Statistical Frequency Analysis 5.1 INTRODUCTION Univariate frequency analysis is widely used for analyzing hydrologic data, including rainfall characteristics, peak discharge series, and low flow records. It is primarily used to estimate exceedance probabilities and variable magnitudes. A basic assumption of frequency analysis is that the vector of data was measured from a temporally or spatially homogeneous system. If measured data are significantly nonhomogeneous, the estimated probabilities or magnitudes will be inaccurate. Thus, changes such as climate or watershed alterations render the data unfit for frequency analysis and other modeling methods. If changes to the physical processes that influence the data are suspected, the data vector should be subjected to statistical tests to decide whether the nonsta- tionarity is significant. If the change had a significant effect on the measured data, it may be necessary to adjust the data before subjecting it to frequency analysis. Thus, the detection of the effects of change, the identification of the nature of any change detected, and the appropriate adjustment of the data are prerequisite steps required before a frequency model can be used to make probability or magnitude estimates. 5.2 FREQUENCY ANALYSIS AND SYNTHESIS Design problems such as the delineation of flood profiles require estimates of discharge rates. A number of methods of estimating peak discharge rates are available. They fall into two basic groups, one used at sites where gaged stream-flow records are available (gaged) and the other at sites where such records are not available (ungaged). Statistical frequency analysis is the most common procedure for the analysis of flood data at a gaged location. It is a general procedure that can be applied to any type of data. Because it is so widely used with flood data, the method is sometimes designated flood frequency analysis . However, statistical frequency analysis can also be applied to other hydrologic variables such as rainfall data for the development of intensity-duration-frequency curves and low-flow discharges for use in water quality control. The variable could also be the mean annual rainfall, the peak discharge, the 7-day low flow, or a water quality parameter. Therefore, the topic will be treated in both general and specific terms. 5 L1600_Frame_C05 Page 77 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC 5.2.1 P OPULATION VERSUS S AMPLE In frequency modeling, it is important to distinguish between the population and the sample. Frequency modeling is a statistical method that deals with a single random variable and thus is classified as a univariate method. The goal of univariate predic- tion is to make estimates of probabilities or magnitudes of random variables. A first step is to identify the population. The objective of univariate data analysis is to use sample information to determine the appropriate population density function, with the probability density function (PDF) being the univariate model from which proba- bility statements can be made. The input requirements for frequency modeling include a data series and a probability distribution assumed to describe the occurrence of the random variable. The data series could include the largest instantaneous peak discharge to occur each year of the record. The probability distribution could be the normal distribution. Analysis is the process of using the sample information to estimate the population. The population consists of a mathematical model that is a function of one or more parameters. For example, the normal distribution is a function of two parameters: the mean µ and standard deviation σ . In addition to identifying the correct PDF, it is necessary to quantify the parameters of the PDF. The population consists of both the probability distribution function and the parameters. A frequently used procedure called the method of moments equates characteris- tics of the sample (e.g., sample moments) to characteristics of the population (e.g., population parameters). It is important to note that estimates of probability and magnitudes are made using the assumed population and not the data sample; the sample is used only in identifying and verifying the population. 