GENERAL SURVEYING 1 OBJECTIVES • At the end of the lecture 4, the students should be able to: • Define measurements and errors and their relationships. • Enumerate the Sources and types of Errors and make proper adjustments • Understand some basic concepts and tools in Statistics OUTLINE • Measurement – Definition – Variability in Repeated Measurements • Errors – Definition – Sources – Types • Concepts and Tools in Statistics – – – – – – General Uses of Statistics Precision versus Accuracy The Concept of Probability Measures of Central Tendency Sample Statistics for Dispersion Measures of Quality Measurement • Process of determining the extent, size or dimensions of a particular quantity in comparison to a given standard • Consists of several physical operations which renders a numerical value • Maybe direct or indirect Measurement and Observation • A measurement entails the entire process of obtaining a desired quantity, including preparations (instrument calibration and setup), pointing, matching, and comparing (reading) • An observation is a single, unadjusted determination of a linear or angular value Variability in Repeated Measurement • Measurements are numerical values for random variables which are subject to statistical fluctuations • An inherent quality of physical properties • Statistical variations due to observational errors • “No exact or true measurement is ever possible and the value of a measured quantity is never known.” Errors and Correction Error • Refer to the difference between the measured or calculated value of a quantity and given or established (“true”) value of that quantity • E =x – t Correction • The negative of error. A • Correction = t – x Sources of Errors Natural Errors • Caused by variations in the phenomena of nature such as changes in magnetic declination, temperature, refraction, etc. Instrumental Errors • Due to imperfections in the instruments used, either from faults in construction or improper adjustments Personal Errors • Arise principally from the limitations of the senses of sight, touch and hearing of the observer Types of Errors • Mistake or Blunders • Systematic or Cumulative Errors • Random or Accidental Errors Mistakes / Blunders • Actually not errors because they are usually so gross in magnitude compared to the other types of errors • One of the most common reasons is simple carelessness on the part of the observer • An observation with a mistake is not useful unless the mistake is removed Common Mistakes / Blunders • • • • • • • Reading the wrong graduation on the tape Omitting a whole length of tape Transposition of figures Reading a scale backward Misplacing a decimal point Incorrect recording of field notes Sighting the wrong target Systematic / Cumulative Errors • So called because they occur according to some deterministic system which, when known, can be expressed by some functional relationship • Caused by physical and natural conditions that vary in accordance with known mathematical or physical laws Types Systematic Error Constant Error • If its magnitude and sign remains the same throughout the measuring process/field conditions are unchanged e.g. tape “too short” or “too long” “TAPING RULES” Counteracting • If its sign changes while its magnitude remains the same • Perhaps due to personal bias of the observer Common Systematic / Cumulative Errors • Equipment out of calibration • Personal biases of the observer Random / Accidental Errors • Produced by irregular causes that are beyond the control of the observer • This variation results from observational errors which have no known functional relationship based upon a deterministic system • Must use probability models Concepts and Tools in Statistics Statistics General Uses of Statistics – Statistics aids in decision making • • • • • Provides comparison Explains action that has taken place Justifies a claim or assertion Predicts future outcome Estimates unknown quantities – Statistics summarizes data for public use Precision and Accuracy Precision • Degree of refinement and consistency of the performance of an operation used to obtain the result • Measure of uniformity or reproducibility of the result Accuracy • Degree of conformity with a standard/accepted value • Denotes how close a given measurement is to the absolute value of the quantity Concept of Probability Probability • Is the likelihood associated with a random event Random Variable • Defined as a variable that takes on any of several possible values, with each of which is associated a probability Random Event • Is one whose relative frequency of occurrence approaches a stable limit as the number of observations/repetitions of an experiment is increased to infinity. Representations of the Probability Density Frequency Diagrams • Histogram – Constructed to represent the probability density of a single random variable. • Stereogram – Constructed to represent the probability density of two random variables. HISTOGRAM STEROGRAM Measures of Central Tendency • • • • Median Mean Mode Midrange Sample Median • Positional middle of the arrayed data. • Characteristics: – Affected by the position of each item but not by the value of each item. • A stable measure of central tendency. Sample Mean • Sum of all the values of the observations divided by the number of observations • Most Probable Value • Characteristics: – Most familiar measure of central tendency used. – Affected by the value of every observation. – In particular, it is strongly influenced by extreme values. • Since it is a calculated number, it may not be an actual number in the data set Sample Mode • Value that occurs most frequently in the sample • Characteristics: – Not always exist. If it does, it may not be unique (2 or more sample modes). – Not affected by extreme values • Easiest to compute. Midrange • Value of observation that is midway along the range • Arithmetic mean of the largest and smallest observations. • Example: 3.1,3.5,3.6,3.9,4.0 Midrange= (4.0+3.1)/2=3.55 *Note: After arranging: 3.1,3.5,3.6,3.9,4.0 Thus, sample median= 3.60 sample midrange= 3.55 Problem Solving • Given at the right is a data of repeated horizontal angular measurements between two lines. Compute for the: 1. 2. 3. 4. Median Mean Mode Midrange of the sample data Sample Statistics for Dispersion • Range – The total spread of the sample – Range = Largest value-Smallest value • Mean Deviation • arithmetic mean of the absolute values of the deviation from any measure of position (usually the mean). Variance • Parameter of dispersion or spread Standard Deviation • Defined as the positive square root of the variance Measures of Quality Weight – Defined as the quantity that is inversely proportional to variance. Relative Error/Precision – Refers to the ratio of the error (measure of precision) to the measured or estimated quantity. Ratio of Misclosure – In surveying, the ratio between the total error and the total length of the survey Mean Square Error – Used as a measure of accuracy References: • Davis, R.E., et. al (1981). Surveying: Theory and Practice. USA: McGraw-Hill, Inc. • La Putt, J.P. (2007). Elementary Surveying. Philippines: National Book Store.
