CL01 - Lecture 2 - Mistakes and Errors.pdf

FUNDAMENTALS OF SURVEYING CESURV30 – ENGR. EARL JAN D. JUGUETA MISTAKES AND ERRORS  EDJ ERRORS The difference between the true value and the measured value of a quan...

FUNDAMENTALS OF SURVEYING CESURV30 – ENGR. EARL JAN D. JUGUETA MISTAKES AND ERRORS  EDJ ERRORS The difference between the true value and the measured value of a quantity. Errors are inherent in all measurements and result from sources which cannot be avoided. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MISTAKES Inaccuracies in measurements which occur because some aspect of a surveying operation is performed by the surveyor with carelessness, inattention, poor judgment, and improper execution A large mistake is referred to as a blunder. Mistakes and blunders are not classified as errors because they usually are so large in magnitude when compared to errors. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING TYPES OF ERRORS  EDJ SYSTEMATIC ERRORS This type of error is one which will always have the same sign and magnitude as long as field conditions remain constant and unchanged The value of these errors may often be calculated and applied as a correction to the measured quantity. Also known as biases, result from factors that comprise the “measuring system” and include the environment, instrument, and observer. Also called cumulative error NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ACCIDENTAL ERRORS The occurrence of such errors are matters of chance as they are likely to be positive or negative, and may tend in part to compensate or average out according to laws of probability Those variates which remain after all other errors have been removed Caused by factors beyond the control of the surveyor and are present in all surveying measurements. Also called random error NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING SOURCES OF ERRORS  EDJ INSTRUMENTAL ERRORS Due to imperfections in the instruments used, either from faults in their construction of from improper adjustments between the different parts prior to their use NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ NATURAL ERRORS Caused by variations in the phenomena of nature such as changes in magnetic declination, temperature, humidity, wind, refraction, gravity and curvature of the earth NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ PERSONAL ERRORS Arise principally from limitations of the senses of sight, touch and hearing of the human observer which are likely to be erroneous of inaccurate NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING THEORY OF PROBABILITY  EDJ PROBABILITY Defined as the number of times something will probably occur over the range of possible occurrences Theory of probability is useful in indicating the precision of results only in so far as they are affected by accidental errors NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ ASSUMPTIONS RELATIVE TO OCCURRENCES OF ERRORS Small errors occur more often than large ones and that they are more probable Large errors happen infrequently and are therefore less probable; for normally distributed errors, unusually large ones may be mistakes rather than accidental errors Positive and negative errors of the same size happen with equal frequency; that is, they are equally probable The mean of an infinite number of observations is the most probable value NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MOST PROBABLE VALUE (MPV) refers to a quantity which, based on available data, has more chance of being correct than