Theory of Errors in Observations PDF

Machine Translated by Google 3 Theory of Errors in Observations ÿ 3.1 INTRODUCTION Making observations (measurements), and subsequent calculations and analyzes using them, are fundamental tasks of surveyors. Good observations require a Combination of human skill and mechanical equipment applied w...

Machine Translated by Google 3 Theory of Errors in Observations ÿ 3.1 INTRODUCTION Making observations (measurements), and subsequent calculations and analyzes using them, are fundamental tasks of surveyors. Good observations require a Combination of human skill and mechanical equipment applied with the utmost judgment. However, no matter how carefully made, observations are never exact and will always contain errors. Surveyors (geomatics engineers), whose work must be performed to exacting standards, should thoroughly under-stand the different kinds of errors, their sources and expected magnitudes under varying conditions, and their manner of propagation. Only then can they select Instruments and procedures necessary to reduce error sizes to within tolerable limits. Of equal importance, surveyors must be capable of assessing the magnitudes of errors in their observations so that either their acceptability can be veri-fied or, if necessary, new ones made. The design of measurement systems is now practiced. Computers and sophisticated software are commonly used tools now Used by surveyors to plan measurement projects and to investigate and distribute errors after results have been obtained. ÿ 3.2 DIRECT AND INDIRECT OBSERVATIONS Observations may be made directly or indirectly. Examples of direct observations They apply a tape to a line, fitting a protractor to an angle, or turning an angle With a total station instrument. An indirect observation is secured when it is not possible to apply a measuring instrument directly to the quantity to be observed. The answer is therefore determined by its relationship to some other observed value or values. As an Machine Translated by Google 46 THEORY OF ERRORS IN OBSERVATIONS Example, we can find the distance across a river by observing the length of a line on one side of the river and the angle at each end of this line to a point on the other side, and then computing the distance by one of the standard trigonometric formulas. Many indirect observations are made in surveying, and since all mea-surements contain errors, it is inevitable that quantities computed from them will also contain errors. The manner by which errors in measurements combine to produce erroneous computed answers is called error propagation. This topic is discussed further in Section 3.17. ÿ 3.3 ERRORS IN MEASUREMENTS By definition, an error is the difference between an observed value for a quantity and its true value, or E=X-X (3.1) X where E is the error in an observation, X the observed value, and its true value. It can be unconditionally stated that (1) no observation is exact, (2) every obser-vation contains errors, (3) the true value of an observation is never known, and, therefore, (4) the exact error present is always unknown. These facts are demon-strated by the following. When a distance is observed with a scale divided into tenths of an inch, the distance can be read only to hundredths (by interpolation). However, if a better scale graduated in hundredths of an inch was available and read under magnification, the same distance might be estimated to thousandths of an inch.And with a scale graduated in thousandths of an inch, a reading to ten-thousandths might be possible. Obviously, accuracy of observations depends on the scale's division size, reliability of equipment used, and human limitations in estimating closer than about one tenth of a scale division. As better equipment is developed, observations more closely approach their true values, but they can never be exact. Note that observations, not counts (of cars, pennies, marbles, or other objects), are under consideration here. ÿ 3.4 MISTAKES These are observer blunders and are usually caused by misunderstanding the problem, carelessness, fatigue, missed communication, or poor judgment. Exam-please include transposition of numbers, such as recording 73.96 instead of the core value of 79.36; reading an angle counterclockwise, but indicating it as a clockwise angle in the field notes; sighting the wrong target; or recording a meas-ured distance as 682.38 instead of 862.38. Large mistakes such as these are not considered in the successful discussion of errors. They must be detected by careful and systematic checking of all work, and eliminated by repeating some or all of the measurements. It is very difficult to detect small mistakes because they merge with errors. When not exposed, these small errors will therefore be incorrectly treated as errors. Machine Translated by Google 3.6 Types of Errors 47 ÿ 3.5 SOURCES OF ERRORS IN MAKING OBSERVATIONS Errors in observations stem from three sources, and are classified accordingly. Natural errors are caused by variations in wind, temperature, humidity, atmospheric pressure, atmospheric refraction, gravity, and magnetic declination. An example is a steel tape whose length varies with changes in temperature. Instrumental errors result from any imperfection in the construction or adjustment of instruments and from the movement of individual parts. For example, the graduations on a scale may not be perfectly spaced, or the scale may be warped.The effect of many instrumental errors can be reduced, or even eliminated, by adopting proper surveying procedures or applying computed corrections. Personal errors arise mainly from limitations of the human senses of sight and touch. As an example, a small error occurs in the observed value of a horizontal angle if the vertical crosshair in a total station instrument is not perfectly aligned on the target, or if the target is the top of a rod that is being held slightly out of plumb. ÿ 3.6 TYPES OF ERRORS Errors in observations are of two types: systematic and random. Systematic errors, also known as biases, result from factors that comprise the “measuring system” and include the environment, instrument, and observer. So long as system conditions remain constant, the systematic errors will likewise remain constant. If conditions errors change, the magnitudes of systematic also change. Because systematic errors tend to accumulate, they are sometimes called cumulative errors. Conditions producing systematic errors conform to physical laws that can be modeled mathematically. Thus, if the conditions are known to exist and can be observed, a correction can be computed and applied to observed values. An example of a constant systematic error is the use of a 100-ft steel tape that has been calibrated and found to be 0.02 ft too long. It introduces a 0.02-ft error each time it is used, but applying a correction readily eliminates the error. An example of variable systematic error is the change in length of a steel tape resulting from temperature differentials that occur during the period of the tape's use. If the temperature changes are observed, length corrections can be computed by a simple formula, as explained in Chapter 6. Random errors are those that remain in measured values after errors and systematic errors have been eliminated. They are caused by factors beyond the control of