Introduction, types of errors, definitions, laws of accidental errors, laws of weights, theory of least squares, rules for giving weights and distribution of errors to the field observations, determination of the most probable values of quantities.
Curves are used in transportation routes to gradually change direction between straight segments. There are several types of curves including simple, compound, reverse, and transition curves. A simple circular curve connects two tangents with a single arc, and is defined by its radius or degree. Transition curves provide a gradual transition between tangents and circular curves to avoid abrupt changes in grade or superelevation that could cause vehicles to overturn. There are several methods for laying out circular curves, including using offset distances from the long chord between tangent points or measuring deflection angles from the initial tangent.
This document discusses the use of a theodolite for surveying. It begins by explaining that a theodolite is needed to precisely measure horizontal and vertical angles, unlike a compass. It then defines theodolite surveying as surveying that measures angles using a theodolite. The document goes on to classify theodolites based on their horizontal axis and method of angle measurement. It describes the basic parts of a transit vernier theodolite and explains terms used in manipulating one. Finally, it discusses methods for measuring horizontal angles, including the general, repetition, and reiteration methods.
1) Contour lines on a map connect points of equal elevation and represent the topography of the land.
2) Contour surveys are conducted at the start of engineering projects to select suitable sites, locate alignments to minimize earthworks, and understand the terrain.
3) Contours are located either directly by tracing lines in the field or indirectly by taking spot levels and interpolating lines on the map. Indirect methods using cross-sections or tacheometry are more efficient for large areas.
1) The presentation summarized the theory of errors, which is the study of how measured quantities contain errors and how those errors propagate through calculations.
2) There are three main types of errors: mistakes, systematic errors, and accidental errors. Accidental errors follow the laws of probability and tend to be small and symmetrically distributed.
3) The presentation covered concepts like probable error, mean square error, weights, and the method of least squares. It also discussed how to determine probable errors for different types of direct and indirect observations and compute most probable values.
This document discusses control surveying and triangulation. It notes that control surveying must account for the curvature of the Earth and refraction, as lines of sight are not entirely straight. It distinguishes between plane and geodetic surveying, with the latter accounting for the spherical shape of the Earth. The document then discusses establishing control points through triangulation, including different classes of triangulation, steps in triangulation like selecting stations, and erecting signals and towers.
This document provides an overview of a total station, including its key components and functions. A total station is an electronic surveying instrument that combines an electronic distance meter and theodolite to measure horizontal and vertical angles and distances. It allows simultaneous measurement of all surveying parameters needed for construction layout and topographic surveys. The total station's main components include an electronic distance measurement system, angle measurement circles, telescope, microprocessor, keyboard, and display. Accessories such as prisms, data collectors, and software enable various surveying tasks.
Curves are used in transportation routes to gradually change direction between straight segments. There are several types of curves including simple, compound, reverse, and transition curves. A simple circular curve connects two tangents with a single arc, and is defined by its radius or degree. Transition curves provide a gradual transition between tangents and circular curves to avoid abrupt changes in grade or superelevation that could cause vehicles to overturn. There are several methods for laying out circular curves, including using offset distances from the long chord between tangent points or measuring deflection angles from the initial tangent.
This document discusses the use of a theodolite for surveying. It begins by explaining that a theodolite is needed to precisely measure horizontal and vertical angles, unlike a compass. It then defines theodolite surveying as surveying that measures angles using a theodolite. The document goes on to classify theodolites based on their horizontal axis and method of angle measurement. It describes the basic parts of a transit vernier theodolite and explains terms used in manipulating one. Finally, it discusses methods for measuring horizontal angles, including the general, repetition, and reiteration methods.
1) Contour lines on a map connect points of equal elevation and represent the topography of the land.
2) Contour surveys are conducted at the start of engineering projects to select suitable sites, locate alignments to minimize earthworks, and understand the terrain.
3) Contours are located either directly by tracing lines in the field or indirectly by taking spot levels and interpolating lines on the map. Indirect methods using cross-sections or tacheometry are more efficient for large areas.
1) The presentation summarized the theory of errors, which is the study of how measured quantities contain errors and how those errors propagate through calculations.
2) There are three main types of errors: mistakes, systematic errors, and accidental errors. Accidental errors follow the laws of probability and tend to be small and symmetrically distributed.
3) The presentation covered concepts like probable error, mean square error, weights, and the method of least squares. It also discussed how to determine probable errors for different types of direct and indirect observations and compute most probable values.
This document discusses control surveying and triangulation. It notes that control surveying must account for the curvature of the Earth and refraction, as lines of sight are not entirely straight. It distinguishes between plane and geodetic surveying, with the latter accounting for the spherical shape of the Earth. The document then discusses establishing control points through triangulation, including different classes of triangulation, steps in triangulation like selecting stations, and erecting signals and towers.
This document provides an overview of a total station, including its key components and functions. A total station is an electronic surveying instrument that combines an electronic distance meter and theodolite to measure horizontal and vertical angles and distances. It allows simultaneous measurement of all surveying parameters needed for construction layout and topographic surveys. The total station's main components include an electronic distance measurement system, angle measurement circles, telescope, microprocessor, keyboard, and display. Accessories such as prisms, data collectors, and software enable various surveying tasks.
The document provides an introduction to basic surveying. It defines surveying as determining relative positions of objects on Earth's surface by measuring horizontal distances and preparing maps to scale. The purposes of engineering surveying include determining land areas and volumes needed for construction projects. Objectives of surveying include preparing plans for infrastructure like roads, buildings, and canals, as well as measuring land areas. Surveying principles include considering parts in relation to the whole and locating points using two or more measurements.
Distance Measurement & Chain Surveying
Contents
• Introduction About Surveying
.
• Primary Division Of Surveying • Classification Of Surveying • Distance Measurement And Chain Surveying • Principle Of Surveying • Types Of Tapes Based On The Materials Used • Erecting And Dropping A Perpendicular • Obstacle In Chain Survey • Types Of Errors • Corrections of Tape • Off –Sets • Ranging • Conclusion . • Homework And Next Lecture . • References.
-Definition of Surveying.
Types of Surveying
1. Plane Surveying
2. Geodetic Survey
3. Cadastral surveying
4. Aerial Surveying
5. Hydro graphic Surveying (Hydro-Survey)
6. Topographical Survey
7. Engineering Survey.
Primary division of Surveying
Reconnaissance.