5.2.2 A NALYSIS VERSUS S YNTHESIS As with many hydrologic methods that have statistical bases, the terms analysis and synthesis apply to the statistical frequency method. Frequency analysis is “breaking down” data in a way that leads to a mathematical or graphical model of the rela- tionship between flood magnitude and its probability of occurrence. Conversely, synthesis refers to the estimation of (1) a value of the random variable X for some selected exceedance probability or (2) the exceedance probability for a selected value of the random variable X . In other words, analysis is the derivation of a model that can represent the relation between a random variable and its likelihood of occurrence, while synthesis is using the resulting relation for purposes of estimation. It is important to point out that frequency analysis may actually be part of a more elaborate problem of synthesis. Specifically, separate frequency analyses can be performed at a large number of sites within a region and the value of the random variable X for a selected exceedance probability determined for each site; these values can then be used to develop a regression model using the random variable X as the criterion or dependent variable. As an example, regression equations that relate peak discharges of a selected exceedance probability for a number of sites to water- shed characteristics are widely used in hydrologic design. This process is called regionalization . These equations are derived by (1) making a frequency analysis of annual maximum discharges at a number ( n ) of stream gage stations in a region; L1600_Frame_C05 Page 78 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC (2) selecting the value of the peak discharge from each of the n frequency curves for a selected exceedance probability, say the l00-year flood; and (3) developing the regression equation relating the n values of peak discharge to watershed char- acteristics for the same n watersheds. 5.2.3 P ROBABILITY P APER Frequency analysis is a common task in hydrologic studies. A frequency analysis usually produces a graph of the value of a single hydrologic variable versus the probability of its occurrence. The computed graph represents the best estimate of the statistical population from which the sample of data was drawn. Since frequency analyses are often presented graphically, a special type of graph paper, which is called probability paper, is required. The paper has two axes. The ordinate is used to plot the value of the random variable, that is, the magnitude, and the probability of its occurrence is given on the abscissa. The probability scale will vary depending on the probability distribution used. In hydrology, the normal and Gumbel extreme-value distributions are the two PDFs used most frequently to define the probability scale. Figure 5.1 is on normal probability paper. The probability scale represents the cumulative normal distribution. The scale at the top of the graph is the exceedance probability, that is, the probability that the random variable will be equaled or exceeded in one time period. It varies from 99.99% to 0.01%. The lower scale is the nonexceedance probability, which is the probability that the correspond- ing value of the random variable will not be exceeded in any one time period. This scale extends from 0.01% to 99.99%. The ordinate of probability paper is used for FIGURE 5.1 Frequency curve for a normal population with µ = 5 and σ = 1. L1600_Frame_C05 Page 79 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC the random variable, such as peak discharge. The example shown in Figure 5.1 has an arithmetic