Adama

University
Civil Engineering and
Architecture Department
 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.
May 6, 2023 4
 Systems of measurement

♣ There are two main systems of

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

May 6, 2023 5
 ….cont’d
i. Metric System
♣ First adopted by France in 1791
♣ The International System of
Units (abbreviated SI from the French
Le Système International d'Unités)
d'Unités is the
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 symbol Quantity Symbol

Meter m length l

Kilogram kg mass m

Second s time t

Ampere A electric current I

Kelvin K thermodynamic temperature T

Candela cd luminous intensity Iv

Mole mol amount of substance n

May 6, 2023 7
 ….cont’d
♣ It is the world's most widely used system
used of
measurement, both in everyday 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
♣ The older metric system included several
groups of units. The SI was developed
from the old meter-kilogram-second
system, rather than the centimeter-gram-
second system, which, in turn, had a few
variants. Because the SI is not static, units
are created and definitions are modified
through international agreement among
many nations as the technology of
measurement progresses, and as the
precision of measurements improves.
A standard set of prefixes in powers of ten may
be used to derive larger and smaller units from
the base units
 ….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

May 6, 2023 11
 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
May 6, 2023 12
 ….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
May 6, 2023 13
 ….cont’d
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)
May 6, 2023 14
 ….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)

May 6, 2023 15
 ….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
May 6, 2023 16
 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. 

May 6, 2023 17
 ….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,
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
May 6, 2023 18
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

May 6, 2023 19
 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
May 6, 2023 21
 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.
May 6, 2023 22
 ….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.
May 6, 2023 23
 ….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

May 6, 2023 24
May 6, 2023 25
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

May 6, 2023 26
 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*****
May 6, 2023 27
 ….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
May 6, 2023 28
 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.

May 6, 2023 29
 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

May 6, 2023 30
 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
May 6, 2023 31
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
May 6, 2023 32
 Sources of errors
♠ As discussed above, errors stem from three
sources, which are classified as instrumental,
natural, and personal:
a. Instrumental errors: are caused by imperfections
in instrument construction or adjustment.
Examples of instrument error are:
 Imperfect linear or angular scales
 Instrument axes are not perfectly parallel or
perpendicular to each other
 Misalignment of various part of the instrument
 Optical distortions causing “what you see is not
exactly what you are supposed to see”

May 6, 2023 33
 ….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.
May 6, 2023 35
 ….cont’d
b. Natural errors: are caused by changes in
natural phenomena, such as temperatures,
winds, humidity, refraction, and magnetic field.
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 6, 2023 36
 ….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

May 6, 2023 37
 ….cont’d
c. Personal error: occur due to human
limitations, such as sense of sight and touch.
-----Personal errors can be characterized as either
systematic or random.
-----Personal systematic errors are caused by an
observer tendency to react the same way
under the same conditions (inconsistency).
When there is no such tendency, the personal
errors are considered to be 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.
May 6, 2023 38
 Types of errors

In surveying, errors can be broadly classified

into the following two types.
1.Systematic or cumulative errors
2.Accidental or random errors

May 6, 2023 39
 1.Systematic /cumulative error

♠ A systematic error is an error that will

always have the same magnitude and the
same algebraic sign under the same
conditions.
♠ In most cases, systematic errors are
caused by physical and natural conditions
that vary in accordance with fixed
mathematical or physical laws.

May 6, 2023 40
 ….cont’d

♠ Systematic errors follow some well-defined

mathematical or physical law or system.
♠ The magnitude and the sign of the
systematic errors can be determined and a
suitable correction can be applied to the
measured quantity.
♠ So, systematic errors can be corrected or
eliminated
May 6, 2023 41
 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 6, 2023 42
 ….cont’d
♠ Theoretically, an accidental error has an
equal chance of being negative or positive.
Thus, errors of this type tend to be
compensating.
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 6, 2023 43
 ….cont’d

Accidental errors occur due to:

-imperfection in the instruments
-human limitation or
-Change in atmospheric conditions

May 6, 2023 44
 Accuracy and Precision
♠ 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 or is the degree of refinement with
which a given quantity is measured.
♠ In other words, it is the closeness of one
measurement to another.
♠ If a quantity is measured several times and the
values obtained are very close to each other, the
precision is said to be high.