• This is preliminary survey of the land to be surveyed. It may be either
1-Ground reconnaissance 2- Aerial reconnaissance survey.
Objectives of Reconnaissance
1. To ascertain the possibility of building or constructing route or track through the area.
Classification of Surveying:
1- Classification based on the instruments used:
A. Chain Surveying.
B. Compass Surveying.
C. Theodolite Surveying.
D. Tachometric Surveying .
E. Trigonometric Surveying.
F. Total station and GPS.
G. Photogrammetric and Aerial Surveying.
H. Plan Table .
2- According to the method used:
i. Traversing .
ii. Triangulation .
iii. Tacheometric.
iv. Trigonometric.
3- According to the Purpose of surveying:
i. Engineering survey.
ii. Military survey.
iii. Geological survey .
iv. Topographical survey
Chain and Tape Survey
-Length& Distance Measurements.
-Distance Measurement and Chain Surveying.
• In general there are two methods:
1- Direct methods of measuring lengths
2- Indirect methods of measuring distances.
There are two kinds of measurements used in plane surveying.
*Linear measurements
*Angular measurements
-Instruments used in Chain Surveying.
Types of tapes based on the materials used.
.......
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.
.
.
.
.
.
.
.
Asst. Prof. Salar K.Hussein
Mr. Kamal Y.Abdullah
Asst.Lecturer. Dilveen H. Omar
Erbil Polytechnic University
Technical Engineering College
Civil Engineering Department
This document discusses methods for setting out simple circular curves in surveying. There are two main methods: linear and angular.
The linear method uses only a tape or chain and does not require angle measurement. It includes setting out curves by offsets from the long chord, by successive bisection of arcs, and by offsets from the tangents.
The angular method is used for longer curves and involves measuring deflection angles. It includes Rankine's method of tangential angles using a tape and theodolite to measure deflection angles from the back tangent to points on the curve. The two theodolite method also uses angle measurement between two theodolites.
This document discusses trigonometric levelling, which is a method of determining elevation differences between stations using vertical angles and known distances. It presents three cases for determining the elevation of a point using a theodolite: 1) when the base of the object is accessible, 2) when the base is inaccessible and instrument stations are in the same vertical plane, and 3) when the base is inaccessible and instrument stations are not in the same vertical plane. Equations for calculating relative heights are provided for each case using trigonometric functions of the vertical angles and distances between points. Corrections may be needed for long distances to account for earth's curvature and refraction.
Introduction to surveying, ranging and chainingShital Navghare
This presentation contains the complete introduction of surveying. It also includes all the instrucments used in linear measurement and the terms related to Ranging and Chaining
The document is a presentation on total stations. It introduces total stations as an electronic combination of a theodolite, electronic distance measuring device, and microprocessor. It then lists the contents which will be covered, including the introduction, advantages, disadvantages, precautions, and conclusion. The main body explains what a total station is and how it works. It provides details on the advantages like quick setting, on-board computations, and automation. Disadvantages include costs and needing skilled personnel. Precautions when using a total station are also outlined.
Metric Chain : It Consists of galvanized mild steel wire of 4mm diameter known as link.
It is available in 20m, 30m, 50m length which consists of 100 links.
Gunter’s Chain : A 66 feet long chain consists of 100 links, each of 0.66 feet, it is known as Gunter’s chain.
This chain is suitable for taking length in miles.
Engineer’s Chain : A 100 feet long chain consisting of 100 links each of 1 feet is known as engineer’s chain.
This chain is used to measure length in feet and area in sq.yard.
Revenue Chain : it is 33 feet long chain consisting of 16 links.
This chain is used for distance measurements in feet & inches for smaller areas.
Tacheometric surveying uses a tacheometer to determine horizontal and vertical distances through angular measurements. A tacheometer is a theodolite fitted with stadia hairs and an anallatic lens. The tacheometric formula relates the staff intercept, focal length, stadia interval and additive constant to calculate horizontal distances. Methods include stadia, fixed/movable hair, and non-stadia techniques. Determining the tacheometer constant involves measuring distances and staff intervals at stations to solve equations. Errors arise from incorrect stadia intervals or graduations. Tacheometric surveying provides distances in rough terrain but with less precision than other methods.
This document discusses contouring and contour maps. It defines a contour line as a line connecting points of equal elevation. The vertical distance between consecutive contours is called the contour interval, which depends on factors like the nature of the ground and the map scale. Contour maps show the topography of an area and can be used for engineering projects, route selection, and estimating earthworks. Methods of plotting contours include direct methods using levels or hand levels, and indirect methods like gridding, cross-sectioning, and radial lines. Characteristics of contours provide information about the landscape.
This ppt presentation covers compass surveying, which explains principal of compass surveying, Types of compass, Difference between compass, Bearing, Definitions related to compass surveying etc.
Traverse surveying involves using instruments to measure distance and direction to create a network of points. There are two main types of traverses - open and closed. Open traverses extend in one direction while closed traverses form a closed loop. Common surveying instruments and methods used in traverse surveying include chain, compass, theodolite, and plane table. Key terms in traverse surveying include bearings, meridians, and reductions of bearings. Traverse calculations involve adjusting angles or directions to ensure closure of the network of points. Sample problems are provided to demonstrate conversions between whole circle bearings, reduced bearings, and fore and back bearings.
The document defines levelling as determining the relative heights of points. It discusses the principle of obtaining a horizontal line of sight and objectives of finding point elevations and establishing points at required elevations. Different types of levels, staffs, benchmarks, and adjustments are described. Various levelling classifications are defined including simple, differential, profile, check, reciprocal and precise levelling. The key principle of levelling is to obtain a horizontal line of sight to measure staff readings and determine reduced levels of points.
This document summarizes methods for setting out simple circular curves based on linear and angular methods. The linear methods discussed are by offsets from the long chord, successive bisection of arcs, offsets from tangents, and offsets from chords produced. The angular methods discussed are Rankine's method of tangential angles, the two theodolite method, and the tacheometric method. Each method is briefly described in one or two sentences.
This document discusses the principles and classification of triangulation, which is a surveying method used to determine distances based on geometry. It describes three orders or classifications of triangulation: primary, secondary, and tertiary. Primary triangulation establishes the most precise control points over large areas. Secondary triangulation uses smaller triangles within the primary framework, while tertiary triangulation establishes intermediate control for detailed surveys using even smaller triangles. Specifications for each order are provided, such as average triangle size, expected errors, and instrumentation precision.