the observer, obey the laws of probability, and are sometimes called accidental errors. They are present in all surveying observations. The magnitudes and algebraic signs of random errors are matters of chance. There is no absolute way to compute or eliminate them, but they can be estimated using adjustment procedures known as least squares (see Section 3.21 and Chapter 16). Random errors are also known as compensating errors, since they tend to partially cancel themselves in a series of observations. For example, a person interpolating to hundredths of a foot on a tape graduated only to tenths, or reading a level rod marked in hundredths, will presumably estimate too high on Machine Translated by Google 48 THEORY OF ERRORS IN OBSERVATIONS some values and too low on others. However, individual personal characteristics may nullify such partial compensation since some people are inclined to interpo-late high, others interpolate low, and many favor certain digits—for example, 7 instead of 6 or 8, 3 instead of 2 or 4, and particularly 0 instead of 9 or 1. ÿ 3.7 PRECISION AND ACCURACY A discrepancy is the difference between two observed values of the same quan-tity. A small discrepancy indicates there are probably no mistakes and random Errors are small. However, small discrepancies do not preclude the presence of systematic errors. Precision refers to the degree of refinement or consistency of a group of ob-servations and is evaluated on the basis of discrepancy size. If multiple observa-tions are made of the same quantity and small discrepancies result, this indicates high precision. The degree of attainable precision is dependent on equipment sensitivity and observer skill. Accuracy denotes the absolute proximity of observed quantities to their true values. The difference between precision and accuracy is perhaps best illustrated With reference to target shooting. In Figure 3.1(a), for example, all five shots exist in a small group, indicating a precise operation; That is, the shooter was able to re-peat the procedure with a high degree of consistency. However, the shots are far from the bull's-eye and therefore not accurate. This probably results from mis-aligned rifle sights. Figure 3.1(b) shows randomly scattered shots that are neither precise nor accurate. In Figure 3.1(c), the closely spaced grouping, in the bull's-eye, It represents both precision and accuracy. The shooter who obtained the results in (a) was perhaps able to produce the shots of (c) after aligning the rifle sights. In survey-ing, this would be equivalent to the calibration of observing instruments. As with the shooting example, a survey can be precise without being accu-rate. To clarify, if refined methods are employed and readings taken carefully, say to 0.001 ft, but there are instrumental errors in the measuring device and cor-rections are not made for them, the survey will not be accurate. As a numerical Example, two observations of a distance with a tape assumed to be 100,000 ft long, that is actually 100.050 ft, might give results of 453.270 and 453.272 ft. These Values are precise, but they are not accurate, since there is a systematic error of approximately 4.53 * 0.050 = 0.23 ft in each. The precision obtained would be expressed as which is excellent, but (453.272 - 453.270)>453.271 = 1>220,000, Figure 3.1 Examples of precision and accuracy. (a) Results are precise but not accurate. (b) Results are neither precise nor more accurate. (c) Results are both precise and accurate. (a) (b) (c) Machine Translated by Google 3.9 Probability 49 Accuracy of the distance is only in 2000. Also, a survey may 0.23>453.271 = 1 part appear to be accurate when rough observations have been taken. For example, the Angles of a triangle may be read with a compass to only the nearest degree and 1>4 yet produce a sum of exactly 180°, or a zero misclosure error. On good surveys, Precision and accuracy are consistent throughout. ÿ 3.8 ELIMINATING MISTAKES AND SYSTEMATIC ERRORS All field operations and office calculations are governed by a constant effort to Eliminate mistakes and systematic errors. Of course it would be preferable if mis-takes never occurred, but because humans are fallible, this is not possible. In the field, experienced observers who alertly perform their observations using stan-dardized repetitive procedures can minimize mistakes. Mistakes that do occur can be corrected only if discovered. Comparing several observations of the same Quantity is one of the best ways to identify mistakes. Making a common sense esti-mate and analysis is another. Assume that five observations of a line are recorded as follows: 567.91, 576.95, 567.88, 567.90, and 567.93. The second value disagrees with the others, apparently because of a transposition of figures in reading or recording. Either casting out the doubtful value, or preferably repeating the ob-servation can erase this mistake. When a mistake is detected, it is usually best to repeat the observation. However, if a sufficient number of other observations of the quantity are avail- able and in agreement, as in the foregoing example, the widely divergent result may be discarded. Serious consideration must be given to the effect on an average age before discarding a value. It is seldom safe to change a recorded number, Even though there appears to be a simple transposition in figures. Tampering with physical data is always a bad practice and will certainly cause trouble, even if done infrequently. Systematic errors can be calculated and proper corrections applied to the observations. Procedures for making these corrections to all basic surveying ob-servations are described in the chapters that follow. In some instances, it may be possible to adopt a field procedure that automatically eliminates systematic er-rors. For example, as explained in Chapter 5, a leveling instrument out of adjustment causes incorrect readings, but if all backsights and foresights are made the same length, the errors cancel in differential leveling. ÿ 3.9 PROBABILITY At one time or another, everyone has had an experience with games of chance, Such as coin flipping, card games, or dice, which involves probability. In basic Mathematics courses, laws of combinations and permutations are introduced. It is shown that events that happen randomly or by chance are governed by mathematical principles referred to as probability. Probability may be defined as the ratio of the number of times a result should occur to its total number of possibilities. For example, in the toss of a fair die there is a one-sixth probability that a 2 will come up. This simply means that There are six possibilities, and only one of them is a 2. In general, if a result may Machine Translated by Google 50 THEORY OF ERRORS IN OBSERVATIONS occur in m ways and fail to occur in n ways, then the probability of its occurrence is The probability that any result will occur is a fraction between 0 m>(m + n). and 1; 0 indicating impossibility and 1 denoting absolute certainty. Since any The result must either occur or fail, the sum of the probabilities of occurrence and failure is 1. Thus if is the1>6 probability of throwing a 2 with one toss of a die, then is(1-1>6), or that a 2 will not come up. the probability 5>6 The theory of probability is applicable in many sociological and scientific observations. In Section 