scale. Lognormal probability paper is also available, with the scale for the random variable in logarithmic form. Gumbel and log-Gumbel papers can also be obtained and used to describe the probabilistic behavior of random variables that follow these probability distributions. A frequency curve provides a probabilistic description of the likelihood of occurrence or nonoccurrence of a variable. Figure 5.1 shows a frequency curve, with the value of the random variable X versus its probability of occurrence. The upper probability scale gives the probability that X will be exceeded in one time period, while the lower probability scale gives the probability that X will not be exceeded. For the frequency curve of Figure 5.1, the probability that X will be greater than 7 in one time period is 0.023 and the probability that X will not be greater than 7 in one time period is 0.977. Although a unique probability plotting paper could be developed for each prob- ability distribution, papers for the normal and extreme value distributions are the most frequently used. The probability paper is presented as a cumulative distribution function. If the sample of data is from the distribution function used to scale the probability paper, the data will follow the pattern of the population line when properly plotted on the paper. If the data do not follow the population line, then (1) the sample is from a different population or (2) sampling variation produced a nonrepresentative sample. In most cases, the former reason is assumed, especially when the sample size is reasonably large. 5.2.4 M ATHEMATICAL M ODEL As an alternative to a graphical solution using probability paper, a frequency analysis may be conducted using a mathematical model. A model that is commonly used in hydrology for normal, lognormal, and log-Pearson Type III analyses has the form (5.1) in which X is the value of the random variable having mean and standard deviation S , and K is a frequency factor. Depending on the underlying population, the specific value of K reflects the probability of occurrence of the value X . Equation 5.1 can be rearranged to solve for K when X , , and S are known and an estimate of the probability of X occurring is necessary: (5.2) In summary, Equation 5.1 is used when the probability is known and an estimation of the magnitude is needed, while Equation 5.2 is used when the magnitude is known and the probability is needed. XXKS=+ X X K XX S = − L1600_Frame_C05 Page 80 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC 5.2.5 P ROCEDURE In a broad sense, frequency analysis can be divided into two phases: deriving the population curve and plotting the data to evaluate the goodness of fit. The following procedure is often used to derive the frequency curve to represent the population: 1. Hypothesize the underlying density function. 2. Obtain a sample and compute the sample moments. 3. Equate the sample moments and the parameters of the proposed density function. 4. Construct a frequency curve that represents the underlying population. This procedure is referred to as method-of-moments estimation because the sample moments are used to provide numerical values for the parameters of the assumed population. The computed frequency curve representing the population can then be used to estimate magnitudes for a given return period or probabilities for specified values of the random variable. Both the graphical frequency curve and the mathe- matical model of Equation 5.1 are the population. It is important to recognize that it is not necessary to plot the data points in order to make probability statements about the random variable. While the four steps listed above lead to an estimate of the population frequency curve, the data should be plotted to ensure that the population curve is a good representation of the data. The plotting of the data is a somewhat separate part of a frequency analysis; its purpose is to assess the quality of the fit rather than act as a part of the estimation process. 