This document discusses the topic of chain surveying for a civil engineering class project. It provides definitions of chain surveying, noting that it involves measuring linear distances between survey stations to divide an area into triangles without taking angular measurements. It then outlines the key principles and terms of chain surveying, such as defining main stations, subsidiary stations, tie stations, main survey lines, base lines, check lines, and tie lines. Finally, it provides the basic procedures for conducting a chain survey between two stations.
1) Curves are gradual bends provided in transportation infrastructure like roads, railways and canals to allow for a smooth change in direction or grade.
2) There are two main types of curves - horizontal curves which provide a gradual change in direction, and vertical curves which provide a gradual change in grade.
3) Curves are needed to safely guide vehicles and traffic when changing directions or grades, to improve visibility, and to prevent erosion of canal banks from water pressure.
1) Levelling is the process of determining the relative elevations of points on or near the earth's surface. It is important for engineering projects to determine elevations along alignments.
2) Levelling is used to prepare contour maps, determine altitudes, and create longitudinal and cross sections needed for projects.
3) Key terms include bench mark, datum, reduced level, line of collimation, and height of instrument. Different types of levelling include simple, differential, fly, longitudinal, and cross-sectional levelling.
This document provides information about circular curves used in highways and railways. It discusses the different types of curves including simple, compound, and reverse curves. It defines key elements of circular curves such as radius, deflection angle, tangent length, and mid-ordinate. It presents the relationships between radius and degree of curvature. Finally, it describes various methods for setting out circular curves in the field, including linear methods using offsets and angular methods using a theodolite.
This document discusses the theory of errors in survey measurements. It begins with an introduction to the types of errors that can occur, including mistakes, systematic errors, and accidental errors. It then provides definitions for key terms like direct observation, indirect observation, conditioned quantity, true value, most probable value, true error, and residual error. The document goes on to explain statistical formulas used to calculate the probable error of single observations, the probable error of the mean, standard deviation, standard error, and precision. It also covers the law of accidental errors and laws of weights. Finally, it discusses the theory of least squares and provides examples of determining probable errors and finding the most probable value of quantities.
This document provides an overview of the theory of errors in survey measurements. It discusses the different types of errors including mistakes, systematic errors, and accidental errors. It defines key terms like true value, most probable value, true error, and residual error. Statistical formulas for probable error, standard deviation, and standard error are presented. Laws of accidental errors and weights that describe the distribution and weighting of errors are covered. The principle of least squares and its application in determining most probable values from observations is explained. Methods for calculating error in sums, error of the mean, and errors in indirect and conditional observations are also summarized.
The document provides an introduction to basic surveying. It defines surveying as determining relative positions of objects on Earth's surface by measuring horizontal distances and preparing maps to scale. The purposes of engineering surveying include determining land areas and volumes needed for construction projects. Objectives of surveying include preparing plans for infrastructure like roads, buildings, and canals, as well as measuring land areas. Surveying principles include considering parts in relation to the whole and locating points using two or more measurements.
Distance Measurement & Chain Surveying
Contents
• Introduction About Surveying
.
• Primary Division Of Surveying • Classification Of Surveying • Distance Measurement And Chain Surveying • Principle Of Surveying • Types Of Tapes Based On The Materials Used • Erecting And Dropping A Perpendicular • Obstacle In Chain Survey • Types Of Errors • Corrections of Tape • Off –Sets • Ranging • Conclusion . • Homework And Next Lecture . • References.
-Definition of Surveying.
Types of Surveying
1. Plane Surveying
2. Geodetic Survey
3. Cadastral surveying
4. Aerial Surveying
5. Hydro graphic Surveying (Hydro-Survey)
6. Topographical Survey
7. Engineering Survey.
Primary division of Surveying
Reconnaissance.
• This is preliminary survey of the land to be surveyed. It may be either
1-Ground reconnaissance 2- Aerial reconnaissance survey.
Objectives of Reconnaissance
1. To ascertain the possibility of building or constructing route or track through the area.
Classification of Surveying:
1- Classification based on the instruments used:
A. Chain Surveying.
B. Compass Surveying.
C. Theodolite Surveying.
D. Tachometric Surveying .
E. Trigonometric Surveying.
F. Total station and GPS.
G. Photogrammetric and Aerial Surveying.
H. Plan Table .
2- According to the method used:
i. Traversing .
ii. Triangulation .
iii. Tacheometric.
iv. Trigonometric.
3- According to the Purpose of surveying:
i. Engineering survey.
ii. Military survey.
iii. Geological survey .
iv. Topographical survey
Chain and Tape Survey
-Length& Distance Measurements.
-Distance Measurement and Chain Surveying.
• In general there are two methods:
1- Direct methods of measuring lengths
2- Indirect methods of measuring distances.
There are two kinds of measurements used in plane surveying.
*Linear measurements
*Angular measurements
-Instruments used in Chain Surveying.
Types of tapes based on the materials used.
.......
.
.
.
.
.
.
.
.
.
.
Asst. Prof. Salar K.Hussein
Mr. Kamal Y.Abdullah
Asst.Lecturer. Dilveen H. Omar
Erbil Polytechnic University
Technical Engineering College
Civil Engineering Department
This document discusses methods for setting out simple circular curves in surveying. There are two main methods: linear and angular.
The linear method uses only a tape or chain and does not require angle measurement. It includes setting out curves by offsets from the long chord, by successive bisection of arcs, and by offsets from the tangents.
The angular method is used for longer curves and involves measuring deflection angles. It includes Rankine's method of tangential angles using a tape and theodolite to measure deflection angles from the back tangent to points on the curve. The two theodolite method also uses angle measurement between two theodolites.
This document discusses trigonometric levelling, which is a method of determining elevation differences between stations using vertical angles and known distances. It presents three cases for determining the elevation of a point using a theodolite: 1) when the base of the object is accessible, 2) when the base is inaccessible and instrument stations are in the same vertical plane, and 3) when the base is inaccessible and instrument stations are not in the same vertical plane. Equations for calculating relative heights are provided for each case using trigonometric functions of the vertical angles and distances between points. Corrections may be needed for long distances to account for earth's curvature and refraction.
