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To Hydrologie Dota

Agriculture Handbook No.259

Agricultural Research Service U.S. DEPARTMENT OF AGRICULTURE

in cooperation with the Ohio Agricultural Experiment Station
CONTENTS

Page

Introduction 3

Testing for Representativeness of the Sample 4

Modified Log-Normal Distribution 5

Fitting Equations for Known Statistical Distributions 6

Normal Distribution Q

Extreme-Value Distribution 7

Log-Normal Distribution 7

Discussion g

Selected References 8

Symbols 9

Appendix 10

Tables 10

Figures 17

Washington, D.C. Issued April 1964

Approved for reprinting August 1972
SIMPLIFIED TECHNIQUES

To Hydrologie Data

By J. L. McGuiNNESs, analytical statiHicia^i, and D. L. BRAKENSIEK, hydraulic

engineer, Soil and Water Conservation Research Division, Agricultural Research

Service

INTRODUCTION

Within the past decade, the technique of fitting frequency distributions to hydrologie data for small areas has become generally

accepted (Í7, ^4)-^ For a frequency analysis, the hydrologie data are

treated as statistical variables associated with an assumed statistical

distribution. The empirical frequency distribution of the data is then

used to determine the magnitude of the variable to be expected in a

given recurrence interval or return period. As an illustration, from

a set of 19 annual-maximum peak-runoff events, the hydrologist may

require an estimate of the magnitude of the peak that will be equaled

or exceeded once in 25 years on the average. Procedures for fitting

frequency distributions to data are available but are spread through

various technical publications and textbooks. Thus, it may be inconvenient for the hydrologie analyst to find the fitting procedures for the

distribution he chooses.

This handbook brings together simplified procedures for fitting the

most commonly used distributions in hydrologie frequency analysis—

the normal, extreme-value, log-normal, and modified log-normal distributions. The modified log-normal distribution is well suited for the

case where a specific statistical distribution is not or cannot be postu-

^ Italic numbers in parentheses refer to Selected References, p. 8.

lated. In addition to fitting procedures, a simple test of the hydrologie

representativeness of a set of hydrologie data is outlined.

The theories behind the several frequency distributions, including

their advantages and limitations, are not discussed in this report. The

references contained herein will serve to introduce the interested reader

to the voluminous and sometimes contradictory writings on these

subjects.

The question of the appropriate formula to use in computing the

frequency plotting positions for data is one illustration of the disagreement that generally exists. The authors prefer the formula m/(n+1),

where m is the order number of the point and n is the number of points

to be fitted. This formula was derived by Gumbel (l^) and Chow (8)

and is distribution-free. Yet, an earlier formula developed empirically

by Hazen (16)^ m/{2n-l), is still favored by some workers {19). After

investigating the plotting-position formulas, E. L. Neff, of the Soil

and Water Conservation Research Division, in an unpublished report,

concluded that the quality and representativeness of the record are much

more important than the choice of a plotting-position formula in the

development of meaningful frequency lines.

There has also been some debate in the past as to whether only the

maximum of annual values or all values above a certain base value

should be selected for a frequency analysis. Chow {6) has shown that

for recurrence intervals of about 10 years or longer there is essentially

no difference between the two methods of selecting data. Since the

series of maximum values is simpler and theoretically more suited for

frequency analysis, the authors prefer it to the partial-duration series.

3
AGRICULTURE HANDBOOK 259, U.S. DEPARTMENT OF AGRICULTURE

The hydrologist using frequency distributions to describe his data

should recognize the inherent limitations of the method. The data

constitute a sample only. Samples, particularly small samples, are

noted for the way in which they may vary from their parent populations. To obtain the true 50-year peak rate estimate for a watershed

with 20 years of record, it would be necessary to show—

(1) That the 20-year record is completely representative of the

long-term (but unknown) peak rate statistical distribution; and

(2) That the statistical distribution used to describe the 20-year

record is the true distribution that exactly describes the longterm (unknown) peak rate frequency distribution for the watershed under study.

The hydrologist may take steps as described below to provide a satisfactory answer to the first of these requirements. The second must

rest on assumptions that generally cannot be proved. Despite the

limitation, however, frequency analyses are a valuable tool in hydrologie

studies of small areas.