3.6, it was pointed out that random errors exist in all sur-veying work. This can perhaps be better appreciated by considering the measur-ing process, which generally involves executing several elementary tasks. Besides Instrument selection and calibration, these tasks may include setting up, center-ing, aligning, or pointing the equipment; setting, matching, or comparing index marks; and reading or estimating values from graduated scales, dials, or gauges. Because of equipment and observer imperfections, exact observations cannot be made, so they will always contain random errors. The magnitudes of these errors, and the frequency with which errors of a given size occur, follow the laws of probability. For convenience, the term error will be used to mean only random error for the remainder of this chapter. It will be assumed that all mistakes and systematic Errors have been eliminated before random errors are considered. ÿ 3.10 MOST PROBABLE VALUE It has been stated earlier that in physical observations, the true value of any quan-tity is never known. However, its most probable value can be calculated if redun- dant observations have been made. Redundant observations are measurements in excess of the minimum needed to determine a quantity. For a single unknown, Such as a line length that has been directly and independently observed a number of times using the same equipment and procedures,1 the first observation estab-lishes a value for the quantity and all additional observations are redundant. The Most probable value in this case is simply the arithmetic meaning, or M= ©M n (3.2) the sum of the individual where isMthe most probable value of the quantity, ©M measurements M, and n the total number of observations. Equation (3.2)can be derived using the principle of least squares, which is based on the theory of probability. As discussed in Chapter 16, in more complicated problems, where the Observations are not made with the same instruments and procedures, or if several interrelated quantities are being determined through indirect observations, Most probable values are calculated by employing least-squares methods. The 1 The significance of using the same equipment and procedures is that observations are of equal reliaability or weight. The subject of unequal weights is discussed in Section 3.20. Machine Translated by Google 3.12 Occurrence of Random Errors 51 Treatment here relates to multiple direct observations of the same quantity using the same equipment and procedures. ÿ 3.11 RESIDUALS Having determined the most probable value of a quantity, it is possible to calculate residuals. A residual is simply the difference between the most probable value and any observed value of a quantity, which in equation form is n=M-M (3.3) where v is the residual in any observation M, and is the mostM probable value for the quantity. Residuals are theoretically identical to errors, with the exception that residuals can be calculated whereas errors cannot because true values are never known. Thus, residuals rather than errors are the values actually used in the analysis and adjustment of survey data. ÿ 3.12 OCCURRENCE OF RANDOM ERRORS To analyze the manner in which random errors occur, consider the data of Table 3.1, which represents 100 repetitions of an angle observation made with a Precise total station instrument (described in Chapter 8). Assume these observa-tions are free from mistakes and systematic errors. For convenience in analysis the data, except for the first value, only the seconds' portions of the observations are tabulated. The data have been rearranged in column (1) so that entries begin With the observed smallest value and are listed in increasing size. If a certain value was obtained more than once, the number of times it occurred, or its frequency, is tabulated in column (2). From Table 3.1, it can be seen that the dispersion (range in observations However, it is difficult to ana-lyze from smallest to largest) is 30.8 - 19.5 = 11.3 sec. the distribution pattern of the observations by simply scanning the tabular values; that is, beyond assessing the dispersion and noticing a general trend for observations toward the middle of the range to occur with greater frequency. To assist in studying the data, a histogram can be prepared. This is simply a bar graph showing the sizes of the observations (or their residuals) versus their frequency of occurrence. It gives an immediate visual impression of the distribution pattern of the observations (or their residuals). For the data of Table 3.1, a histogram showing the frequency of occurrence of the residuals has been developed and is plotted in Figure 3.2. To plot a his-togram of residuals, it is first necessary to compute the most probable value for the observed angle. This has been done with Equation (3.2). As shown at the bot-Then using tom of Table 3.1, its value is 27°43¿24.9–. Equation (3.3), residuals for All observed values are computed. These are tabulated in column (3) of Table 3.1. The residuals vary from 5.4– -5.9–. (The sum of the absolute value of these two extremes is the dispersion, or 11.3–. to ) To obtain a histogram with an appropriate number of bars for portraying the distribution of residuals adequately, the interval of residuals represented by Machine Translated by Google 52 THEORY OF ERRORS IN OBSERVATIONS TABLE 3.1 ANGLE OBSERVATIONS FROM PRECISE TOTAL STATION INSTRUMENT Observed Value (1) No. Residual Observed (Sec) Value (2) (3) (1 Cont.) 1 5.4 20.0 1 4.9 25.2 20.5 1 4.4 20.8 1 21.2 1 21.3 (sec) (3 Cont.) (2.Cont.) 3 -0.2 1 -0.3 25.4 1 -0.5 4.1 25.5 2 -0.6 3.7 25.7 3 -0.8 1 3.6 25.8 4 -0.9 21.5 1 3.4 25.9 2 -1.0 22.1 2 2.8 26.1 1 -1.2 22.3 1 2.6 26.2 2 -1.3 22.4 1 2.5 26.3 1 -1.4 22.5 2 2.4 26.5 1 -1.6 22.6 1 2.3 26.6 3 -1.7 22.8 2 2.1 26.7 1 -1.8 23.0 1 1.9 26.8 2 -1.9 23.1 2 1.8 26.9 1 -2.0 23.2 2 1.7 27.0 1 -2.1 23.3 3 1.6 27.1 3 -2.2 23.6 2 1.3 27.4 1 -2.5 23.7 2 1.2 27.5 2 -2.6 23.8 2 1.1 27.6 1 -2.7 23.9 3 1.0 27.7 2 -2.8 24.0 5 0.9 28.0 1 -3.1 24.1 3 0.8 28.6 2 -3.7 24.3 1 0.6 28.7 1 -3.8 24.5 2 0.4 29.0 1 -4.1 24.7 3 0.2 29.4 1 -4.5 24.8 3 0.1 29.7 1 -4.8 24.9 2 0.0 30.8 1 -5.9 25.0 2 27°43¿19.5–27°43¿19.5 -0.1 27°43¿25.1– Residential No. © = 2494.0 © = 100 Mean = 2494.0>100 = –24.9 Most Probable Value = 27°43¿24.9– Machine Translated by Google 3.12 Occurrence of Random Errors 53 16 14 Frequency polygon Normal distribution curve 12 10 Histogram 8 6 Figure 3.2 Histogram, frequency polygon, and normal distribution curve of 2 59.5+ 52.5+ 55.4+ 58.3+ 51.3+ 54.2+ 57.1+ 50.1+ 00.– 0 53 5.30.