Errors And Statistics & Probable Errors Lecture 2 PDF

ERRORS AND STATISTICS & PROBABLE ERRORS LECTURE 2 OBJECTIVES Identify errors associated with common field operations and field data accordingly; Be familiar with sample statistics in surveying; and Gain knowledge on the concept of probable error. MEASUREMENT Process of de...

ERRORS AND STATISTICS & PROBABLE ERRORS LECTURE 2 OBJECTIVES Identify errors associated with common field operations and field data accordingly; Be familiar with sample statistics in surveying; and Gain knowledge on the concept of probable error. 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 render a numerical value Maybe direct or indirect MEASUREMENT ERROR AND CORRECTION Error Refers to the difference between the measured or calculated value of a quantity and given or established (“true”) value of that quantity e = C -t ERROR AND CORRECTION Correction The negative of error correction = t - C SOURCES OF ERROR 1. Natural errors Caused by variations in the phenomena of nature such as changes in magnetic declination, temperature, refraction, etc. 2. Instrumental errors Due to imperfections in the instruments used, either from faults in construction or improper adjustments 3. Personal errors Arise principally from the limitations of the senses of sight, touch and hearing of the observer TYPES OF ERRORS 1. Mistake or 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 TYPES OF ERRORS Mistake or Blunders COMMON EXAMPLES 1. Reading the wrong graduation on the tape 2. Omitting a whole length of tape 3. Transposition of figures 4. Reading a scale backward 5. Misplacing a decimal point 6. Incorrect recording of field notes 7. Sighting the wrong target TYPES OF ERRORS 2. Systematic or 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 OF ERRORS Systematic or Cumulative Errors TYPES a. 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” b. Counteracting if its sign changes while its magnitude remains the same TYPES OF ERRORS Systematic or Cumulative Errors COMMON EXAMPLES 1. Equipment out of calibration 2. Personal biases of the observer 3. Use of incorrect units (feet instead of meters) TYPES OF ERRORS 3. 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 STATISTICS General Uses of Statistics Statistics aids in decision making Provides comparison Explains the action that has taken place Justification of a claim or assertion Prediction of future outcome Estimation of unknown quantities Statistics summarizes data for public use PRECISION VS. 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 PRECISION VS. ACCURACY GOOD ACCURACY POOR ACCURACY & GOOD PRECISION & GOOD PRECISION GOOD ACCURACY BAD ACCURACY & BAD PRECISION & BAD PRECISION MEASURES OF CENTRAL TENDENCY 1. 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 MEASURES OF CENTRAL TENDENCY 1. Mean MEASURES OF CENTRAL TENDENCY 2. 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. MEASURES OF CENTRAL TENDENCY 2. Median MEASURES OF CENTRAL TENDENCY 3. 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 MEASURES OF CENTRAL TENDENCY 3. Mode MEASURES OF CENTRAL TENDENCY 4. Midrange Value of observation that is midway along the range Arithmetic mean of the largest and smallest observations STATISTICS FOR DISPERSION 1. Range The total spread of the sample Range = Largest value - Smallest value 2. Variance parameter of dispersion or spread STATISTICS FOR DISPERSION 3. Standard Deviation defined as the positive square root of the variance RESIDUAL Sometimes called the deviation Defined as the difference between any measured quantity and its most probable value (MPV) V = X -XO PROBABLE ERROR A quantity which, when added to and subtracted from the MPV, defines a range within which there is 50% chance that the true value lies inside (or outside) the limits thus set A logical estimate based upon the methods and equipment used, upon the experience of the observers, and upon the field conditions existing during the measurement PROBABLE ERROR The value of PE is derived from the method of least squares: RELATIVE PRECISION Expressed by a fraction having the magnitude of the error in the numerator and the magnitude of a measured quantity in the denominator Both quantities should be in the same units and the numerator is reduced to 1 to provide easy comparison with other measurements RELATIVE PRECISION RELATIVE PRECISION RELATIVE PRECISION +0.13 -0.05 -0.07 +0.03 -0.05 +0.01 RELATIVE (ERROR) PRECISION RELATIVE (ERROR) PRECISION WEIGHTED OBSERVATION Degree of reliability Usually based upon: judgment of the surveyor number of measurements taken for a particular quantity the assumption that it is inversely proportional to the square of the probable error e.g., computation of GWAs WEIGHTED OBSERVATION INTERRELATIONSHIP OF ERRORS INTERRELATIONSHIP OF ERRORS INTERRELATIONSHIP OF ERRORS INTERRELATIONSHIP OF ERRORS 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