Introduction to surveying, ranging and chainingShital Navghare
This presentation contains the complete introduction of surveying. It also includes all the instrucments used in linear measurement and the terms related to Ranging and Chaining
The document is a presentation on total stations. It introduces total stations as an electronic combination of a theodolite, electronic distance measuring device, and microprocessor. It then lists the contents which will be covered, including the introduction, advantages, disadvantages, precautions, and conclusion. The main body explains what a total station is and how it works. It provides details on the advantages like quick setting, on-board computations, and automation. Disadvantages include costs and needing skilled personnel. Precautions when using a total station are also outlined.
Metric Chain : It Consists of galvanized mild steel wire of 4mm diameter known as link.
It is available in 20m, 30m, 50m length which consists of 100 links.
Gunter’s Chain : A 66 feet long chain consists of 100 links, each of 0.66 feet, it is known as Gunter’s chain.
This chain is suitable for taking length in miles.
Engineer’s Chain : A 100 feet long chain consisting of 100 links each of 1 feet is known as engineer’s chain.
This chain is used to measure length in feet and area in sq.yard.
Revenue Chain : it is 33 feet long chain consisting of 16 links.
This chain is used for distance measurements in feet & inches for smaller areas.
Tacheometric surveying uses a tacheometer to determine horizontal and vertical distances through angular measurements. A tacheometer is a theodolite fitted with stadia hairs and an anallatic lens. The tacheometric formula relates the staff intercept, focal length, stadia interval and additive constant to calculate horizontal distances. Methods include stadia, fixed/movable hair, and non-stadia techniques. Determining the tacheometer constant involves measuring distances and staff intervals at stations to solve equations. Errors arise from incorrect stadia intervals or graduations. Tacheometric surveying provides distances in rough terrain but with less precision than other methods.
This document discusses contouring and contour maps. It defines a contour line as a line connecting points of equal elevation. The vertical distance between consecutive contours is called the contour interval, which depends on factors like the nature of the ground and the map scale. Contour maps show the topography of an area and can be used for engineering projects, route selection, and estimating earthworks. Methods of plotting contours include direct methods using levels or hand levels, and indirect methods like gridding, cross-sectioning, and radial lines. Characteristics of contours provide information about the landscape.
This ppt presentation covers compass surveying, which explains principal of compass surveying, Types of compass, Difference between compass, Bearing, Definitions related to compass surveying etc.
Traverse surveying involves using instruments to measure distance and direction to create a network of points. There are two main types of traverses - open and closed. Open traverses extend in one direction while closed traverses form a closed loop. Common surveying instruments and methods used in traverse surveying include chain, compass, theodolite, and plane table. Key terms in traverse surveying include bearings, meridians, and reductions of bearings. Traverse calculations involve adjusting angles or directions to ensure closure of the network of points. Sample problems are provided to demonstrate conversions between whole circle bearings, reduced bearings, and fore and back bearings.
The document defines levelling as determining the relative heights of points. It discusses the principle of obtaining a horizontal line of sight and objectives of finding point elevations and establishing points at required elevations. Different types of levels, staffs, benchmarks, and adjustments are described. Various levelling classifications are defined including simple, differential, profile, check, reciprocal and precise levelling. The key principle of levelling is to obtain a horizontal line of sight to measure staff readings and determine reduced levels of points.
This document summarizes methods for setting out simple circular curves based on linear and angular methods. The linear methods discussed are by offsets from the long chord, successive bisection of arcs, offsets from tangents, and offsets from chords produced. The angular methods discussed are Rankine's method of tangential angles, the two theodolite method, and the tacheometric method. Each method is briefly described in one or two sentences.
This document discusses the principles and classification of triangulation, which is a surveying method used to determine distances based on geometry. It describes three orders or classifications of triangulation: primary, secondary, and tertiary. Primary triangulation establishes the most precise control points over large areas. Secondary triangulation uses smaller triangles within the primary framework, while tertiary triangulation establishes intermediate control for detailed surveys using even smaller triangles. Specifications for each order are provided, such as average triangle size, expected errors, and instrumentation precision.
This document discusses the topic of chain surveying for a civil engineering class project. It provides definitions of chain surveying, noting that it involves measuring linear distances between survey stations to divide an area into triangles without taking angular measurements. It then outlines the key principles and terms of chain surveying, such as defining main stations, subsidiary stations, tie stations, main survey lines, base lines, check lines, and tie lines. Finally, it provides the basic procedures for conducting a chain survey between two stations.
1) Curves are gradual bends provided in transportation infrastructure like roads, railways and canals to allow for a smooth change in direction or grade.
2) There are two main types of curves - horizontal curves which provide a gradual change in direction, and vertical curves which provide a gradual change in grade.
3) Curves are needed to safely guide vehicles and traffic when changing directions or grades, to improve visibility, and to prevent erosion of canal banks from water pressure.
1) Levelling is the process of determining the relative elevations of points on or near the earth's surface. It is important for engineering projects to determine elevations along alignments.
2) Levelling is used to prepare contour maps, determine altitudes, and create longitudinal and cross sections needed for projects.
3) Key terms include bench mark, datum, reduced level, line of collimation, and height of instrument. Different types of levelling include simple, differential, fly, longitudinal, and cross-sectional levelling.
This document provides information about circular curves used in highways and railways. It discusses the different types of curves including simple, compound, and reverse curves. It defines key elements of circular curves such as radius, deflection angle, tangent length, and mid-ordinate. It presents the relationships between radius and degree of curvature. Finally, it describes various methods for setting out circular curves in the field, including linear methods using offsets and angular methods using a theodolite.
This document discusses the theory of errors in survey measurements. It begins with an introduction to the types of errors that can occur, including mistakes, systematic errors, and accidental errors. It then provides definitions for key terms like direct observation, indirect observation, conditioned quantity, true value, most probable value, true error, and residual error. The document goes on to explain statistical formulas used to calculate the probable error of single observations, the probable error of the mean, standard deviation, standard error, and precision. It also covers the law of accidental errors and laws of weights. Finally, it discusses the theory of least squares and provides examples of determining probable errors and finding the most probable value of quantities.
This document provides an overview of the theory of errors in survey measurements. It discusses the different types of errors including mistakes, systematic errors, and accidental errors. It defines key terms like true value, most probable value, true error, and residual error. Statistical formulas for probable error, standard deviation, and standard error are presented. Laws of accidental errors and weights that describe the distribution and weighting of errors are covered. The principle of least squares and its application in determining most probable values from observations is explained. Methods for calculating error in sums, error of the mean, and errors in indirect and conditional observations are also summarized.