+ 0 50.1- 57.1- 54.2- 51.3- 58.3- 55.4- 52.5- 59.5- eyccnneeruruqcecrfF o 4 Size of residual Each bar, or the class interval, was chosen as –0.7. This produced 17 bars on the graph. The range of residuals covered by each interval, and the number of residuals that occur within each interval, are listed in Table 3.2. By plotting class inter-vals on the abscissa against the number (frequency of occurrence) of residuals in each interval on the ordinate, the histogram of Figure 3.2 was obtained. If the adjacent top center points of the histogram bars are connected with straight lines, the so-called frequency polygon is obtained. The frequency polygon for the data of Table 3.1 is superimposed as a heavy dashed blue line in Figure 3.2. It graphically displays essentially the same information as the histogram. If the number of observations being considered in this analysis were increased progressively, and accordingly the histogram's class interval took smaller and smaller, ultimately the frequency polygon would approach a smooth continuous curve, symmetrical about its center like the one shown with the heavy solid blue line in Figure 3.2. For clarity, this curve is shown separately in Figure 3.3. The curve's “bell shape” is characteristic of a normally distributed group of errors, and thus it is often referred to as the normal distribution curve. Statisticians frequently call it the normal density curve, since it shows the densities of errors having vari-ous sizes. In surveying, normal or very nearly normal error distributions are expected, and henceforth in this book that condition is assumed. In practice, histograms and frequency polygons are seldom used to represent error distributions. Instead, normal distribution curves that approximate them are preferred. Note how closely the normal distribution curve superimposed on Figure 3.2 agrees with the histogram and the frequency polygon. As demonstrated with the data of Table 3.1, the histogram for a set of observations shows the probability of occurrence of an error of a given size graphically by bar areas. For example, 14 of the 100 residuals (errors) in Figure 3.2 are between -0.35– and +0.35–. This represents 14% of the errors, and the center histogram bar, which corresponds to this interval, is 14% of the total area of all residuals from angle measurements made with total station. Machine Translated by Google 54 THEORY OF ERRORS IN OBSERVATIONS TABLE 3.2 RANGES OF CLASS INTERVALS AND NUMBER OF RESIDUALS IN EACH INTERVAL Histogram Interval (Sec) Number of Residents in Interval -5.95 to -5.25 1 -5.25 to -4.55 1 -4.55 to -3.85 2 -3.85 to -3.15 3 -3.15 to -2.45 6 -2.45 to -1.75 8 -1.75 to -1.05 10 -1.05 to -0.35 11 -0.35 to +0.35 14 +0.35 to +1.05 12 +1.05 to +1.75 11 +1.75 to +2.45 8 +2.45 to +3.15 6 +3.15 to +3.85 3 +3.85 to +4.55 2 +4.55 to +5.25 1 +5.25 to +5.95 1 © = 100 bars. bars. Likewise, the area between ordinates constructed at any two abscissas of a normal distribution curve represents the percent probability that an error of that size exists. Since the area sum of all bars of a histogram represents all errors, it therefore represents all probabilities, and thus its sum equals 1. Likewise, the total area beneath a normal distribution curve is also 1. If the same observations of the preceding example had been taken using better equipment and more caution, smaller errors would be expected and the normal distribution curve would be similar to that in Figure 3.4(a). Compared to Figure 3.3, this curve is taller and narrower, showing that a greater percentage of values have smaller errors, and fewer observations contain large ones. For this com-parison, the same ordinate and abscissa scales must be used for both curves. Thus, the observations of Figure 3.4(a) are more precise. For readings taken less pre-cisely, the opposite effect is produced, as illustrated in Figure 3.4(b), which shows a shorter and wider curve. In all three cases, however, the curve maintained its characteristic symmetric bell shape. From these examples, it is seen that relative precisions of groups of observations become readily apparent by comparing their normal distribution curves. The Machine Translated by Google 3.14 Measures of Precision 55 – 0.6+ 0.5+ 0.4+ 0.3+ 0.2+ 0.1+ 0.1- 0.2- 0.3- 0.4- 0.5- 0.6- eyccnneeruruqcecrfF o Inflection point + -1.65 +1.65 (–E90) (E90) -1.96 +1.96 (–E95) (E95) Size of residual Normal distribution curve for a set of observations can be computed using parame-ters derived from the residuals, but the procedure is beyond the scope of this text. ÿ 3.13 GENERAL LAWS OF PROBABILITY From an analysis of the data in the preceding section and the curves in Figures 3.2 through 3.4, some general laws of probability can be stated: 1. Small residuals (errors) occur more often than large ones; That is, they are more probable. 2. Large errors happen infrequently and are therefore less probable; for nor-mally distributed errors, unusually large ones may be mistakes rather than random errors. 3. Positive and negative errors of the same size occur with equal frequency; That is, they are equally probable. [This enables an intuitive deduction of Equation (3.2) to be made: that is, the most probable value for a group of repeated observations, made with the same equipment and procedures, is the mean.] ÿ 3.14 MEASURES OF PRECISION As shown in Figures 3.3 and 3.4, although the curves have similar shapes, there are significant differences in their dispersions; that is, their abscissa widths differ. The magnitude of dispersion is an indication of the relative precisions of the observa-tions. Other statistical terms are more commonly used to express precisions of groups Figure 3.3 Normal distribution curve. Machine Translated by Google 56 THEORY OF ERRORS IN OBSERVATIONS 24 22 20 18 16 Inflection point 14 12 10 8 6 4 2 – + (a) 14 12 10 Inflection point 8 6 Figure 3.4 Normal distribution curves for: (a) increased precision, (b) decreased precision. 4 2 – + (b) of observations are standard deviation and variance. The equation for the standard deviation is s = ;A©n2n - 1 (3.4) s the standard deviation of a group of observations of the same quanwhere is ©n2 ntity, the residual of an individual observation, the sum of squares of the s2 , individual residuals, and n the number of observations. Variance is equal to the square of the standard deviation. Note that in Equation (3.4), the standard deviation has both plus and minus values. On the normal distribution curve, the numerical value of the standard deviation is the abscissa at the inflection points (locations where the curve Machine Translated by Google 3.14 Measures of Precision 57 changes from concave downward to concave upward). In Figures 3.3 and 3.4, these inflection points are shown. Note the closer spacing between them for the more precise observations of Figure 3.4(a) as compared to Figure 3.4(b). Figure 3.5 is a graph showing the percentage of the total area under a nor-mal distribution curve that exists between ranges of residuals (errors) having equal positive and negative values. The abscissa scale is shown in multiples of the standard deviation. From this curve, the area between residuals +s of and -s equals approximately 68.3% of the total area under the normal distribution curve. Hence, it gives the range of residuals that can be expected to occur 68.3% of the time. This relationship is shown more clearly on the curves in Figures 3.3 ;s are shown shaded. The percentages shown and 3.4, where the areas between in Figure 3.5 apply to all normal distributions; Regardless of curve shape or the numerical value of the standard deviation. 