5.2.6 S AMPLE M OMENTS For the random variable X , the sample mean ( ), standard deviation ( S ), and stan- dardized skew ( g ) are, respectively, computed by: (5.3a) (5.3b) (5.3c) X X n X i i n = = ∑ 1 1 S n XX i i n = − − = ∑ 1 1 2 1 05 () . g nXX nnS i i n = − −− = ∑ () ()( ) 3 1 3 12 L1600_Frame_C05 Page 81 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC For use in frequency analyses where the skew is used, Equation 5.3c represents a standardized value of the skew. Equations 5.3 can also be used when the data are transformed by taking the logarithms. In this case, the log transformation should be done before computing the moments. 5.2.7 P LOTTING P OSITION F ORMULAS It is important to note that it is not necessary to plot the data before probability statements can be made using the frequency curve; however, the data should be plotted to determine how well they agree with the fitted curve of the assumed population. A rank-order method is used to plot the data. This involves ordering the data from the largest event to the smallest event, assigning a rank of 1 to the largest event and a rank of n to the smallest event, and using the rank ( i ) of the event to obtain a probability plotting position; numerous plotting position formulas are available. Bulletin l7B (Interagency Advisory Committee on Water Data, 1982) provides the following generalized equation for computing plotting position prob- abilities: (5.4) where a and b are constants that depend on the probability distribution. An example is a = b = 0 for the uniform distribution. Numerous formulas have been proposed, including the following: Weibull: (5.5a) Hazen: (5.5b) Cunnane: (5.5c) in which i is the rank of the event, n is the sample size, and p i values give the exceedance probabilities for an event with rank i. The data are plotted by placing a point for each value of the random variable at the intersection of the value of the random variable and the value of the exceedance probability at the top of the graph. The plotted data should approximate the population line if the assumed population model is a reasonable assumption. The various plotting position formulas provide different probability estimates, especially in the tails of the distributions. The fol- lowing summary shows computed probabilities for each rank for a sample of nine using the plotting position formulas of Equations 5.5. P ia nan i = − −−+1 P i n i = +1 P i n i n i = − = −21 2 05. P i n i = − + 04 02 . . L1600_Frame_C05 Page 82 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC The Hazen formula gives smaller probabilities for all ranks than the Weibull and Cunnane formulas. The probabilities for the Cunnane formula are more dispersed than either of the others. For a sample size of 99, the same trends exist as for n = 9. 5.2.8 R ETURN P ERIOD The concept of return period is used to describe the likelihood of flood magnitudes. The return period is the reciprocal of the exceedance probability, that is, p = 1/ T . Just as a 25-year rainfall has a probability of 0.04 of occurring in any one year, a 25-year flood has a probability of 0.04 of occurring in any one year. It is incorrect to believe that a 25-year event will not occur again for another 25 years. Two 25- year events can occur in consecutive years. Then again, a period of 100 years may pass before a second 25-year event occurs. Does a 25-year rainfall cause a 25-year flood magnitude? Some hydrologic models make this assumption; however, it is unlikely to be the case in actuality. It is a reasonable assumption for modeling because models are based on the average of expectation or on-the-average behavior. In actuality, a 25-year flood magnitude will not occur if a 25-year rainfall occurs on a dry watershed. Similarly, a 50-year flood could occur from a 25-year rainfall if the watershed was saturated. Modeling often assumes that a T -year rainfall on a watershed that exists in a T -year hydrologic condition will produce a T -year flood. 