has any other Σ𝑋 𝑋1 + 𝑋2 + 𝑋3 + 𝑋𝑛 𝑚𝑝𝑣 = = 𝑛 𝑛 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MOST PROBABLE VALUE (MPV) Sample Problem: A surveying instructor sent our six groups of students to measure a distance between two points marked on the ground. The students came up with the following six different values; 250.25, 250.15, 249.90, 251.04, 250.50 and 251.22 meters. Assuming these values are equally reliable and that variations result from accidental errors, determine the most probable value of the distance measured. Σ𝑋 𝑋1 + 𝑋2 + 𝑋3 + 𝑋𝑛 𝑚𝑝𝑣 = = 𝑛 𝑛 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MOST PROBABLE VALUE (MPV) Given: 250.25m, 250.15m, 249.90m, 251.04m, 250.50m and 251.22m Σ𝑋 250.25 + 250.15 + 249.90 + 251.04 + 250.50 + 251.22 𝑚𝑝𝑣 = = 𝑛 6 𝑚𝑝𝑣 = 250.51𝑚 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MOST PROBABLE VALUE (MPV) Sample Problem: The angles about a point Q have the following observed values. 1300 15′ 20′′ , 1420 37′ 30′′ and 870 07′ 40′′. Determine the most probable value of each angle. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MOST PROBABLE VALUE (MPV) Given: 𝜃1 = 1300 15′ 20′′ , 𝜃2 = 1420 37′ 30′′ and 𝜃3 = 870 07′ 40′′. 𝑆𝑢𝑚 = 1300 15′ 20′′ + 1420 37′ 30′′ + 870 07′ 40′′ 𝑆𝑢𝑚 = 3600 00′ 30′′ 𝐷𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦 = 3600 − 3600 00′ 30′′ 𝐷𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦 = −30′′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ MOST PROBABLE VALUE (MPV) 𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = 𝐷𝑖𝑠𝑐𝑒𝑝𝑎𝑛𝑐𝑦/3 30′′ 𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 = − = −10′′ 3 𝜃1 = 1300 15′ 20′′ − 10′′ = 𝟏𝟑𝟎𝟎 𝟏𝟓′ 𝟏𝟎′′ 𝜃2 = 1420 37′ 30′′ − 10′′ = 𝟏𝟒𝟐𝟎 𝟑𝟕′ 𝟐𝟎′′ 𝜃3 = 870 07′ 40′′ − 10′′ = 𝟖𝟕𝟎 𝟎𝟕′ 𝟑𝟎′′ NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ RESIDUAL The difference between any measured value of a quantity and its most probable value Σ𝑋 𝑣=𝑋− 𝑛 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ PROBABLE ERROR The quantity which, when added to and subtracted from the most probable value, defines a range within which there is a 50% chance that the true value of the measured quantity lies inside (or outside) the limits thus set Σ𝑣 2 𝑃𝐸𝑠 = ±0.6745 𝑛−1 Σ𝑣 2 𝑃𝐸𝑚 = ±0.6745 𝑛(𝑛 − 1) NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ RELATIVE PRECISION The total amount of error in a given measurement should relate to the magnitude of the measured quantity in order to indicate the accuracy of measurement. It is expressed by a fraction having the magnitude of the error in the numerator and the magnitude of the measured quantity in the denominator It is necessary to express both quantities in the same units, and the numerator is reduced to unity or 1 in order to provide an easy comparison with other measurements. 𝑃𝐸 𝑅𝑃 = 𝑚𝑝𝑣 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ SAMPLE Sample Problem: The following values were determined in a series of tape measurements of a line: 1000.58, 1000.40, 1000.38, 1000.48, 1000.40 and 1000.46 Determine the following: a. Most probable value of the measured length b. Probable error of single measurement and probable error of the mean c. Relative Precision of the measurement NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ SAMPLE a. Most probable value Given: 1000.58m, 1000.40m, 1000.38m, 1000.48m, 1000.40m and 1000.46m Σ𝑋 𝑚𝑝𝑣 = 𝑛 1000.58 + 1000.40 + 1000.38 + 1000.48 + 1000.40 + 1000.46 = 6 𝒎𝒑𝒗 = 𝟏𝟎𝟎𝟎. 