100 99.7 1.9599 95 90 1.6449 90 1.4395 80 1.2816 1.1503 1.0364 70 68.27 0.9346 0.8416 60 eg ytailtin be reecb a avdro e ren urfP oc a u p 0.7554 50 0.6745 50 0.5978 0.5244 40 0.4538 30 0.3853 0.3186 0.2534 20 0.1891 Figure 3.5 Relation between error and 0.1257 10 0.0627 0 0 0.5 1.0 1.5 2.0 Error 2.5 3.0 3.5 percentage of area under normal distribution curve. Machine Translated by Google 58 THEORY OF ERRORS IN OBSERVATIONS ÿ 3.15 INTERPRETATION OF STANDARD DEVIATION It has been shown that the standard deviation establishes the limits within which Observations are expected to fall 68.3% of the time. In other words, if an observa-tion is repeated ten times, it will be expected that about seven of the results will fall within the limits established by the standard deviation, and conversely about Three of them will fall anywhere outside these limits. Another interpretation is that one additional observation will have a 68.3% chance of falling within the limits set by the standard deviation. When Equation (3.4) is applied to the data of Table 3.1, a standard deviation of ;2.19 is obtained. In examining the residuals in the table, 70 of the 100 values, or 70%, are actually smaller than 2.19 sec. This illustrates that the theory of prob-ability closely approximates reality. ÿ 3.16 THE 50, 90, AND 95 PERCENT ERRORS From the data given in Figure 3.5, the probability of an error of any percentage probability can be determined. The general equation is EP = CPs (3.5) where EP, is a certain percentage error and CP, the corresponding numerical factor taken from Figure 3.5. By Equation (3.5), after extracting appropriate multipliers from Figure 3.5, the following are expressions for errors that have a 50%, 90%, and 95% chance of occurring: E50 = 0.6745s (3.6) E90 = 1.6449s (3.7) E95 = 1.9599s (3.8) is the so-called probable error. It establishes The 50 percent error, or E50, limits within which the observations should fall 50% of the time. In other words, an observation has the same chance of coming within these limits as it has of falling outside of them. The 90 and 95 percent errors are commonly used to specify precisions required on surveying (geomatics) projects. Of these, the 95 percent error, too Frequently called the two-sigma error, (2s) is most often specified. As an exam-ple, a particular project may call for the 95 percent error to be less than or equal to a certain value for the work to be acceptable. For the data of Table 3.1, applying-ing and Equations (3.7) and (3.8), the 90 and 95 percent errors are ;3.60 ;4.29 sec , respectively. These errors are shown graphically in Figure 3.3. The so-called three-sigma error (3s) is also often used as a criterion for Rejecting individual observations from sets of data. From Figure 3.5, there is a 99.7% probability that an error will be less than this amount. Thus, within a group is considered to be a misof observations, any value whose residual exceeds 3s take, and either a new observation must be taken or the calculations based on one less value. Machine Translated by Google 3.16 The 50, 90, and 95 Percent Errors 59 The x-axis is asymptote of the normal distribution curve, so the 100 per-cent error cannot be evaluated. This means that no matter what size error is found, a larger one is theoretically possible. ÿ Example 3.1 To clarify definitions and use the equations given in Sections 3.10 through 3.16, Suppose that a line has been observed 10 times using the same equipment and procedures. The results are shown in column (1) of the following table. It is It was assumed that no errors exist, and that the observations have already been made corrected for all systematic errors. Compute the most probable value for the line length, its standard deviation, and errors having 50%, 90%, and 95% probability. Length(ft) (1) N Residual(ft) N (2) 2 (3) 538.57 +0.12 0.0144 538.39 538.37 -0.06 0.0036 -0.08 0.0064 538.39 538.48 -0.06 0.0036 +0.03 538.49 538.33 +0.04 0.0009 0.0016 -0.12 0.0144 538.46 +0.01 0.0001 538.47 +0.02 0.0004 538.55 +0.10 0.0100 © = 5384.50 © = 0.00 ©n2 = 0.0554 Solution By Equation (3.2), M= 5384.50 10 = 538.45 ft By Equation (3.3), the residuals are calculated. These are tabulated in column (2) and their squares listed in column (3). Note that in column (2) the algebraic sum of residuals is zero. (For observations of equal reliability, except for round off, this column should always total zero and thus provide a computational check.) By Equation (3.4), s = ; B©n2n - 1 9 = B 0.0554 = ;0.078 = ;0.08 ft. By Equation (3.6), E50 = ;0.6745s = ;0.6745(0.078) = ;0.05 ft. By Equation (3.7), E95 = ;1.6449(0.078) = ;0.13 ft. By Equation (3.8), E99 = ;1.9599(0.078) = ;0.15 ft. Machine Translated by Google 60 THEORY OF ERRORS IN OBSERVATIONS The following conclusions can be drawn concerning this example. 1. The most probable line length is 538.45 ft. 2. The standard deviation of a single observation is 0.08 ft. Accordingly, The normal expectation is that 68% of the time a recorded length will lie between 538.45 - 0.08 and 538.45 + 0.08 or between 538.37 and 538.53 ft; that is, about seven values should lie within these limits. (Actually seven of them do.) 3. The probable error is(E0.05 Therefore, it can be anticipated 50) ft. Half, or five of the observations, will fall in the interval 538.40 to 538.50 ft. (Four values do.) 4. The 90% error is ;0.13 ft, And thus nine of the observed values can be expected to be within the range of 538.32 and 538.58 ft. 5.The error is 95% ;0.15 ft, so the length can be expected to lie between 538.30 and 538.60, 95% of the time. (Note that all observations indeed are Within the limits of both the 90 and 95 percent errors.) ÿ 3.17 ERROR PROPAGATION It was stated earlier that because all observations contain errors, any quantities computed from them will likewise contain errors. The process of evaluating errors in quantities computed from observed values that contain errors is called error propagation. The propagation of random errors in mathematical formulas can be computed using the general law of the propagation of variances. Typically in sur-veying (geomatics), this formula can be simplified since the observations are usu-ally mathematically independent. For example, let a, b, c, and be observed , n , En, respectively. Also let Z be a quantity Values containing errors Ea, Eb, Ec, Á derived by computation using these observed quantities in a function f, such that Z = f(a, b, c, Á, n) (3.9) , n are independent observations, the error in the Then assuming that a, b, c, Á computed quantity Z is 2 2 2 2 EZ = ;A a 0f 0a Ea b + a 0f0b Eb b + a 0f0c Ecb + a + a 0f 0n En b (3.10) where the terms are the0f>0b, partial0f>0c, derivatives of the 0f>0a, Á, 0f>0n function f with respect to the variables a, b, c, Á , n. In the subsections that follow- low, specific cases of error propagation common in surveying are discussed, and Examples are presented. 