TESTING FOR REPRESENTATIVENESS OF THE SAMPLE

For a frequency analysis of a short set of data to have meaning, it

must be shown that the years of record are representative of what

might be expected on a long-term basis. In practice, this usually

means a check on the representativeness of some observed climatic element, such as precipitation, as these records are usually long enough

to define a representative period. If a long-term runoff record is

available, it can be used instead of a precipitation record in the test

proposed in this section.

The U-test [18) is suggested as a simple, distribution-free method

of checking representativeness. The following example uses the annual precipitation amounts for the Coshocton, Ohio, Weather Bureau

station for the period 1909-59. The ¿7-test is here used to determine

whether the frequency distribution of a given period—for instance,

1939-59—is identical with that of the longer period of rainfall record.

If it is the same, then conclusions drawn from the 1939-59 record are

based on a weather sample that is representative of a much longer

period of time.

To perform the ¿/-test, the 51 annual precipitation totals for the

1909-59 period at Coshocton are ranked in decreasing order. (See

table 1, Appendix.) The sum of the rank numbers {Ty for the 1939-

59 period is found to be 556. If the number of years in the 1939-59

period is denoted as a=21, and the remaining years as J=30, then

the statistic U may be computed as

ü=a J+ [a(a + l)/2]-7^. (1)

When the hypothesis of the two distributions being identical is true

(for a and h both larger than 8), the random variable {U) will have

approximately a normal distribution with the mean

ü=a 1/2

and variance

<rt72=a6(& + a4-l)/12.

For the city of Coshocton data, r/=315, crc7=52.2, and

(2)

(3)

í=(í7-Z7)/au=-0.19. (4)

The probability of a value of t greater than ±0.19 is approximately 85

percent {3, 22).

The conclusion is that the 1939-59 period at the city of Coshocton

has the same frequency function as the remaining period. Therefore,

this period is an entirely representative period at Coshocton station

with respect to the annual precipitation amounts examined, since precipitation totals at the station are highly correlated with the city of

Coshocton totals. Thus, a frequency analysis estimate of the annual

precipitation that would be equaled or exceeded once in 25 years at the

Coshocton station should have a real meaning, because it is based on

a 1939-59 record that is representative of the long-term climate of the

area. Since annual precipitation and annual runoff are also correlated

in the Coshocton area, the extension to frequency analysis of annual runoff amounts is fairly safe.

The problem of judging representativeness is more complex for such

variâtes as peak runoff rates. Concerning such rates, it would be unusual to have a long-term runoff record available for a comparable stream

in the area. However, correlation analysis may show that the shortterm record in question is significantly related to some precipitation parameter, such as maximum 30-minute intensity. In this case, it may be

' The symbols are given on p. 9,
FITTING FREQUENCY DISTRIBUTIONS TO HYDROLOGIC DATA

possible to obtain a long-term record of maximum 30-minute intensities from a nearby Weather Bureau station. If there is significant

correlation between the study record and the U.S. Weather Bureau

intensity record for the short-time period, the long-term intensity

record could be used for the representativeness test. The reliability of

the representativeness of this test as an inference to peak rates will, in

this case, depend upon the degree of correlation found between the peak

rates and the short-term intensities.

In the event that the test shows a short-term record to be nonrepresentative, it would be idle to fit a frequency curve to the data. The

resulting curve would have little predictive value for future events.

The test might be of use, however, in selecting parts of a record for

a specific characteristic such as low flow periods. Here, the analyst

would be gaining some idea of how certain parts of a record act in

comparison with normal flow expectations.

After a given record is found to be representative of long-term expectations for the area, the next step is to fit a frequency distribution

to the data. As mentioned earlier, some assumption must be made

about the true frequency distribution that would describe the record.

A. wide range of distributions is available, but there is no way to prove

that one is generally superior to another.

MODIFIED LOG-NORMAL DISTRIBUTION

Chow (9) has shown that the modified log-normal distribution has

the widest application in hydrologie analysis. The distribution is based

on estimation of a transformation constant to obtain data described by

a log-normal distribution. Whereas the log-normal distribution will

plot aö a straight line on log-normal graph paper, the modifi^ed lognormal distribution may plot as a concave curve or as a convex curve.

The generality of the modified log-normal distribution is emphasized

by the fact that it includes the normal, log-normal, and approximate

extreme-value distributions as special cases. Perhaps the main reason

it has not been more widely adopted by hydrologists is the difficulty of

computing the parameters needed to obtain an exact fit. Development

of the mathematical concepts that lead to an exact fit of the modified

log-normal distribution is presented in a later section.