𝟒𝟓 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ SAMPLE b. Probable Error Given: 𝑣1 = 1000.58 − 1000.45 = +0.13 𝑣2 = 1000.40 − 1000.45 = −0.05 𝑣3 = 1000.38 − 1000.45 = −0.07 𝑣4 = 1000.48 − 1000.45 = +0.03 𝑣5 = 1000.40 − 1000.45 = −0.05 𝑣6 = 1000.46 − 1000.45 = +0.01 𝑣1 2 = (+0.13)2 = 0.0169 𝑣4 2 = (+0.03)2 = 0.0009 𝑣2 2 = (−0.05)2 = 0.0025 𝑣5 2 = (−0.05)2 = 0.0025 𝑣3 2 = (−0.07)2 = 0.0049 𝑣6 2 = (+0.01)2 = 0.0001 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ SAMPLE Σ𝑣 2 = 𝑣1 2 + 𝑣2 2 + 𝑣3 2 + 𝑣4 2 + 𝑣5 2 + 𝑣6 2 Σ𝑣 2 = 0.0169 + 0.0025 + 0.0049 + 0.0009 + 0.0025 + 0.0001 Σ𝑣 2 = 0.0278 Σ𝑣 2 0.0278 𝑃𝐸𝑠 = ±0.6745 = ±0.6745 = ±𝟎. 𝟎𝟓𝒎 𝑛−1 6−1 Σ𝑣 2 0.0278 𝑃𝐸𝑚 = ±0.6745 = ±0.6745 = ±𝟎. 𝟎𝟐𝒎 𝑛(𝑛 − 1) 6(6 − 1) NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ SAMPLE c. Relative Precision 𝑃𝐸𝑠 0.05 𝟏 𝑅𝑃𝑠 = = = 𝑚𝑝𝑣 1000.45 𝟐𝟎𝟎𝟎𝟎 𝑃𝐸𝑚 0.02 𝟏 𝑅𝑃𝑚 = = = 𝑚𝑝𝑣 1000.45 𝟓𝟎𝟎𝟎𝟎 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ WEIGHTED OBSERVATIONS It is evident that some observations are more precise than others because of better equipment, improved techniques, and superior field conditions. In making adjustments, it is consequently desirable to assign relative weights to individual observations. It can logically be concluded that if an observation is very precise, it will have a small standard deviation or variance, and thus should be weighted more heavily in an adjustment than an observation of lower precision. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ WEIGHTED MEASUREMENTS Sample Problem: Four measurements of a distance were recorded as 284.18, 284.19,284.22 and 284.20 meters and given weights of 1, 3, 2 and 4 respectively. Determine the weighted mean. NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ WEIGHTED MEASUREMENTS Solution: Measured Length Assigned Weight P = X(W) (X) (W) 284.18 1 284.18 284.19 3 852.57 284.22 2 568.44 284.20 4 1136.80 Sums ΣW = 10 ΣP = 2841.99 Weighted Mean = ΣP / ΣW = 2841.99 / 10 = 284.20 m NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ INTERRELATIONSHIP OF ERRORS Summation of Errors If several measured quantities are added, each of which is affected by accidental errors, the probable error of the sum is given by the square root of the sum of the squares of the separate probable errors arising from the several sources 𝑃𝐸𝑠 = ± 𝑃𝐸1 2 + 𝑃𝐸2 2 + 𝑃𝐸3 2 + 𝑃𝐸𝑛 2 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ SUMMATION OF ERRORS Sample Problem: The three sides of a triangular-shaped tract of land is given by the following measurements and corresponding probable errors: a = 162.54 ± 0.03m, b = 234.26 ± 0.05m and c = 195.70 ± 0.04m. Determine the probable error of the sum and the most probable value of the perimeter. 𝑃𝐸𝑅 = 𝑎 + 𝑏 + 𝑐 = 162.54 + 234.26 + 195.70 = 𝟓𝟗𝟐. 𝟓𝟎𝒎 𝑷𝑬𝑺 = ± 𝑃𝐸1 2 + 𝑃𝐸2 2 + 𝑃𝐸3 2 = ± 0.032 + 0.052 + 0.042 = ±0.07m mpv = 592.50 ± 0.07m NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ INTERRELATIONSHIP OF ERRORS Product of Errors For a measured quantity which is determined as the product of two other independently measured quantities such as 𝑄1 and 𝑄2 , the probable error of the product is given by 𝑃𝐸𝑃 = ± (𝑄1 × 𝑃𝐸2 )2 + (𝑄2 × 𝑃𝐸1 )2 NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING  EDJ PRODUCT OF ERRORS Sample Problem: The two sides of a rectangular lot were measured with certain estimated probable errors as follows: W = 253.36 ± 0.06m and L = 624.15 ± 0.08m. Determine the area of the lot and the probable error in the resulting calculation. 𝐴𝑟𝑒𝑎 = 𝐿 × 𝑊 = 624.15 × 253.36 = 𝟏𝟓𝟖𝟏𝟑𝟒. 𝟔𝟒 𝒔𝒒𝒎 𝑷𝑬𝒑 = ± (𝐿 × 𝑃𝐸𝑊 )2 +(𝑊 × 𝑃𝐸𝐿 )2 = ± (624.15 × 0.06)2 +(253.36 × 0.08)2 = ±42.58 sqm mpv = 158134.64 ± 42.58sqm NATIONAL UNIVERSITY DEPARMENT OF CIVIL ENGINEERING