3.17.1 Error of a Sum Assume the sum of independently observed observations a, b, c, . . . is Z.The formula for the computed quantity Z is Z=a+b+c+Á Machine Translated by Google 3.17 Error Propagation 61 The partial derivatives of Z with respect to each observed quantity are Substituting these partial derivatives into 0Z>0a = 0Z>0b = 0Z>0c = Á = 1. Equation (3.10), the following formula is obtained, which gives the propagated error in the sum of quantities, each of which contains a different random error: ESum = ; 2E2 a + E2b + E2c +Á (3.11) s , E50, E90, or E95 ), where E represents any specified percentage error (such as and a, b, and c are the separate, independent observations. The error of a sum can be used to explain the rules for addition and sub-traction using significant figures. Recall the addition of 46.7418, 1.03, and 375.0 from Example (a) from Section 2.4. Significant figures indicate that there is Uncertainty in the last digit of each number. Thus, assume estimated errors of ;0.0001, ;0.01, and ;0.1 respectively for each number. The error in the sum of 20.00012 + these three numbers are . The sum of three 0.012 + 0.12 = ;0.1 numbers is 422.7718, which was rounded, using the rules of significant figures, to 422.8. Its precision matches the estimated accuracy produced by the error in the sum of the three numbers. Note how the least accurate number controls the accuracy in the summation of the three values. ÿ Example 3.2 Assume that a line is observed in three sections, with the individual parts equal to (753.81, ;0.012), (1238.40, ;0.028), (1062.95, and ;0.020) respectively. ft, Determine the line's total length and its anticipated standard deviation. Solution Total length = 753.81 + 1238.40 + 1062.95 = 3055.16 ft. By Equation (3.11), ESum = ; 20.0122 + 0.0282 + 0.0202 = 0.036 ft 3.17.2 Error of a Series Sometimes a series of similar quantities, such as the angles within a closed poly-gon, are read with each observation being in error by about the same amount. The total error in the sum of all observed quantities of such a series is called the Error of the series, designated as ESeries. If the same error E in each observation is assumed and Equation (3.11) applied, the series error is ESeries = ; 2E2 + E2 + E2 + Á = ; 2nE2 = ;E2n where E represents the error in each individual observation and n the number of observations. This equation shows that when the same operation is repeated, random Errors tend to balance out and the resulting error of a series is proportional to the square root of the number of observations. This equation has extensive use—for instance, to determine the allowable misclosure error for angles of a traverse, as discussed in Chapter 9. (3.12) Machine Translated by Google 62 THEORY OF ERRORS IN OBSERVATIONS ÿ Example 3.3 Assume that any distance of 100 ft can be taped with an error of 0.02 ft if certain techniques are employed. Determine the error in taping 5000 ft using these skills. Solution Since the number of 100 ft lengths in 5000 ft is 50 then by Equation (3.12) ESeries = ;E2n = ;0.02250 = ;0.14 ft ÿ Example 3.4 A distance of 1000 ft is to be taped with an error of not more than 0.10 ft. Determine how accurately each 100 ft length must be observed to ensure that the error will not exceed the permissible limit. Solution 10, Since by Equation (3.12), 100 ft is ESeries = ;E2n n =and ESeries E = ; ; 2n 0.10 the allowable error E in = ;0.03 ft 210 ÿ Example 3.5 Suppose it is required to tape a length of 2500 ft with an error of not more than ;0.10 ft. How accurately must each tape length be observed? Solution Since 100 ft is again considered the unit length, n = 25, and by Equation (3.12), the allowable error E in 100 ft is E=; 0.10 = ;0.02 ft 225 Analyzing Examples 3.4 and 3.5 shows that the larger the number of possibilities, the greater the chance for errors to cancel out. Machine Translated by Google 3.17 Error Propagation 63 3.17.3 Error of a Product Ea Eb The equation for propagated AB, where and are the respective errors in A and B, is Eprod = ; 2A2E2 _ b + B2 E2a (3.13) The physical significance of the error propagation formula for a product is illustrated in Figure 3.6, where A and B are shown to be observed sides of a rec- Ea Eb tangular parcel of land with errors and respectively. The product AB is the 2A2 E2 b represents either of the longer parcel area. In Equation (3.13), = AEb b or (horizontal) crosshatched bars and is the error caused by either +Eb -E 2B2 E2 The term = BEa a is represented by the shorter (vertical) crosshatched . bars, which is the error resulting from either +Ea -E a or . ÿ Example 3.6 For the rectangular lot illustrated in Figure 3.6, observations of sides A and B with their 95% errors are (252.46, ;0.053) and (605.08, ;0.072) ft, respectively. Calculate the parcel area and the expected 95% error in the area. Solution Area = 252.46 * 605.08 = 152,760 ft2 By Equation (3.13), E95 = ; 2(252.46)2 (0.072)2 + (605.08)2 (0.053)2 = ;36.9 ft2 Example 3.6 can also be used to demonstrate the validity of one of the rules of significant figures in computation. The computed area is actually 152,758.4968 ft2 . However, the rule for significant figures in multiplication (see Section 2.4) states that there cannot be more significant figures in the answer A -Ea +Ea B -Eb +Eb Figure 3.6 Error of area. Machine Translated by Google 64 THEORY OF ERRORS IN OBSERVATIONS than in any of the individual factors used. Accordingly, the area should be rounded off to 152,760 (five significant figures). From Equation (3.13), with an 2 the answer could be error of or;36.9 fromft 152,721.6 , to 152,758.4968 ; 36.9, 152,795.4 ft2 . Thus, the fifth digit in the answer is seen to be questionable, and Hence the number of significant figures specified by the rule is verified. 3.17.4 Error of the Mean Equation (3.2) stated that the most probable value of a group of repeated observations of equal weight is the arithmetic mean. Since the meaning is computed from Individual observed values, each of which contains an error, the meaning is also sub-ject to error. By applying Equation (3.12), it is possible to find the error for the sum of a series of observations where each one has the same error. Since the sum Divided by the number of observations gives the mean, the error of the mean is found by the relationship Eseries Em = n Substituting Equation (3.12) for Eseries Em = E2n n = E 2n (3.14) the correwhere E is the specified percentage error of a single observation, Em sponding percentage error of the mean, and n the number of observations. The error of the mean at any percentage probability can be determined and Applied to all criteria that have been developed. For example, the standard devia- (E68)m sm tion of the meaning, or is s (E68)m = sm = 2n = ;A©n2 n(n - 1) (3.15a) and the 90 and 95 percent errors of the mean are (E90)m (E95)m E90 = 2n = ;1.6449A©n2 n(n - 1) (3.15b) = ;1.9599A ©n2 n(n - 1). (3.15c) E95 = 2n These equations show that the error of the mean varies inversely as the square root of the number of repetitions. Thus, to double the accuracy—that is, to Reduce the error by one half—four times as many observations must be made. Machine Translated by Google 3.19 Conditional Adjustment of Observations 65 ÿ Example 3.7 Calculate the standard deviation of the mean and the 90% error of the mean for the observations of Example 3.1. Solution By Equation (3.15a), sm = s 0.078 2n 210 = ;0.025 ft = ; Also, by Equation (3.15b), (E90)m = ;1.6449(0.025) = ;0.041 ft These values show the error limits of 68% and 90% probability for the line's length. It can be said that the true line length has a 68% chance of being within ;0.025 of the mean, and a 90% likelihood of falling not farther than ;0.041 ft from the mean. ÿ 3.18 APPLICATIONS The preceding example problems show that the equations of error probability are applied in two ways: 1. To analyze observations already made, for comparison with other results or with specification requirements. 