5.3 POPULATION MODELS Step 1 of the frequency analysis procedure indicates that it is necessary to select a model to represent the population. Any probability distribution can serve as the model, but the lognormal and log-Pearson Type III distributions are the most widely used in hydrologic analysis. They are introduced subsequent sections, along with the normal distribution or basic model. n = 9 n = 99 Rank p w p h p c Rank p w p h p c 1 0.1 0.05 0.065 1 0.01 0.005 0.006 2 0.2 0.15 0.174 2 0.02 0.015 0.016 3 0.3 0.25 0.283 . 4 0.4 0.35 0.391 . 5 0.5 0.45 0.500 . 6 0.6 0.55 0.609 98 0.98 0.985 0.984 7 0.7 0.65 0.717 99 0.99 0.995 0.994 8 0.8 0.75 0.826 9 0.9 0.85 0.935 L1600_Frame_C05 Page 83 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC 5.3.1 NORMAL DISTRIBUTION Commercially available normal probability paper is commonly used in hydrology. Following the general procedure outlined above, the specific steps used to develop a curve for a normal population are as follows: 1. Assume that the random variable has a normal distribution with population parameters µ and σ . 2. Compute the sample moments and S (the skew is not needed). 3. For normal distribution, the parameters and sample moments are related by µ = and σ = S. 4. A curve is fitted as a straight line with plotted at an exceedance probability of 0.8413 and at an exceedance probability of 0.1587. The frequency curve of Figure 5.1 is an example for a normal distribution with a mean of 5 and a standard deviation of 1. It is important to note that the curve passes through the two points: and . It also passes through the point defined by the mean and a probability of 0.5. Two other points that could be used are and . Using the points farther removed from the mean has the advantage that inaccuracies in the line drawn to represent the population will be smaller than when using more interior points. The sample values should then be plotted (see Section 5.2.7) to decide whether the measured values closely approximate the population. If the data provide a reasonable fit to the line, one can assume that the underlying population is the normal distribution and the sample mean and standard deviation are reasonable estimates of the location and scale parameters, respectively. A poor fit indicates that the normal distribution is not appropriate, that the sample statistics are not good estimators of the population parameters, or both. When using a frequency curve, it is common to discuss the likelihood of events in terms of exceedance frequency, exceedance probability, or the return period (T) related to the exceedance probability (p) by p = l/T, or T = l/p. Thus, an event with an exceedance probability of 0.01 should be expected to occur 1 time in 100. In many cases, a time unit is attached to the return period. For example, if the data represent annual floods at a location, the basic time unit is 1 year. The return period for an event with an exceedance probability of 0.01 would be the l00-year event (i.e., T = 1/0.01 = 100); similarly, the 25-year event has an exceedance probability of 0.04 (i.e., p = 1/25 = 0.04). It is important to emphasize that two T-year events will not necessarily occur exactly T years apart. They can occur in successive years or may be spaced three times T years apart. On average, the events will be spaced T years apart. Thus, in a long period, say 10,000 years, we would expect 10,000/T events to occur. In any single 10,000-year period, we may observe more or fewer occurrences than the mean (10,000/T). Estimation with normal frequency curve — For normal distribution, estimation may involve finding a probability corresponding to a specified value of the random variable or finding the value of the random variable for a given probability. Both problems can be solved using graphical analysis or the mathematical