PCE1 Fundamentals of Surveying - Chapter 1 PDF

PCE1 | FUNDAMENTALS OF SURVEYING CHAPTER 1 (THEORIES OF ERRORS AND MISTAKES) A. PRECISION AND ACCURACY Most probable value (𝑥̅) is the best estimate of the true Precision is the degree of closeness or conformity of...

PCE1 | FUNDAMENTALS OF SURVEYING CHAPTER 1 (THEORIES OF ERRORS AND MISTAKES) A. PRECISION AND ACCURACY Most probable value (𝑥̅) is the best estimate of the true Precision is the degree of closeness or conformity of value. repeated measurements of the same quantity to each ∑ 𝑛𝑥 𝑥̅ = other whereas accuracy is the degree of conformity of a ∑𝑛 measurement to its true value. Where: 𝑛𝑥 = product of observed value and number of B. THEORY OF PROBABILITY observations Probability is the number of times something will 𝑛 = total number of observations probably occur over the range of possible occurrences. Theory of probability is useful in indicating the precision The residual, which is sometimes referred to as the of results only in so far as they are affected by accidental deviation, is defined as the difference between any errors. It is based upon the following assumptions measured value of a quantity and its most probable relative to the occurrences of error. value. 𝑣 = 𝑥 − 𝑥̅ Laws of Probability Small errors occur more than large ones and E. STANDARD DEVIATION they are more probable. Standard Deviation, also called Root-Mean Square Error, Large errors happen infrequently and are is a measure of spread of distribution for a population, therefore less probable. assuming the observations are of equal reliability. Positive and negative errors of the same size happen with equal frequency, that is, they are ∑ (𝜇 − 𝑥)2 equally probable. The mean of an infinite δn = ± √ n number of observations is the most probable value. However, 𝜇 cannot be determined from a sample of C. ERROR TYPES observations. Instead, the arithmetic mean 𝑥̅ is accepted C.1 Gross errors are, in fact, not errors at all, but result of as the most probable value and the population standard mistakes that are due to the carelessness of the observer. deviation is estimated as The gross errors must be detected and eliminated from the survey measurements before such measurements ∑ (𝑥 − 𝑥̅ )2 can be used. 𝑆𝐷 = ± √ 𝑛−1 C.2 Systematic errors follow some pattern and can be expressed by functional relationships based on some F. STANDARD ERROR OF MEAN deterministic system. Like the gross errors, the systematic errors must also be removed from the measurements by applying necessary corrections. ∑ (𝑥 − 𝑥̅ )2 𝑆𝐸𝑚 = ±√ C.3 Random errors are treated using probability model. 𝑛 (𝑛 − 1) Theory of errors deals only with such types of observational errors. Hence, the precision of the mean is enhanced with respect to that of a single observation. There are n D. MOST PROBABLE VALUE deviations/residuals from the mean of the sample and A fixed value of a quantity may be considered as its true their sum will be zero. Thus, knowing (n - 1) deviations value T. The difference between the measured quantity the surveyor could deduce the remaining deviation, and and its true value is known as error ( e = x – T). it may be said that there are (n - 1) degrees of freedom. Since the true value of a measured quantity cannot be The number used when estimating the population is determined, the exact value of e can never be found out. standard deviation. PCE1 | FUNDAMENTALS OF SURVEYING CHAPTER 1 (THEORIES OF ERRORS AND MISTAKES) G. PROBABLE ERROR OF SINGLE MEASUREMENT The most probable error is defined as the error for which there are equal chances of the true error being less and greater than probable error. ∑ ( x − x̄ )𝟐 𝑃𝐸𝑆 = ±0.6745 √ (𝑛 − 1) H. PROBABLE ERROR OF MEAN MEASUREMENT ∑ ( x − x̄ )2 𝑃𝐸𝑚 = ±0.6745 √ 𝑛(𝑛 − 1) I. VARIANCE Variance (V) is used as a measure of dispersion or spread of a distribution. ∑ (𝑥 − 𝑥̅ )2 𝑉= 𝑛−1 J. RELATIVE PRECISION Relative Error, sometimes called Relative Precision, is expressed by a fraction having the magnitude of a measured quantity in the denominator. It is also used to determine the degree of refinement obtained: 𝑃𝐸𝑚 𝑅𝑃 = 𝑥̅ K. WEIGHTED OBSERVATION This quantity W is known as weight of the measurement indicated the reliability of a quantity. Adjustments of Weighted Observations a. The weights are inversely proportional to the square of the corresponding probable errors. 𝑘 𝑊= 𝑃𝐸 2 b. The weights are proportional to the number of observations c. The weights are applied to the individual measurements of unequal reliability to reduce them to one standard. The most probable value is then the weighted mean 𝑥̅𝑚 of the measurements. Thus, ∑ 𝑊𝑋 𝑥̅𝑚 = ∑𝑊