2. To establish procedures and specifications in order that the required results will be obtained. The application of the various error probability equations must be tempered with judgment and caution. Recall that they are based on the assumption that the errors conform to a smooth and continuous normal distribution curve, which in turn is based on the assumption of a large number of observations. Frequently in surveying only a few observations—often from two to eight—are taken. If these conform to a normal distribution, then the answer obtained using probability equations will be reliable; if they do not, the conclusions could be misleading. In the absence of knowledge to the contrary, however, an assumption that the errors are normally distributed is still the best available. ÿ 3.19 CONDITIONAL ADJUSTMENT OF OBSERVATIONS In Section 3.3, it was emphasized that the true value of any observed quantity is never known. However, in some types of problems, the sum of several observations must equal a fixed value; For example, the sum of the three angles in a plane triangle has a total of 180°. In practice, therefore, the observed angles are adjusted to make them add to the required amount. Correspondingly, distances —either horizontal or vertical—must often be adjusted to meet certain conditional requirements. The methods used will be explained in later chapters, where the operations are taken up in detail. Machine Translated by Google 66 THEORY OF ERRORS IN OBSERVATIONS ÿ 3.20 WEIGHTS OF OBSERVATIONS It is evident that some observations are more precise than others because of bet-ter equipment, improved techniques, and superior field conditions. In making adjustments, it is consequently desirable to assign relative weights to the 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 (held closer to its observed value) in an adjustment than an observa-tion of lower precision. From this reasoning, it is deduced that weights of obser-vations should bear an inverse relationship to precision. In fact, it can be shown that relative weights are inversely proportional to variances, or 1 Wa r (3.16) s2a s2a . where isWthe a weight of an observation a, which has a variance of Thus, the The higher the precision (the smaller the variance), the larger should be the relative weight of the observed value being adjusted. In some cases, variances are unknown Originally, weights must be assigned to observed values based on estimates of their relative precision. If a quantity is observed repeatedly and the individual Observations have varying weights, the weighted mean can be calculated from the expression ©WM MW = where MW is the weighted mean, ©WM their corresponding observations, and ©W ©W (3.17) the sum of the individual weights times the sum of the weights. ÿ Example 3.8 Suppose four observations of a distance are recorded as 482.16, 482.17, 482.20, and 482.18 and given weights of 1, 2, 2, and 4, respectively, by the surveyor. Determine the weighted meaning. Solution By Equation (3.17) 482.16 + 482.17(2) + 482.20(2) + 482.14(4) MW = = 482.16 ft 1+2+2+4 In computing adjustments involving unequally weighted observations, correc-tions applied to observed values should be made inversely proportional to the relative weights. Machine Translated by Google 3.21 Least-Squares Adjustment 67 ÿ Example 3.9 Assume the observed angles of a certain plane triangle, and their relative weights, are A = 49°51¿15–, W =a1; B = 60°32¿08–, Wb = 2; and C = 69°36¿33–, Wc = 3. Compute the weighted mean of the angles. Solution The sum of the three angles is computed first and found to be less than4the Required geometrical condition of exactly 180°. The angles are therefore adjusted in inverse proportion to their relative weights, as illustrated in the accompanying tabulation. Angle C with the greatest weight (3) gets the smallest correction, 2x; B receives 3x; and A, 6x. Observed Angle Wt Correction Numerical Rounded Adjusted Corr. Corr. Angle A 49°51¿15– 1 6x +2.18– +2– 49°51¿17– B 60°32¿08– 2 3x +1.09– +1– 60°32¿09– C 69°36¿33– 3 2x +0.73– +1– 69°36¿34– Sum 179°59¿56– g = 6 11x +4.00– +4– 180°00¿00– 11x = 4– and x = +0.36– It must be emphasized again that adjustment calculations are based on the theory of probability are valid only if systematic errors and employing proper procedure-dures , equipment, and calculations eliminate errors. ÿ 3.21 LEAST-SQUARES ADJUSTMENT As explained in Section 3.19, most surveying observations must conform to cer-tain geometrical conditions. The amounts by which they fail to meet these condi-tions are called misclosures, and they indicate the presence of random errors. In 4–. procedures are used to Example 3.9, for example, the misclosure was Various distribute these misclosure errors to produce mathematically perfect geometrical conditions. Some simply apply corrections of the same size to all observed values, where each correction equals the total misclosure (with its algebraic sign changed), divided by the number of observations. Others introduce corrections in proportion to assigned weights. Still others employ rules of thumb, for example, the “compass rule” described in Chapter 10 for adjusting closed traverses. Because random errors in surveying conform to the mathematical laws of probability and are “normally distributed,” the most appropriate adjustment pro-cedure should be based upon these laws. Least squares is such a method. It is not a new procedure, having been applied by the German mathematician Karl Gauss as early as the latter part of the 18th century. However, until the advent of com-puters, it was only used sparingly because of the lengthy calculations involved. Machine Translated by Google 68 THEORY OF ERRORS IN OBSERVATIONS Least squares are suitable for adjusting any of the basic types of surveying observations described in Section 2.1, and are applicable to all of the commonly used surveying procedures. The method enforces the condition that the sum of the weights of the observations times their corresponding squared residuals is minimized. This fundamental condition, which is developed from the equation for the normal error distribution curve, provides the most probable values for the adjusted quantities. In addition, it also (a) enables the computation of precisions of the adjusted values, (b) reveals the presence of errors so steps can be taken to eliminate them, and (c) makes possible the optimum design of survey procedures in the office before going to the field to take observations. The basic assumptions that underlie least-squares theory are as follows: (1) mistakes and systematic errors have been eliminated so only random errors remain; (2) the number of observations being adjusted is large; and (3) the frequency distribution of errors is normal. Although these assumptions are not always met, the least-squares adjustment method still provides the most rigorous error treatment available, and hence it has become very popular and important in modern surveying. A more detailed discussion of the subject is presented in Chapter 16. ÿ 3.22 USING SOFTWARE Computations such as those in Table 3.1 can be long and tedious. Fortunately, spreadsheet software often has the ability to calculate the mean and stan-dard deviation of a group of observations. For example, in Microsoft Excel®, the mean of a set of observations can be determined using the average() function and the standard deviation can be determined using the stdev() function. Likewise, histograms of data can also be plotted once the data is organized into classes. The reader can download all of the Excel files for this book by downloading the file Excel Spreadsheets.zip from the companion website for this book at http:// www.pearsonhighered.com/ghilani. The spreadsheet