models of X X ()XS− ()XS+ (, .)XS− 0 8413 (, .)XS+ 0 1587 (,.)XS+ 2 0 0228 (, . ) XS− 2 0 9772 L1600_Frame_C05 Page 84 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC Equations 5.1 and 5.2. A graphical analysis estimation involves simply entering the probability and finding the corresponding value of the random variable or entering the value of the random variable and finding the corresponding exceedance proba- bility. In both cases, the fitted line (population) is used. The accuracy of the estimated value will be influenced by the accuracy used in drawing the line or graph. Example 5.1 Figure 5.2 shows a frequency histogram for the data in Table 5.1. The sample consists of 58 annual maximum instantaneous discharges, with a mean of 8620 ft 3 /sec, a standard deviation of 4128 ft 3 /sec, and a standardized skew of 1.14. In spite of the large skew, the normal frequency curve was fitted using the procedure of the pre- ceding section. Figure 5.3 shows the cumulative normal distribution using the sample mean and the standard deviation as estimates of the location and scale parameters. The population line was drawn by plotting X + S = 12,748 at p = 15.87% and – S = 4492 at p = 84.13%, using the upper scale for the probabilities. The data were plotted using the Weibull plotting position formula (Equation 5.5a). The data do not provide a reasonable fit to the population; they show a significant skew with an FIGURE 5.2 Frequency histograms of the annual maximum flood series (solid line) and logarithms (dashed line) based on mean (and for logarithms) and standard deviations (S x and for logarithms S y ): Piscataquis River near Dover-Foxcroft, Maine. X L1600_Frame_C05 Page 85 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC TABLE 5.1 Frequency Analysis of Peak Discharge Data: Piscataquis River Rank Weibull Probability Random Variable Logarithm of Variable 1 0.0169 21500 4.332438 2 0.0339 19300 4.285557 3 0.0508 17400 4.240549 4 0.0678 17400 4.240549 5 0.0847 15200 4.181844 6 0.1017 14600 4.164353 7 0.1186 13700 4.136721 8 0.1356 13500 4.130334 9 0.1525 13300 4.123852 10 0.1695 13200 4.120574 11 0.1864 12900 4.110590 12 0.2034 11600 4.064458 13 0.2203 11100 4.045323 14 0.2373 10400 4.017034 15 0.2542 10400 4.017034 16 0.2712 10100 4.004322 17 0.2881 9640 3.984077 18 0.3051 9560 3.980458 19 0.3220 9310 3.968950 20 0.3390 8850 3.946943 21 0.3559 8690 3.939020 22 0.3729 8600 3.934499 23 0.3898 8350 3.921686 24 0.4068 8110 3.909021 25 0.4237 8040 3.905256 26 0.4407 8040 3.905256 27 0.4576 8040 3.905256 28 0.4746 8040 3.905256 29 0.4915 7780 3.890980 30 0.5085 7600 3.880814 31 0.5254 7420 3.870404 32 0.5424 7380 3.868056 33 0.5593 7190 3.856729 34 0.5763 7190 3.856729 35 0.5932 7130 3.853090 36 0.6102 6970 3.843233 37 0.6271 6930 3.840733 38 0.6441 6870 3.836957 39 0.6610 6750 3.829304 40 0.6780 6350 3.802774 41 0.6949 6240 3.795185 L1600_Frame_C05 Page 86 Friday, September 20, 2002 10:12 AM © 2003 by CRC Press LLC [...]... 1931 1932 1933 1934 19 35 1936 1937 1938 1939 1940 1941 1942 1943 1944 19 45 21 21 22 22 23 23 24 24 25 25 26 27 29 30 32 33 1870 153 0 1120 1 850 4890 2280 1700 2470 50 10 2480 1280 2080 2320 4480 1860 2220 1946 1947 1948 1949 1 950 1 951 1 952 1 953 1 954 1 955 1 956 1 957 1 958 1 959 1960 1961 35 37 39 41 43 45 45 45 45 45 45 45 45 45 45 45 1600 3810 2670 758 1630 1620 3811 3140 2140 1980 455 0 3090 4830 3170 1710... (D) T = 2-yr f (a) (b) Rs T = 10-yr T = 100-yr f f Rs Rs U (%) 10 20 30 40 50 1.22 1.47 1.72 1.98 2.27 0.0 25 0.0 25 0.026 0.029 1.13 1.28 1. 45 1.63 1.81 0.0 15 0.017 0.018 0.018 1.08 1.19 1.31 1.44 1 .56 0.011 0.012 0.013 0.012 D (%) 10 20 30 40 50 1. 35 1. 75 2. 35 3.20 4.20 0.040 0.060 0.0 85 0.100 1.18 1.40 1.80 2.30 3.22 0.022 0.040 0. 050 0.092 1. 15 1. 25 1 .50 2.00 2 .55 0.010 0.0 25 0. 050 0. 055 5. 4.2 METHOD... (ft3/sec) (5) log10Qa (6) Qa (ft3/sec) (7) 0.99 −2.6 857 2 2.7386 54 8 2.8006 632 0.90 −1.32309 2.9989 998 3.0439 1106 0.70 −0. 