Debre Markos University

Civil Engineering Dep’t

By: Demeke S.
 Chapter Two
Units of measurement and theory
of errors

Out line
Introduction to units of measurement
Conversion of units
Scale
Theory of errors
 What is measurement?
♣ The word measurement is derived from the
Greek word "metron," which means a
limited proportion
♣ In science, measurement is the process of
estimating or determining the magnitude of
a quantity
♣ The term measurement can also be refers to
a specific result obtained from a quantity
 Units of measurement
♣ A unit of measurement is a definite
magnitude of a physical quantity
♣ Defined and adopted by convention and/or
by law
♣ It is used as a standard for measurement of
the same physical quantities
---For example, length is a physical quantity
and meter is a unit of length that represents a
definite predetermined length.
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 Systems of measurement

♣ There are two main systems of

measurements. These are:
i. Metric system (SI = System International)
ii. British (Imperial) system

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 ….cont’d
i. Metric System
♣ First adopted by France in 1791
♣ The International System of
Units (abbreviated SI from the French
d'Unités is the
Le Système International d'Unités)
modern form of the metric system and is
generally a system of units of measurement
devised around seven base units and the
convenience of the number ten
Table 2.1 The Seven SI base Units

Name/ unit Unit Quantity Symb

symbol ol
Meter m length l

Kilogram kg mass m

Second s time t

Ampere A electric current I

Kelvin K thermodynamic T
temperature
Candela cd luminous intensity Iv

Mole mol amount of substance n

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 ….cont’d
♣ It is the most widely used system of
measurement, both commerce and science
♣ Metric units are universally used in scientific
work, and widely used around the world for
personal and commercial purposes
♣ The goal of the metric system is to have a
single unit for every physical quantity and
to avoid the need for conversion factors
when making calculations with physical
quantities
 ….cont’d

ii. British (Imperial) system

♣ The imperial system is one of many
systems of English or foot-pound-second
units, so named because of the base units
of length, mass and time

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A standard set of prefixes in powers of ten may
be used to derive larger and smaller units from
the base units
 SI units of different measurement
i. The SI unit of length is the base unit meter (m)
1 Decameter = 101m = 1dam
1 hectometer = 102m = 1hm
1 kilometer = 103m = 1km
1 mega meter = 106m = 1Mm
1 giga meter = 109m = 1Gm
1 tetrameter = 1012m = 1Tm
1 Decimeter = 10-1m = 1dm
1 centimeter = 10-2m = 1cm
1 Millimeter = 10-3m = 1mn
1 micrometer = 10-6m = 1Mm
1 nano meter = 10-9m = 1nm
1 Pico meter = 10-12m = 1pm
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 ….cont’d
ii. The SI unit for area, is the derived units meter
square (m2)
1 hectare = 104m2 = 100 X 100 m = 1 ha
1 square kilometer = 106 m 2 = 1000m X 1000m
= 1km2 = 100 ha
iii. The SI unit for volume is the derived unit cubic
meter (m3)
1000 cu millimeters = 1 cubic centimeter meter
1000 cu centimeters = 1 cubic decimeter
1000 cu decimeters = 1 cubic meter
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 iv. The SI units for plane Angles
♣ There are three systems in use for angular
unit, namely sexagesimal graduation,
centesimal graduation and radian

a. Sexagesimal graduation: a circle is divided in

to 360 parts
1 full circle = 3600 (degrees)
10 = 1/360 full circle = 2π/360 rad = π/180 rad
10 = 60’ (minutes) and 1’ = 60 ‘’ (seconds)

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 ….cont’d
b. Centesimal graduation: The circle is
divided in to 400 parts.
1 full circle = 400 gon/grad
1 gon = 1/400 full circle = 2π/400rad =
π/200rad
1 gon = 100 c gon (centigon)
1 c gon = 10 milli gon (milligon)
1 mgon = 10cc (centicentigon)

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 ….cont’d
c. Radian: - The radian (rad is the basic unit
of measurement of angles; one radian is
defined as the angle subtended at the
center of a circle by an arc length exactly
equal to the radius of the circle. The
circumference of a circle equals 2π
radians in a circle
3600 = 2π radian and
1 radian = 57.300
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 Conversion of Units
♣ We can convert measurements from one unit to
another unit within the same system (English or
Metric) or between the two systems.
♣ To convert measurements, it is necessary to
know conversion factors between
measurements.
♣ A conversion factor is a clever way of writing 1
as a fraction in which the numerator is equal to
the denominator but the numerator and the
denominator have different units.