c3.xls demonstrates the use of the functions mentioned previously and also demonstrates the use of a spreadsheet to solve the example problems in this chapter. Also on the companion website for this book is the software STATS. This software can read a text file of data and compute the statistics demonstrated in this chapter. Furthermore, STATS will histogram the data using a user-specified number of class intervals. The help file that accompanies this software describes the file format for the data and the use of the software. For those having the software Mathcad® version 14.0 or higher, an accompanying e-book is available on the companion website. This e-book is in the file Mathcad files.zip on the companion website. If this book is decompressed in the Mathcad subdirectory handbook, the e-book will be available in the Mathcad help system. This e-book can also be accessed by selecting the file elemsurv.hbk in your Windows directory and has a worksheet that demonstrates the examples presented in this chapter. For those who do have Mathcad version 14.0 or higher, a set of hypertext markup language (html) files of the e-book are available on the companion website. These files can be accessed by opening the file index.html in your browser. Machine Translated by Google Problems 69 PROBLEMS Asterisks (*) indicate problems that have partial answers given in Appendix G. 3.1 Explain the difference between direct and indirect measurements in surveying. Give two examples of each. 3.2 Define the term systematic error, and give two surveying examples of a systematic error. 3.3 Define the term random error, and give two surveying examples of a random error. 3.4 Explain the difference between accuracy and precision. 3.5 Discuss what is meant by the precision of an observation. A distance AB is observed repeatedly using the same equipment and procedures, and the results, in meters, are listed in Problems 3.6 through 3.10. Calculate (a) the line's most probable length, (b) the standard deviation, and (c) the standard deviation of the mean for each set of results. 3.6* 65.401, 65.400, 65.402, 65.396, 65.406, 65.401, 65.396, 65.401, 65.405, and 65.404 3.7 Same as Problem 3.6, but discard one observation, 65.396. 3.8 Same as Problem 3.6, but discard two observations, 65.396 and 65.406. 3.9 Same as Problem 3.6, but include two additional observations, 65.398 and 65.408. 3.10 Same as Problem 3.6, but include three additional observations, 65.398, 65.408, and 65.406. In Problems 3.11 through 3.14, determine the range within which observations should fall (a) 90% of the time and (b) 95% of the time. List the percentage of values that actually fall within these ranges. 3.11* For the data of Problem 3.6. 3.12 For the data of Problem 3.7. 3.13 For the data of Problem 3.8. 3.14 For the data of Problem 3.9. In Problems 3.15 through 3.17, an angle is observed repeatedly using the same equipment and procedures. Calculate (a) the angle's most probable value, (b) the stan-dard deviation, and (c) the standard deviation of the mean. 3.15* 23°30¿00–, 23°29¿40–, 23°30¿15–, and 23°29¿50– . 3.16 Same as Problem 3.15, but with three additional observations, 23°29¿55–, 23°30¿05–, and 23°30¿20– . 3.17 Same as Problem 3.16, but with two additional observations, 23°29¿55– . 23°30¿05– and 3.18* A field party is capable of making taping observations with a standard deviation of 0.010 ft per 100 ft tape length. What standard deviation would be expected in a distance of 200 ft taped by this party? 3.19 Repeat Problem 3.18, except that the standard deviation per 30-m tape length is ;0.003 m and a distance of 120 m is taped. What is the expected 95% error in 120 m? 3.20 A distance of 200 ft must be taped in a manner to ensure a standard deviation smaller than 0.04 ft. What must be the standard deviation per 100 ft tape length to achieve the desired precision? 3.21 Lines of levels were run requiring instrument setups. If the rod reading for each backsight and foresight s, has a standard deviation what is the standard deviation in each of the following level lines? (a) n = 26, s = ;0.010 ft (b) n = 36, s = ;3 mm Machine Translated by Google 70 THEORY OF ERRORS IN OBSERVATIONS 3.22 A line AC was observed in 2 sections AB and BC, with lengths and standard deviations listed below. What is the total length of AC, and its standard deviation? *(a) AB = 60.00 ; 0.015 ft; BC = 86.13 ; 0.018 ft (b) AB = 60,000 ; 0.008m; 35.413; 0.005 m 3.23 Line AD is observed in three sections, AB, BC, and CD, with lengths and standard deviations as listed below. What is the total length AD and its standard deviation? (a) AB = 572.12 ; 0.02 ft; BC = 1074.38 ; 0.03 ft; CD = 1542.78 ; 0.05 ft (b) AB = 932.965 ; 0.009 m; BC = 945.030 m ; 0.010 m; CD = 652.250 m ; 0.008 m 3.24 A distance AB was observed four times as 236.39, 236.40, 236.36, and 236.38 ft. The The observations were given weights of 2, 1, 3, and 2, respectively, by the observer. *(a) Calculate the weighted mean for distance AB. (b) What difference results if Later judgment revises the weights to 2, 1, 2, and 3, respectively? 3.25 Determine the weighted mean for the following angles: (a) 89°42¿45–, wt 2; 89°42¿42–, wt 1; 89°42¿44–, wt 3 (b) 36°58¿32– ; 3–; 36°58¿28– ; 2–; 36°58¿26– ; 3–; 36°58¿30– ; 13.26 Specifications for observing angles of an n-sided polygon limit the total angular mis-closure to E. How accurately each angle must be observed for the following values of n and E? (a) n = 10, E = 8– (b) n = 6, E = 14– 3.27 What is the area of a rectangular field and its estimated error for the following recorded values: *(a) 243.89 ; 0.05 ft, by 208.65 ; 0.04 ft (b) 725.33; 0.08 ft by 664.21 ; 0.06 ft (c) 128.526 ; 0.005 m, by 180.403 ; 0.007 m 3.28 Adjust the angles of triangle ABC for the following angular values and weights: *(a) A = 49°24¿22–, wt 2; B = 39°02¿16–, wt 1; C = 91°33¿00–, wt 3 (b) A = 80°14¿04–, wt 2; B = 38°37¿47–, wt 1; C = 61°07¿58–, wt 3 3.29 Determine relative weights and perform a weighted adjustment (to the nearest sec-ond) for angles A, B, and C of a plane triangle, given the following four observations for each angle: Angle A Angle B Angle C 38°47¿58– 71°22¿26– 69°50¿04– 38°47¿44– 71°22¿22– 69°50¿16– 38°48¿12– 71°22¿12– 69°50¿30– 38°48¿02– 71°22¿12– 69°50¿10– 3.30 A line of levels was run from benchmarks A to B, B to C, and C to D. The elevation The differences obtained between benchmarks, with their standard deviations, are listed below. What is the difference in elevation from benchmark A to D and the standard deviation of that elevation difference? (a) BM A to BM B = +34.65 ; 0.10 ft; BM B to BM C = -48.23 ; 0.08 ft; BM C to BM D = -54.90 ; 0.09 ft (b) BM A to BM B = +27.823 ; 0.015 m; BM B to BM C = +15.620 ; 0.008m; and BM C to BM D = +33.210 ; 0.011 m (c) BM A to BM B = -32.688 ; 0.015 m; BM B to BM C = +5.349 ; 0.022 m; and BM C to BM D = -15.608 ; 0.006 m 3.31 Create a computational program that solves Problem 3.9. 3.32 Create a computational program that solves Problem 3.17. 3.33 Create a computational program that solves Problem 3.29. and Machine Translated by Google Bibliography 71 BIBLIOGRAPHY Alder, K. 2002. The Measure of All Things—The Seven-Year Odyssey and Hidden Error that Transformed the World. New York: The Free Press. Bell, J. 2001. “Hands On: TDS for Windows CE On the Ranger.” Professional Surveyor 21 (No. 1): 33. Buckner, R.B