458 12 3.1642 1 459 3.1983 157 9 0 .50 0.08302 3.2676 1 852 3.2949 1972 0.0 65 0.20 0. 856 53 3.4 153 2602 3.4329 2710 0.041 0.10 1.21618 3.4840 3048 3.4971 3142 0.031 0.04 1 .56 740 3 .55 11 355 7 3 .55 98 3629 0.020 0.02 1.77716 3 .59 12 3901 3 .59 73 3 956 0.014 0.01 1. 954 72 3.6 251 4218 3.6290 4 256 0.009... 4700 4380 51 90 3960 56 00 4670 7080 4640 53 6 6680 8360 18700 52 10 Log Q Rank 3.9420 4.1903 3.60 85 3.7993 3.4 955 3.6191 3.8261 4. 350 2 3 .58 88 3.9 058 3.6042 3.2041 3.6493 3.6263 3.4786 3.9614 3.7076 3.9921 3.7924 4.0294 3 .58 88 3 .53 40 3 .51 05 3.83 25 3 .57 29 3.6721 3.64 15 3.7 152 3 .59 77 3.7482 3.6693 3. 850 0 3.66 65 2.7292 3.8248 3.9222 4.2718 3.7168 7 3 27 14 35 26 12 1 30 9 28 37 23 25 36 6 19 5 15 4 31 33... the 20-year annual maximum; (b) the exceedance probability and return period for an event of 55 00 cfs; and (c) the index ratio of the 100-year flood to the 2-year flood 4210 2010 50 10 4290 3720 2920 250 0 2110 3110 2420 4280 3860 3 150 4 050 3260 7230 51 70 255 0 52 20 3 650 359 0 650 5- 1 6 Given a mean and standard deviation of the base-10 logarithms of 0.36 and 0. 15, respectively, find the 1 0-, 5 0-, and 100-year... 1912 .5 1734.1 27 95. 3 2192.2 55 5.1 1094.9 2332.4 3068.9 2220.6 2331.3 1303.4 1989 .5 3004.8 2764.2 28 05. 1 9 95. 8 912.0 1212.6 1192 .5 158 6.0 2040.0 2300.0 2041.0 2460.0 2890.0 254 0.0 Rank 47 22 44 32 14 25 34 36 8 28 42 21 37 5 23 31 33 10 20 48 43 16 4 19 17 38 29 6 11 9 45 46 40 41 35 27 18 26 15 7 13 Exceedance Probability 0. 959 2 0.4490 0.8980 0. 653 1 0.2 857 0 .51 02 0.6939 0.7347 0.1633 0 .57 14 0. 857 1 0.4286... 1967 1937 1 953 19 65 1 950 1 952 1947 1933 19 75 1966 1964 1973 1 955 1940 1943 1944 1938 1930 1934 19 45 1963 1932 19 35 1 954 1971 1941 1961 1936 1962 Exceedance Probability 0.0204 0.0408 0.0612 0.0816 0.1020 0.1224 0.1429 0.1633 0.1837 0.2041 0.22 45 0.2449 0.2 653 0.2 857 0.3061 0.32 65 0.3469 0.3673 0.3878 0.4082 0.4286 0.4490 0.4694 0.4898 0 .51 02 0 .53 06 0 .55 10 0 .57 14 0 .59 18 0.6122 0.6327 0. 653 1 0.67 35 0.6939... L1600_Frame_C 05 Page 94 Friday, September 20, 2002 10:12 AM TABLE 5. 2 Annual Maximum Floods for Back Creek Year 1929 1930 1931 1939 1940 1941 1942 1943 1944 19 45 1946 1947 1948 1949 1 950 1 951 1 952 1 953 1 954 1 955 1 956 1 957 1 958 1 959 1960 1961 1962 1963 1964 19 65 1966 1967 1968 1969 1970 1971 1972 1973 Q 8 750 155 00 4060 6300 3130 4160 6700 22400 3880 8 050 4020 1600 4460 4230 3010 9 150 51 00 9820 6200... 64.17 50 .28 51 .91 38.23 5- 2 4 Compute a log-Pearson Type III frequency analysis for the data of Problem 5- 2 0 Compare the fit with the lognormal and the estimates of parts (a), (b), and (c) 5- 2 5 Using the data of Problem 5- 1 5, perform a log-Pearson Type III analysis Make the three estimations indicated in Problem 5- 1 5, and compare the results for the normal, lognormal (Problem 5- 2 2), and LP3 analyses 5- 2 6... 1.708 1 .55 7 1.732 1.706 1.881 1.788 2.0 75 1.846 2.044 1.881 1.7 65 1. 855 1.890 1.943 1.713 1.838 1.984 1.814 1.920 1.648 1.822 1.830 1.863 1.672 1. 757 2.123 1.969 1.740 1 .58 3 1.748 1.722 1.900 1.806 879.3 21 75. 6 1037.1 1889.3 255 1.4 2069.4 1694.8 1 451 .4 2838.1 2016.7 1173.8 2194.3 1 354 .0 3037.7 2114 .5 1910.3 1733 .5 27 95. 3 2191.6 55 5.1 1094.7 2332.7 3068.9 2220 .5 2331.0 1303.2 1989.1 47 22 44 32 13 25 34 . 0.923 1 951 9 150 3.9614 6 0. 154 1 952 51 00 3.7076 19 0.487 1 953 9820 3.9921 5 0.128 1 954 6200 3.7924 15 0.3 85 1 955 10700 4.0294 4 0.103 1 956 3880 3 .58 88 31 0.7 95 1 957 3420 3 .53 40 33 0.846 1 958 3240 3 .51 05. 0.7 458 59 60 3.7 752 46 45 0.7627 55 90 3.747412 46 0.7797 53 00 3.724276 47 0.7966 52 50 3.720 159 48 0.8136 51 50 3.711807 49 0.83 05 5140 3.710963 50 0.84 75 4710 3.673021 51 0.8644 4680 3.670246 52 . 8040 3.9 052 56 27 0. 457 6 8040 3.9 052 56 28 0.4746 8040 3.9 052 56 29 0.49 15 7780 3.890980 30 0 .50 85 7600 3.880814 31 0 .52 54 7420 3.870404 32 0 .54 24 7380 3.868 056 33 0 .55 93 7190 3. 856 729 34 0 .57 63 7190