MODULE 02 - ERROR CE 214 Fundamentals of Surveying Lec 1st Semester, AY 2022-2023 Engr. Karen S. Galoso Instructor Civil Engineering Department OUTLINE βͺ Accuracy & Precision βͺ Definition, Types and Sources of Errors βͺ Theory of Errors & Adjustments CE 214 FUNDAMENTALS OF SURVEYING LEC Accuracy and Precision ACCURACY the degree of conformity of a given measurement with a standard value. PRECISION the extent to which a given set of measurements agree with their mean. Precision without accuracy Neither precision nor accuracy CE 214 FUNDAMENTALS OF SURVEYING LEC Precision and accuracy Definition, Types and Sources of Errors ERRORS a difference between a measured or observed value and the “true” value. SOURCES • NATURAL ERRORS due to variations in the environment such as magnetic declination, temperature, humidity, wind etc. • INSTRUMENTAL ERRORS due to limitations in the instruments used • PERSONAL ERRORS due to the limitations of senses of sight, touch and hearing. MISTAKES or gross error; the inaccuracies in measurements which happen due to carelessness, inattention, poor judgment and wrong execution. TYPES SYSTEMATIC ERRORS repeated; errors for which the magnitude and algebraic sign can be determined; also known as cumulative errors CE 214 FUNDAMENTALS OF SURVEYING LEC ACCIDENTAL ERRORS random; likely to be positive or negative Theory of Errors & Adjustments MOST PROBABLE VALUE ΰ΄₯= πΏ σπ π RESIDUAL ΰ΄₯ −π π= πΏ v – residual πΰ΄€ – the most probable value x – individual quantity (observation) n – number of observation ΰ΄₯= πΏ σπ β π π For a single quantity multiple measurement taken as π₯1 , π₯2 , π₯3 … with weight π€1 , π€2 , π€3 … PROBABLE ERROR OF SINGLE OBSERVATION π·π¬π = π. ππππ σ ππ π−π PROBABLE ERROR OF THE MEAN σ ππ π·π¬π = π. ππππ π(π − π) CE 214 FUNDAMENTALS OF SURVEYING LEC Percent Error 50% 0.6745 90% 1.645 95.5% 2.00 95% 1.96 99.9% 3.29 Theory of Errors & Adjustments ERROR PROPAGATION Case I. Sum of Errors π¬πππ = π¬π π + π¬π π + π¬π π + β― + π¬π π Case II. Error in a product π¬ππππ πππ = (πΈπ β π¬π )π + (πΈπ β π¬π )π CE 214 FUNDAMENTALS OF SURVEYING LEC Theory of Errors & Adjustments Sample Problem 1 A surveying instructor sent out seven groups of students to measure a distance between two points marked on the ground. The students came up with the following measured values: 240.25 m, 240.15 m, 239.90 m, 241.04 m, 240.50m, 241.22m and 242m. Find the most probable value of the measured distance. Sample Problem 2 The angles about a point Q have the following observed values : 120Λ15’20”, 152Λ37’30” and 87Λ07’40”. Determine the most probable value for each angle. Solution Solution ΰ΄₯ = πππ. ππ π πΏ Most probable values are: π1 = 120° 15′ 10′′ π2 = 152° 37′ 20′′ π3 = 87° 07′ 30′′ CE 214 FUNDAMENTALS OF SURVEYING LEC σ1 σ3 σ2 Theory of Errors & Adjustments Sample Problem 3 Given are the interior angles of a triangle. Determine the discrepancy for the given observation and the most probable value of each angle. π΄ = 33° 14′ 37′′ π΅ = 98° 30′ 09′′ πΆ = 48° 15′ 05′′ Solution Most probable values are: 33° 14′ 40′′ π΄= π΅ = 98° 30′ 12′′ πΆ = 48° 15′ 08′′ Sample Problem 4 The following values were determined in a series of tape measurements of a line: 1001.58 m, 1001.40 m, 1001.38 m, 1001.48 m, 1001.40 m, 1001.46 m. Determine the ff: 1) Most Probable Value 2) Probable Error of a single measurement 3) Probable Error of the mean 4) Final expression of the most probable length Solution ΰ΄₯ = ππππ. ππ π πΏ π·π¬π = ±π. ππ π π·π¬π = ±π. ππ π π³ = ππππ. π ± π. ππ π CE 214 FUNDAMENTALS OF SURVEYING LEC Theory of Errors & Adjustments Sample Problem 5 Four measurements of a distance were recorded as tabulated: Measured Length 285.18 m 285.19 m 285.22 m 285.20 m Weight 1 3 2 4 What is the most probable value of the measurement? Sample Problem 6 The three sides of a triangular lot has the following sides and corresponding probable errors. Determine the probable error of the sum and the most probable value of the perimeter. Solution π·π¬πππ = ±π. ππ π π· = πππ. ππ ± π. ππ π Solution ΰ΄₯ = πππ. πππ π πΏ CE 214 FUNDAMENTALS OF SURVEYING LEC π = 152.64 ± 0.03 π π = 253.25 ± 0.05 π π = 185.60 ± 0.04 π Theory of Errors & Adjustments Sample Problem 7 The sides of a rectangular lot has the following sides and corresponding probable error. π = 564.75 ± 0.08 π πΏ = 235.75 ± 0.06 π Determine the probable error of the product and the most probable value of the area. Solution π·π¬ππππ πππ = ±ππ. ππ ππ π¨ = πππ, πππ. ππ ± ππ. ππ ππ CE 214 FUNDAMENTALS OF SURVEYING LEC