This document discusses uncertainties and errors in physical measurements. It explains that there are two types of errors - random errors which are unpredictable, and systematic errors caused by imperfect measuring equipment. Random errors can be reduced by repeating measurements, while systematic errors are reduced by calibrating equipment. Accuracy refers to how close a measurement is to the true value, while precision refers to how close repeated measurements are. The number of significant figures reported in a result should not exceed the least precise value used. The document also discusses determining and expressing uncertainties in measurements, and how to combine uncertainties when performing calculations or graphing data.
The document discusses types of errors that can occur in measurement including mistakes, systematic errors, and accidental errors. It defines key terms related to observations and errors such as direct observations, indirect observations, conditioned quantities, true values, most probable values, true errors, and residual errors. The document also covers laws of weights that describe how the weights of means and sums are determined. It introduces the principle of least squares and the law of accidental errors, explaining how probable errors are computed for different types of observations.
R = R0(1 + α(t - 20))
- The resistance (R) of a copper wire is calculated using a formula that relates it to the resistance at 20°C (R0), the coefficient of resistance (α), and the temperature (t).
- R0 is given as 6Ω with an uncertainty of ±0.3%.
- To determine the uncertainty in R, the uncertainties in R0, α, and t must be determined and propagated through the equation using partial derivatives.
- The overall uncertainty in R combines the individual uncertainties from each variable according to the propagation of uncertainty formula.
This document outlines the key topics in Analytical Chemistry I including significant figures, types of errors, propagation of uncertainty, and systematic vs random errors. It discusses how measurements have uncertainty and errors. There are two main types of errors - systematic errors which affect accuracy and can be discovered and corrected, and random errors which cannot be eliminated and have equal chances of being positive or negative. The document also describes how to calculate the propagation of uncertainty through calculations using addition, subtraction, multiplication, division and other operations. It emphasizes keeping extra digits in calculations to properly account for uncertainty.
This document discusses error in measurement and how to calculate absolute and relative error. It explains that there are two types of errors - systematic errors due to the measuring instrument and random errors due to the person taking the measurement. To reduce random errors, multiple measurements are taken and the average is used. The absolute error is defined as the difference between the true value and measured value, while relative error is absolute error divided by the measurement. Examples are provided to demonstrate calculating error through addition, multiplication, and powers when indirect measurements are involved.
1) The document discusses measurement and error in engineering. It covers characteristics of measuring instruments such as accuracy, precision, sensitivity, and error.
2) Accuracy refers to how close a measurement is to the true value, while precision refers to the reproducibility of measurements. Systematic errors can be corrected, while random errors average out over multiple trials.
3) Significant figures indicate the precision of a measurement. The number of significant figures retained in calculations is determined by the least precise measurement.
This document discusses experimental error in physical measurements. Every measurement has some degree of uncertainty. There are two main types of error - systematic errors which have an assignable cause and tend to be consistent in one direction, and random errors which are natural limitations and occur unpredictably. Accuracy refers to how close a measurement is to the true value, while precision describes the reproducibility of measurements. Proper evaluation of error involves repetition of measurements, use of different methods and people, and statistical analysis of results.
This document discusses experimental error in physical measurements. Every measurement has some degree of uncertainty. There are two main types of error - systematic errors which have an assignable cause and tend to be consistent in one direction, and random errors which are natural and unpredictable. Accuracy refers to how close a measurement is to the true value, while precision refers to the reproducibility of measurements. Proper evaluation of errors involves repetition of measurements, use of different methods, and statistical analysis to determine confidence intervals around results and identify outliers.
- Random uncertainties arise from imprecision in measurements and can cause readings to be above or below the true value. They can be reduced by more precise instruments or repeating measurements.
- Systematic uncertainties result in all readings being consistently too high or too low. They may be due to instrumentation errors or experimental technique and can sometimes be addressed through calibration.
- Uncertainty is incorporated into measurements as a range rather than a single value, and it is important to propagate uncertainties through calculations.
The document discusses experimental data and uncertainty. It explains that all data has some uncertainty due to limitations of instruments and humans. It also discusses accuracy, precision, and significant figures when reporting results. The mean, uncertainty in the mean, and fractional and percentage uncertainties are also covered.
This document defines key terms related to measurement and metrology such as accuracy, precision, sensitivity, and resolution. It provides examples of calculating average values, ranges of error, and combining measurements with associated uncertainties. The key sources of error are defined as gross, systematic, and random errors. Statistical analysis techniques like determining the arithmetic mean and standard deviation are demonstrated. The concept of probability distribution and determining proper error from standard deviation is also explained.
Diploma sem 2 applied science physics-unit 1-chap 2 error sRai University
This document discusses various types of errors that can occur in measurements. It describes instrumental error, observer error, and procedural error as the three main sources of uncertainty. It also defines accuracy as a measure of how close a measurement is to the accepted value, while precision refers to the closeness of repeated measurements. The document provides examples of calculating percentage error, relative error, and discusses significant figures when taking measurements.
Measurement errors, Statistical Analysis, UncertaintyDr Naim R Kidwai
The Presentation covers Measurement Errors and types, Gross error, systematic error, absolute error and relative error, accuracy, precision, resolution and significant figures, Measurement error combination, basics of statistical analysis, uncertainty, Gaussian Curve, Meaning of Ranges
This document discusses terminology and concepts related to measurement and error. It defines true value, accuracy, and precision. There are two types of errors - determinate (systematic) errors which have a known cause, and indeterminate (random) errors which cannot be determined. Accuracy refers to closeness to the true value while precision refers to reproducibility. The standard deviation allows for more variation in a sample compared to the population. When combining uncertainties from multiple measurements, relative uncertainties should be summed for multiplication and division, while absolute uncertainties are summed for addition and subtraction. Significant figures refer to the reliable digits in a measurement and rules govern how many are retained in calculations.
Measurement involves uncertainty from errors and imprecision. An error is the difference between measured and expected values, while uncertainty summarizes the error. Random errors arise from unpredictable fluctuations, and systematic errors are reproducible biases. Accuracy refers to closeness to the true value, while precision reflects consistency of measurements. Uncertainty is quantified from instrument resolution, repeated measurements, or comparison to a standard value. It is calculated and reported with the measurement result.