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 ….cont’d

» Conversion of length » Conversion of Angles

1 inch = 2.54 cm 1 gon = 9/10 deg
1 foot = 0.3048m 1 deg = 10/9 gon,
1 mile = 1.6093 km 400 gon = 3600

» Conversion of area » Conversion of Volume

1 sq. in = 6.4516 sq. cm 1 cu. In = 16.387 cu.cm
1 sq. Ft = 0.0929 sq. cm 1 cu ft = 0.0283 cu.m
1 sq. mile = 2.59 sq. km
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Examples

1. Convert 50.4460 to D-M-S

2. Convert 256016’54” to gon
3. Convert 348 gon to decimal degree
4. Convert 2.5 sq. kilometer to sq. meter
5. Convert 150 feet to yard

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 Scale of a map
♣ The scale of a map is defined as the ratio
of a distance on the map to the
corresponding distance on the ground
♣ The scale of the map permits the user to
convert distance on the map to distance on
the ground or vice versa
♣ Scales of a map are generally classified as
large, medium and small
♣ A large scale map shows the features in a
bigger size than a small scale map
 ….cont’d
♣ Large denominator numbers refer to small
scale, where as small denominator
numbers are indicative of a large scale
-----Large scale:- 1: 1,000 or more
example 1:500
-----Medium scale:- 1:1,000 to 1:10,000
example 1: 5,000
-----Small Scale:- 1: 10,000 or less
example 1:50,000
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 Representations of Scale
Scale of a map can be represented:
i. By statement (e.g.- engineer’s scale)
ii. By representative fraction
iii. By graphical Scale

i. By statement (engineer’s Scale)-According to

this representation, a specified distance on the
map represents the corresponding distance on
the ground.
For example 1cm = 100 meters, that means
1cm on the map represents 100m on the ground.
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 ….cont’d
ii. By representative fraction (RF)-
--------This scale is usually written as a fraction and is called
the representative fraction.
--------The RF is always written with the map distance as 1
and is independent of any unit of measure (yards,
meters, inches, and so forth).

R= dm/da Where:
dm = map distance
da = distance on the ground
Note:
The unit in the numerator and denominator must be the
same (scale is unit less).
E.g. 1:5000 or 1/5000, that means 1 cm on the map
represents 5000 cm (50m) on the ground.
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 ….cont’d
iii. By graphical Scale:
------A graphical Scale is a ruler printed on the map
so that its map distance corresponds to a
convenient unit of length on the ground
------It is used to convert distances on the map to
actual ground distances
------It is also used to determine straight line
distance between two points on a map

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May 31, 2024 24
Reading assignment
------Types of measuring scales
i. Plain Scale
ii. Comparative scale
iii. Diagonal Scale
iv. Vernier Scale
------Conversion of representative scale to
graphical scale

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 Theory of Errors
• The measurement of a quantity is based on
some International fundamental standards.
• These fundamental standards are perfectly
accurate, while others are derived from
these. These derived standards are not
perfectly accurate in spite of all precautions.

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 Theory of error
• In general, measurement of any quantity is
done by comparing with derived standards
which themselves are not perfectly
accurate.
• So, the error in the measurement is not
only due to error in methods but also due
to standards (derived) not being perfectly
accurate. Thus, the measurement with
100% accuracy is not possible with any
method.
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 Theory of errors
♠ Surveying is concerned with measurements of
quantity whose exact or true value may not
determined.
♠ It can be stated unconditionally that
1. no measurement is exact,
2. every measurement contains errors,
3. the true value of a measurement is
never known, and thus
4. the exact sizes of the errors present are
always unknown.
****The true value of a quantity is, a value which is,
absolutely free from all types of errors*****
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 ….cont’d
♠ The true value cannot be determined because
some errors always creep in the measured
quantities
♠ The surveyor must have the skill and judgment
to make very accurate measurement
 As better equipment is developed,
 environmental conditions improve,
 and observer ability increases,
 observations will approach their true values
more closely, but they can never be exact
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 Errors and mistake

♠ Error is the difference, after blunders have

been eliminated, between a measured or
calculated value of a quantity and the true
value of that quantity.

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 Mistakes/ blunder
♠ A blunder, also called a mistake, is an
unpredictable, human mistake. Although a small
blunder may remain undetected and have the
same effect as an error, it is not an error.
♠ Mistakes occur in measurements due to
carelessness, inattention, fatigue, miss-
communication, inexperience or poor judgment
of the surveyor.
For example, recording 79.36 or 73.69 instead
of 73.96

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 Examples (mistakes)
Transposing two numbers
Neglecting to level an instrument
Not placing the sighting point over the
correct point
Misplacing the decimal point
Misunderstanding a call out
Not sighting the point that corresponds to
the point name or number put in the data
collector
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Mistakes are detected and eliminated by
using proper procedures, such as:
Checking each recorded and calculated
value
Making independent and redundant
measure check observations and
measurements
Making redundant measurements that
allow closure computation of sections of
the entire survey
Calculating repeatedly
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Source of Errors:
Depending on sources of origin, errors in measurements
fall into three classes.

They are:
1. Instrumental Errors
2. Natural Errors
3. Personal Errors
 Sources of errors
1. Instrumental errors: are caused by imperfections in

instrument construction or adjustment of

surveying instruments,
 Misalignment of various part of the instrument

 Optical distortions causing “what you see is not

exactly what you are supposed to see”

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 ….cont’d
For example, the divisions on a theodolite or
total station instrument may not be spaced
uniformly.

