we are averaging out the errors in the individual observations.
This is obviously a tedious process and a better method is known as the method of least
squares.
Method of Least Squares
The least square principle states that the most probable value of observed quantities is
that which renders the sum of the squares of residual errors to minimum.
A series of Observed Values Deviation of any particular observed value x from the most
probable value X is,
x-X
X=the most probable value
X1,x2,x3,x4,………….xn
 P a g e | 16

……………………………………………………………………………………………………………………………………
Assignment
1)Acylinder of 80 mm diameter was placed between the micrometer anvils. Due to
inaccurate placement, the angle between the micrometer and cylinder axis was found to
be 1 minute. Calculate the amount of error in the measured diameter of the above
cylinder if the micrometer anvil diameter is 6 mm. Use suitable approximations.
2) A test indicator is used to check the concentricity of a shaft but its stylus is so set that
its movement makes an angle of 350 with the normal to the shaft. If the total indicator
reading is 0.02 mm, calculate the true eccentricity.
 P a g e | 17

3)An error in measuring outside diameter D, inside diameter d and length of a hollow
cylinder was found to be 2%, 3% and 5% respectively. Determine the compound error in
measurement of volume of a cylinder, if D = 50mm, d = 30 mm and L= 80 mm.

5)The diameter of a steel ball is measured five times with a micrometer giving the
following results: 8.011, 8.005, 8.009, 8.014 and 8.011 mm.
(a) Calculate the most probable value of the diameter and its standard deviation and
parameters of the infinite population of similar measurements from which these are
supposed to have come.
(b) Another measurement yields 8.021 mm. Is it wrong? If wrong, comment the reason of
measurement being wrong.
6)The resistance value at a temperature t of a metal wire, Rt is given by the expression;
Rt=Ro(1+αt) where, Ro is the resistance at) 0°C and α is the resistance temperature
coefficient. The resistance values of metal wire at different temperature have been
tabulated as given below. Obtain the values of Ro and α using least square straight-line
fitting.
Temp. (°C) 20 40 60 80 100
Resistance (Ω) 107.5 117.0 117.0 128.0 142.5
P a g e | 18