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This document provides an overview of electronic measurements and instrumentation. It discusses key topics like electronics, measurement processes, instruments, transducers, and performance characteristics of measuring systems. The performance characteristics include static characteristics like accuracy, precision, resolution, and dynamic characteristics like speed of response, fidelity, and lag. Common types of errors in measurement like gross errors, systematic errors and random errors are explained. Statistical analysis methods for random errors are described. Finally, the document discusses the basic meter movement in permanent magnetic moving coil instruments and the D'Arsonval galvanometer mechanism.
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1. 1
PREPARED BY : ASST. PROF. VATSAL D. PATEL
MAHATMA GANDHI INSTITUTE OF
TECHNICAL EDUCATION &
RESEARCH CENTRE, NAVSARI.
2. Measurements of distances and angles are made in various
surveying operations. These survey measurements may
involve several elementary operations, such as setting up,
centering, levelling, bisecting and reading.
In performing these operations, it is impossible to determine
the true values of these quantities (distance or angles) because
some errors always creep in all measurements.
2
3. The errors can be broadly classified into three types :
1. Gross errors or mistakes
2. Systematic or cumulative errors
3. Accidental or random errors
3
5. Under same condition always be the same size and sign.
Always follow some mathematical /physical law.
Correction can be determined and applied.
Effect is +ve/-ve, Cumulative effect.
Constant in character +ve OR –ve ,Results too great/too small.
If undetected systematic errors are very serious.
5
6. SOLUTION OF SYSTEMATIC ERRORS :
Instrument design- errors should auto elimination.
Find out the relationship of error.
6
7. Remains after mistakes and systematic errors have been
eliminated.
Beyond the ability of observer to control.
Represent the limitation of precision.
Obey the laws of chance.
Must handle as per laws of probability.
7
8. Lack of perfection in human eye.
Observation in cm. tape 10mm,9mm,11mm.
May compensate each other- compensating error.
Smaller error – better precision.
8
9. The following definitions of some of the terms should be
clearly understood in adjustment of the measurement :
9
Observation
Observed
value of a
quantity
True value
of a quantity
True error
Most
probable
value (MPV)
Most
probable
error
Residual
error
Observation
equation
Conditioned
equation
Normal
equation
10. Observation :
The measured numerical value of a quantity is known as
observation. For example, 15.36m, 45˚20'15'', etc.
10
Direct observation Indirect observation
If the value of a quantity
is measured directly.
Value of the quantity calculated
indirectly from direct observation
i.e. Angle A= 45˚20'15'' i.e. Angle computed at true station
from satellite station
11. Observed value of a quantity :
The observed value of a quantity is the value obtained from
the observation after applying corrections for systematic errors
and eliminating mistakes.
11
12. Observed value of a quantity :
The observed value of a quantity may be classified as :
12
Independent quantity Dependent or Conditioned
quantity
Whose value is independent
of the value of other quantity
( i.e. R.L. of B.M.)
Whose value is dependent upon
the value of one or more quantity
(i.e. R. L. of other points other
than B.M.)
13. True value of a quantity :
The value which is absolutely free from all the errors
(hypothetical quantity)
True error :
Difference between the true value and observed value.
True error = True value - Observed value
13
14. Most probable value (MPV) :
Is the one which has more chance of being true than has any
other
Most probable error:
Which is added to and subtracted from most probable value
fixes the limits of true value.
Residual error :
Residual error = Observed value – Most probable value
14
15. Observation equation :
Relation between observed quantity and its numerical value
For example, A + B = 78˚15' 20''
Conditioned equation :
Which is expressing the relation between several dependent
quantities.
For example, In triangle ABC (A+ B+ C = 180˚)
15
16. Normal equation :
Multiplying each equation with unknown quantity
No of normal equation= No of unknown
Using normal equation MPV evaluated
16
17. Accidental error follows law of probability.
Occurrence of errors can be expressed by equation.
Equation used to find probable error/precision.
e.g. 45˚22' 30'' ± 2.10''
17
19. Most important features of accidental error :
+ve and –ve errors are equal in size and frequency, as the
curve is symmetrical; that is, they are equally probable.
Small errors are more frequent than large errors; that is, they
are the most probable.
Very large error seldom occur and are impossible.
19
20. It is calculated from the probability curve of errors.
In any large series of observations the probable error is an
error of such a value that the number of errors numerically
greater than it is the same as the number of errors numerically
less than it.
The probable error is indicated along with the value of the
quantity with a ± sign.
20
21. Purposes :
Measure of precision of any series of observation.
Means of assigning weights to two or more quantities.
From the probability curve following six aspects are derived :
21
1
Probable
error of
Single
measurement
2
Probable
error of an
average
3
Probable
error of Sum
of
observations
4
Mean
Square
Error
5
Average
Error
6
Probable error
of Single
measurement
25. Several standards exist for assessing the precision of a set of
observation. The most popular is the standard deviation (σ).
It is a numerical value which indicates the amount of variation
about a central value.
The standard deviation establishes the limits of the error
bound within which 68.3% of the values of the set should lie.
The smaller the value of the standard deviation, the greater the
precision.
25
26. In carrying a line of level across a river, the following eight
readings were taken with level under identical condition.
2.322,2.346,2.352,2.306,2.312,2.300,2.306,2.326. Calculate :
1. The probable error of single observation
2. The probable error of mean
3. Most probable value of staff reading
4. Standard deviation
26
29. 29
Case-1
Direct observation of
equal weight
Probable error of
single observation
of unit weight
Probable error of
single observation
of weight w
Probable error of
single arithmetic
mean
a
b
c
Case-2
Direct observation of
Un-Equal weights
Probable error of
single observation
of unit weight
Probable error of
single observation
of weight w
Probable error of
weighted
arithmetic mean
a
b
c
Case-3
Indirect observation
of Independent
quantity
Case-4
Indirect observation
involving
conditional
Equations
Case-5
Computed
Quantities
31. 31
Case-3
Indirect observation of
independent quantities
Case-4 Indirect
observations involving
conditional Equations
Case-5
Computed
Quantities
The
probable
error of
computed
quantities
may be
calculated
form the
laws…..
32. The observed values of an angle are given below :
Find :
Probable error of single observation value of unit weight.