May 6, 2023 45
 Accurate and precise Accurate but not precise

Precise but not accurate Neither accurate nor precise

Accuracy Vs Precision
May 6, 2023 46
 Basic definitions related to error
i. Most probable value (mean) is the arithmetic mean
or average value of a series of repeated
measurements
------It has more chance to be a true value of a quantity

ii. Residual is the difference between measured value of

a quantity and its most probable value

May 6, 2023 47
 ….cont’d
iii. The standard deviation (σ)
 It is a statistical measure of precision …………
the smaller the value of the standard deviation,
the greater the precision and vise
versa………….
 It is the amount of deviation from mean of any
single measurement
 It shows how much variation or 'dispersion'
there is from the 'average' (mean, or
expected/budgeted value).
 In surveying, it is used to analyze random
errors
May 6, 2023 48
 ….cont’d
Standard deviation (σ ) can be calculated using

May 6, 2023 49
 ….cont’d
iv. Standard error of the mean (m)
The standard error of the mean of a
number of observations of the same
quantity is given by


 v2
n  n  1

May 6, 2023 50
 ….cont’d
v. Maximum error
 In surveying generally 99.9% error (E99.9) is
taken as the maximum error
 It corresponds to a range of +3.29σ and -3.29σ
 The maximum error is often used to separate
mistakes (gross errors) from the random errors
 If any measurement deviates from the mean by
more than ±3.29σ it is considered as a mistake,
and that measurement is rejected

May 6, 2023 51
 ….cont’d

vi. Most probable error of the mean (Em)

Em  0.6745
v 2

n  n  1

May 6, 2023 52
 ….cont’d
vii. Different percentage Errors
Sometimes, the following percentages
of error are also required
(a) 90% Error (E90) = + 1.645σ
(b) 95% Error (E95) = + 1.960σ
(c) 95.5% Error (E95.5) ≈ + 2.0σ
(d) 99.7% Error (E99.7) =+ 2.968σ ≈+3σ

May 6, 2023 53
 Example
Suppose that a line has been measured
10 times using the same equipment and
procedures. It is assumed that no
mistakes exist, and any systematic errors
have been eliminated. Compute the most
probable value, standard deviation and
errors having 50%, 90% & 95% probability
The measured value (538.57, 538.39,
538.37, 538.39, 538.48, 538.49, 538.33,
538.46, 538.47, 538.55) all measurement
are in ft.
May 6, 2023 54
Solution

May 6, 2023 55
 Answer

May 6, 2023 56
Conclusion

May 6, 2023 57
 Relative precision
♠ The relative precision or the degree of
precision is used to express the precision
of the various measurements
♠ It is usually expressed as a ratio of the
standard error of the mean (m) to the
mean value (M) of the quantity
1
Relative Precision = M 
  
 m 

May 6, 2023 58
 ….cont’d
♠ Relative precision =  m /M
♠ It is usually expressed with numerator as
unit
Example - if the standard deviation is ±
0.03m for the mean value of the length of
the line of 615.41m ,
0.03 1
the relative precision = 615.41 
20,500

May 6, 2023 59
 Degree of Accuracy
♠ The degree of accuracy indicates the
accuracy attained in the measurements
♠ It is usually expressed as the ratio of the
error to the measured quantity
For example, a degree of accuracy of 1 in
10,000 indicates that there is an error of 1
unit in 10,000 units.

May 6, 2023 60
 ….cont’d
i. Linear measurements
----The degree of accuracy of the linear
measurement is usually expressed as
the ratio of the standard deviation to the
measured distance
Degree of Accuracy = standard deviation OR
measured distance
s tan dard error
Degree of Accuracy = measured distace
May 6, 2023 61
 ….cont’d
For example if there is a standard deviation
of + 0.05m in a measured distance of
584.65m, the degree of accuracy is 1 in
11700
s tan dard devation
degree of accuracy  Measured dis tan ce
0.05 1
  1
584.65 11693 11700

May 6, 2023 62
 ….cont’d
ii. Angular measurements- For angular
measurements, the degree of accuracy
is usually expressed as k N
W/r N = Number of angles measured

Angular error of closure

K= Number of angles measured

May 6, 2023 63
 ….cont’d
iii. Traverse - the degree of accuracy of a
traverse is usually expressed as the ratio
of the error of closure to the perimeter of
the traverse thus:
Error of closure
D.of . Accu 
Total Perimeter of traverse

May 6, 2023 64
 ….cont’d
iv. Leveling - the degree of accuracy is
usually expressed as
degree of accuracy = K L

Where L= Horizontal length of the route in meter

Error of in elevation
K=
HOrizontal length of route

For example: - if there is an error of 0.2m in a route of 5000m,

what will be the degree of accuracy?

May 6, 2023 65