Another example, if the tape used in measuring

the distance is actually 29.95m long where as
the nominal length is 30m, the instrumental
error occurs because of the imperfect tape.
 ….cont’d
♠ Most instrumental errors are eliminated by using
proper procedures, such as
 observing angles in direct and reverse modes,
 balancing foresights and back sights and
 repeating measurements.
♠ Since not all instrument errors can be
eliminated by procedures, instruments must be
periodically checked, tested and adjusted (or
calibrated.) Instruments must be on a
maintenance schedule to prevent inaccurate
measurements.
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Source of Errors: Natural Errors
2. Natural Errors: These are caused due to variations in
naturl phenomena such as.,
variations in:
o wind,
o temperature,
o humidity,
o refraction,
o gravity and
o magnetic field of the earth.
 ….cont’d
Examples of natural errors are:
 A steel tape whose length varies with changes
in temperature.
 Sun spots activity and its impact on the
ionosphere, hence on GPS surveying.

For example if a tape has been calibrated at 20ºc,

but the field temperature is 30ºc there will be a
natural error due to temperature variation.
May 31, 2024 39
 ….cont’d
♠ Natural errors are mostly systematic and
should be corrected or modeled in the
adjustment.
♠ Some natural errors such as the effect of
curvature and refraction in leveling can be
eliminated by balancing backsight and
foresight

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Source of Errors: Personal Errors
3. Personal Errors:

These arise from limitations of the human senses of:

o sight,
o touch,
o hearing.
 ….cont’d
-----Personal errors can be characterized as either
systematic or random.

----- An example of a personal error is an error in

the measured value of a horizontal angle,
caused by the inability to hold a range pole
perfectly in the direction of the plumb line.

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Source of Errors: Summery
Types of Errors:
Errors are traditionally been classified into three types.

 Gross Error,

 Systematic Error,

 Random Error,
Types of Errors: Gross errors
Gross errors also known as blunders or mistakes,
 Are results from Carelessness on the part of observer
in taking or recording reading;
 For example:
o Faults in equipments,
o Adoption of wrong technique,
o Misinterpretation,

 The blunders or mistakes result into large errors

and thus can easily be detected by comparing with other
types of errors.
b. Systematic error

After mistakes have been detected and eliminated

from the measurements, the remaining errors are usually
classified either as systematic or random error
depending on the characteristics of errors.
Systematic errors occur according to a system.
 These errors follow a definite pattern.
Thus, if an experiment is repeated:
 Under the same conditions,
 Same pattern of systematic errors reoccur.
• Systematic errors are caused by the surveying equipment,
observation methods, and certain environmental factors.
• The following list includes several examples of systematic
errors:
 Using incorrect temperature and/or pressure observations.

 Not applying curvature and refraction constants.

 Using incorrect instrument heights and/or target heights.

 Using an incorrect prism offset.

 Using an imperfectly adjusted instrument.

 Cont…
• The effect of these errors can be minimized by:

1. Properly leveling the survey instrument and targets.

2. Balancing foresight and backsight observations.

3. Entering the appropriate environmental correction factors in the

data collector.

4. Entering the correct instrument heights, targets heights, and

prism offset in the data collector.

5. Periodically calibrating the surveying equipment

So, systematic errors can be corrected or eliminated

2. Random Errors/ Accidental
 After mistakes are eliminated and systematic errors are
corrected, a survey measurement is associated with random
error only.

 This error is small and is equally liable to be plus or minus.

 Random errors are unpredictable and they cannot be

evaluated or quantified exactly.

 Random errors are determined through statistical

analysis least-squares method of adjustment.
 2. Random/accidental errors
♠ These are the errors that remain after all mistakes
and systematic errors have been removed from the
measured values.
♠ Random errors are random in nature and occur
beyond the control of the surveyor.
♠ They usually do not follow any physical law and
therefore must be dealt with according to the
mathematical laws of probability.
♠ It is impossible to avoid random errors in
measurements entirely

May 31, 2024 50

 ….cont’d
Nature of error
• Positive and negative errors equal chance
to occur
• Small errors have most probability to occur.
• Large errors has small chance occurs

May 31, 2024 51

 ….cont’d

Accidental errors occur due to:

-imperfection in the instruments

-human limitation or
-Change in atmospheric conditions

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Accuracy:
The accuracy of a set of repeated observations is
being defined as amount of closeness of their mean to
the population or distribution mean, i.e., closeness of
the mean of observations to the true value.

 Accuracy refers to the degree of perfection obtained

in measurements.
It denotes how close a given measurement is to the true
value of the quantity
Precision:
 Precision pertains to the degree of closeness of
observations among each other in a set of repeated
observations of a measurement.

Thus, if a set of observations for the same parameter

are clustered together, i.e., observations have small
deviations from their sample mean, then the
observations are said to have been obtained with high
precision.
If a quantity is measured several times and the values
obtained are very close to each other, the precision is
said to be high.
it is the closeness of one measurement to another
Accuracy Vs Precision
 Accurate and precise Accurate but not precise

Precise but not accurate Neither accurate nor precise

Accuracy Vs Precision
May 31, 2024 56
Thank You

May 31, 2024 57