Probable error of weighted arithmetic mean
Probable error of single observation of weight 3
32
Angle Weight
85° 40' 20'' 2
85° 40' 18'' 2
85° 40' 19'' 3
35. Weight :
The weight of a quantity is trust worthiness of a quantity.
The relative precision and trustworthiness of an observation as
compared to the precision of other quantities is known as
weight of the observation.
The weights are always expressed in numbers.
35
36. Weight :
Higher number indicate higher precision and trust as compared
to lesser numbers.
36
Law of weight
1
Law of weight
2
Law of weight
3
Law of weight
4
Law of weight
5
Law of weight
6
Law of weight
7
37. Law of weight 1 :
The weight of the arithmetic mean of a number of
observations of unit weight, is equal to the number of the
observations.
37
38. Calculate the weight of the arithmetic mean of the following
observations of an angle of unit weight.
38
Angle Weight
A= 65° 30' 10'' 1
A= 65° 30' 15'' 1
A= 65° 30' 20'' 1
39. The number of observations of unit weight, n=3.
Arithmetic mean
= 65° 30' 15''
From law (1), the weight of the arithmetic mean = n
= 3
Hence, the weight of the arithmetic mean 65° 30' 15'' is 3.
39
40. Law of weight 2 :
The weight of the weighted arithmetic mean of a number of
observations is equal to the sum of the individual weights of
observations.
40
41. An angle A was observed three times as given below with their
respective weights. What is the weight of the weighted
arithmetic mean of the angle ?
41
Angle Weight
A= 40° 15' 10'' 1
A= 40° 15' 14'' 2
A= 40° 15' 12'' 3
42. Weighted arithmetic mean of A
= 40° 15' 12.33''
Sum of the individual weights = 1 + 2 + 3 = 6
From law (2), the weight of the weighted arithmetic mean of
the angle = sum of the individual weights
The weighted arithmetic mean 40° 15' 12.33'‘ has the weight
of 6.
42
43. Law of weight 3 :
The weight of the arithmetic sum of two or more quantities is
equal to the reciprocal of the sum of reciprocals of individual
weights.
43
44. Calculate the weights of (A + B) and (A - B) if the measured
values and the weights of A and B, respectively are :
A = 40° 50' 30'' wt. 3
B = 30° 40' 20'' wt. 4
44
45. A + B = 40° 50' 30'' + 30° 40' 20'' = 71° 30' 50''
A - B = 40° 50' 30'' - 30° 40' 20'' = 10° 10' 10''
From law (3), the weights of (A + B) and (A - B)
= = =
Hence, A + B = 71° 30' 50'' wt.
and A - B = 10° 10' 10'' wt.
45
w1 w2
46. Law of weight 4 :
The weight of the product of any quantity multiplied by a
constant, is equal to the weight of that quantity divided by the
square of that constant.
46
47. What is the weight of 3A if A = 30° 25' 40'' and its weight is
3?
Solution :
From law (4) the weight of (3A) is
Weight of (3A) = 91° 17' 00''
= = =
47
C2
32
48. Law of weight 5 :
The weight of the quotient of any quantity divided by a
constant, is equal to the weight of that quantity multiplied by
the square of that.
48
49. Compute the weight of if A = 36° 20' 40'' of weight 3.
Solution :
The constant C of division is 4 and weight w of the
observation is 3.
From law (5), the weight of is wC2
weight of = = 3 x 42
= 9° 5' 10'' wt. 48
49
50. Law of weight 6 :
The weight of the quotient remains unchanged if all the signs
of the equation are changed or if the equation is added to or
subtracted from a constant.
50
51. If weights of A + B = 76° 20' 30'' is 3, what is the weight of
- (A + B) or 180° - (A + B) ?
Solution :
From law (6), the weights of - (A + B) will remain the same.
Weight of - (A + B) = - 76° 20' 30'' wt. 3 or
Weight of 180° - (A + B) or 103° 39' 30'' is equal to 3.
51
52. Law of weight 7 :
If an equation is multiplied by its own weight, the weight of
the resulting equation is equal to the reciprocal of the weight
of that equation.
52
53. Calculate the weight of the equation (A + B) if weight of
(A + B) is . The observed value of (A + B) is 120° 20' 40''
Solution :
As the equation is being multiplied by its own weight, from
rule (7), the weight of { w (A + B) } will be if the w is the
weight of (A + B).
Weight of [ (A + B) = 90° 15' 30'']= =
53
54. The principle of distributing errors by the method of least
squares is of great help to find the most probable value of a
quantity which has been measured for several times.
It is found from the probability equation that the most
probable values of a series of errors arising from observations
of equal weight are those for which the sum of the squares is a
minimum. The fundamental law of least squares is derived
from this.
54
55. The fundamental principle of the method of least squares may
be stated as follows :
“In observation of equal precision, the most probable value of
the observed quantities are those that render the sum of the
square of the residual error a minimum”
55
56. If the measurement are of equal weight, the most probable
value is that which makes the sum of the square of the residue
(v) a minimum.
Thus, ∑v2 = a minimum
If the measurement are of unequal weight, the most probable
value is that which makes the sum of the products of the
weight (w) and the square of the residuals a minimum.
Thus, ∑w(v)2 = a minimum
56
57. When a quantity is being deduced from a series of
observations, the residual error will be the difference between
the adopted value and the several observed values.
Let X1, X2, X3, X4, …. Be the observed values
Z = most probable value
57
58. Then, Z - X1 = e1
Z – X2 = e2
Z – X3 = e3
... ... ....
Z – Xn = en
Where e’s are the respective residual errors of the observed
values.
58
...................... (1)
59. If M = arithmetic mean, then
=
Where n = number of observed values
59
........(2)
60. From equation (1),
nZ - ∑X = ∑e
or Z = + , but = M from equation (2)
Z = M +
If the number of observations is large, n is very large and e is
kept small by making precise measurement, and the second
term becomes nearly equal to zero.
60
........(3)
61. Thus Z = M
Thus, when the number of observations is large, the arithmetic
is the true value or most probable value.
Z = M
61
62. Now, we calculated the residual errors from the mean value. If
v1, v2, v3 etc. are the residual errors, then
M - X1 = v1
M – X2 = v2
M – X3 = v3
... ... ....
M – Xn = vn
62
...................... (4)
63. Adding the above,
nM - ∑X = ∑v
or M = +
As M = , under the preceding conditions and by preceding
equation and hence = 0
Hence the sum of the residuals equals zero and the sum of plus
residual equals the sum of the minus residuals.
63
64. Rules for giving weights to the field observations :
The weights of an angle varies directly to the number of
observations made on the angle.
The weight of the level lines vary inversely as the lengths of
their routes.
The weights of any angle measured a large number of times, is
inversely proportional to the square of the probable error.
64
65. Rules for giving weights to the field observations :
The correction to be applied to various observed quantities, are
in inverse proportion to their weights.
65
66. Distribution of errors to the field observations :
Whenever observations are made in the field, there is always
some error (accidental error).
It is always necessary to check the observations made in the
field for the closing error, if any.
The sign of correction is opposite to that of the closing error.
The closing error should be distributed to the observed
quantities.
66
67. Distribution of errors to the field observations :
The correction to be applied to an observation is inversely
proportion to the weight of the observation. In other words,
the greater the weight, the smaller the correction.
The correction to be applied to an observation is directly
proportional to the length of the probable error.
67
68. Distribution of errors to the field observations :
In case of line of levels, the correction to be applied is
proportional to the length of the route.
If all the observations are of the same weight, the error is
distributed to observed quantities equally, and therefore the
corrections are equal.
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69. The following are the three angles A,B and C observed at a
station O closing the horizon, along with their probable errors
of measurement. Determine their corrected values.
A = 82° 15' 20'' ± 2''
B = 128° 26' 10'' ± 4''
C = 149° 18' 15'' ± 3''
69
70. Sum of three angles = 359° 59' 45''
Discrepancy = 360° 00' 00'' - 359° 59' 45'' =15''
Hence each angle is to be increased, and the error of 15'' is to
be distributed in proportionate to the square of the probable
error.
70
71. 71
Hence Corrected angles
are
A 82° 15' 20'' + 2.07'' 82° 15' 22.07''
B 128° 26' 10'' + 8.28'' 128° 26' 18.28''
C 149° 18' 15'' + 4.65'' 149° 18' 19.65''
TOTAL 360° 00' 00'' OK
72. Adjust the following angles closing the horizon
72
Angle Weight
A = 112° 20' 47'' 2
B = 90° 30' 15'' 3
C = 58° 12' 05'' 1
D = 98° 57' 00'' 4
73. Sum of four angles = 360° 00' 08''
Discrepancy = 360° 00' 00'' - 360° 00' 08'' = - 08''
Hence each angle is to be decreased, and the error of 08'' is to
be distributed in to the angles in an inverse proportionate to
their weights.
73
74. 74
?
* 4
Hence
Corrected
angles are
A 112° 20' 47'' - 0.96'' 112° 20' 45.04''
B 90° 30' 15'' - 0.64'' 90° 30' 13.36''
C 58° 12' 05'' - 1.92'' 58° 12' 2.08''
D 98° 57' 00'' - 0.48'' 98° 56' 59.92''
TOTAL 360⁰ 00’ 00”
75. MPV : The most probable value of a quantity is the value
which has more chances of being true than any other value.
(close to the true value)
It can be determined form principle of least square.
If systematic errors are eliminated from the observations, the
arithmetic mean will be the most probable value of the
quantity being observed.
75
76. Methods of determination of MPV :
76
Determination of MPV
Direct observations
Direct observations of
equal weights
Direct observation of
unequal weights
Indirect observations
Indirect observation
involving unknown of
equal weights
Indirect observation
involving unknown of
unequal weights
Observation equations
accompanied by
condition equations
The Normal equation
The method of differences
The method of correlates
77. CASE-1 Direct observation of equal weight :
The MPV of directly observed quantity of observation of
equal weight is the arithmetic mean of observations.
Thus, if X1, X2, X3, X4, …… Xn , X is Most probable value.
n = No. of observation
77
78. Following direct measurements of a base line were taken :
2523.32 m; 2523.25 m; 2523.17 m; 2523.38 m; 2523.47 m;
2523.68 m. Calculate the most probable value of the length of
the base line.
Solution : MPV = Arithmetic mean M
78
= 2523.378 m. MPV of base line length X = 2523.378 m
79. CASE-2 Direct observation of unequal weight :
The MPV of directly observed quantity of observation of
unequal weigh is weighted arithmetic mean of observations.
Thus, if X1, X2, X3, X4, …… Xn , X is Most probable value.
W = weight of observation w₁, w₂, w₃….
n = No. of observation
79
80. Find the most probable value of the angle from the following
observations :
Angle A = 76° 35' 00'' wt. 1
Angle B = 76° 33' 40'' wt. 2
Solution : MPV = The weighted arithmetic mean of the
observed quantities
80
= 76° 33' 6.67''
81. CASE-3 Indirectly observed quantities involving
unknowns of equal weights :
Most probable value found by method of normal equations.
To form normal equations, multiply equations by their
coefficients of unknowns and add the result.
81
82. Find the most probable value of the angle A from the
following observation equations :
Angle A = 30° 28' 40'' weight 2.
Angle 3A = 91° 25' 55'' weight 3.
82
83. MPV = The normal equation of A
Multiply first eq. by 2 = (1 * 2) and second eq. by 9 = (3 *3)
2*A = 2 * 30° 28' 40'' 2A = 60° 57' 20'' ____(1)
9*3A = 9* 91° 25' 55'' 27A= 822° 53' 15''____(2)
Sum (1) + (2) = 29A = 883° 50' 35''
So Angle A = 883° 50' 35'' / 29 = 30° 28' 38.45''
Most probable value of angle A = 30° 28' 38.45''
83
84. CASE-4 Indirectly observed quantities involving
unknowns of unequal weights :
Most probable value found by method of normal equations.
To form normal equations, “multiply each observation
equation by the product of the algebraic coefficient of the
unknown quantity in the equation and the weight of the that
observation and add the equations thus formed”
84
85. CASE-5 Observation equations accompanied by condition
equations, conditioned quantities :
One or more conditional equations are available.
Methods to compute MPV
1. The Normal equations
2. The Methods of differences
3. The Method of correlates
85
86. The Normal equations :
By avoiding the conditioned equations and forming the normal
equations of the unknowns.
The Methods of differences :
Direct method of forming the normal equations for the
observed quantities is suitable for simple cases.
86
87. Method of correlates :
The method of correlates is also called the method of
condition equation or the method of Lagrange multiplier.
87