A technique of adding a constant to rectify skewed data was recommended by a Subcommittee of the Joint Division Committee on Floods,

American Society of Civil Engineers (2) which reviewed flood-frequency methods. Procedures given herein make possible direct

determination of this rectifying constant, thus eliminating the need for

a trial-and-error process.

The graphical fitting procedure, presented in this section, is exact

enough for the usual hydrologie analysis. One requirement of the

graphical procedure is a plotting of the data. This plotting should

always be made prior to any fitting. From the plotting, a tentative

idea can be obtained of whether a normal, extreme-value, log-normal,

or modified log-normal fit would be more appropriate.

Thus, the graphical procedure as outlined in this handbook is useful

for any or all of the following :

(1) To obtain estimates of the magnitude of a variate at selected

frequencies of occurrence.

(2) To evaluate directly the constant that will rectify a skewed set

of data.

(3) To provide a means of determining which of several distributions will best fit the data and obtain estimates of the parameters

of this distribution.

To fit a modified log-normal distribution, the first step of the graphical procedure is to list the original data (x) in decreasing order,

compute the mean (x) of the series, and divide each original variate

by the mean to form new variâtes (x/x), (See table 2.) Plotting

positions are obtained from table 3. The x/x values are plotted at their

computed plotting positions on transparent log-normal probability

paper (fig. 1, Appendix).

The sheet with the plotted data is then superimposed on the graphs

of the Cv curve set (figs. 2 to 11). It will be noted that as Ov increases,

the curves become steeper. From these 10 figures, it will not be too

difficult to find a graphical estimate of a value of Cv that most nearly

fits the data. At the same time, interpolation among the curves on the

Cr sheet will provide an initial estimate of the value of CV,

To refine the initial estimate of CV. the data sheet is superimposed

on figures 12 to 22, a set of CV curve sheets. Whereas the Cv sheets

differentiate as to slope, the CV sheets differentiate as to curvature.

Superimposing the data sheet on the CV graphs and following the

curves for the particular Cv value previously obtained will enable the

user to obtain a graphical estimate of the value of CV.
6 AGRICULTURE HANDBOOK 259, U.S. DEPARTMENT OF AGRICULTURE

After the graphical estimates of both Cv and GV are obtained, the

theoretical modified log-normal frequency curve is calculated from the

formula

x/x=\-VC,K, (5)

(See Chow, 7.)

Table 4 gives the K values corresponding to nine different probabilities for a range of values of CV. The graphical estimate of CV is

used to enter a specific line of table 4. The K values on this line are

substituted in turn in equation 5 together with the graphical estimate

of Cv to calculate the theoretical x/x values (see table 2). These nine

values of x/x define the theoretical probability curve that fits the data.

The observed data and theoretical curve points can be reconverted to

the original units of measurement by multiplication by a?, as shown in

the last column of the lower part of table 2.

As with any graphical method, increased familiarity with the procedure will result in more exact fitting. The following points may be

helpful in deciding on a final fit. Increasing the value of C^ will tend

to make the curve steeper. Thus, if the fit developed graphically

tends to underestimate the high points and overestimate the lower

values, it would indicate that a higher value of Cv should be used. An

increase in the value of CV, for a fixed C^, will tend to make the theoretical curve more concave upward. Decreasing the Cv and CV values

has the reverse effect.

It is possible that in some problems the user would want to give more

weight to a part of the data. In a minimum water-yield study, the

lower values might be considered as more important. It would be

possible to fit the lower tail of the theoretical curve more closely by the

use of the Cv and 6^F properties discussed above while giving less weight

to the fit of the higher values.

Experience has shown that it is more important to obtain a good first

graphical estimate of Cv than of CV, If the correct Cv is selected, the

slope of the theoretical curve will be correct. An incorrect CV value

would result in either too much or too little curvature. This situation

can be corrected by making the appropriate change in CV only and

recomputing the curve. If Cv has been incorrectly estimated, it will

mean first that the appropriate change in Cv be made to correct the

slope and then CV reestimated to obtain the proper curvature.

There is another feature of this procedure that is of practical use.

It is possible to compute a constant that will transform a curved frequency line to a straight line. The exception is the case of ¿7y=0,

where the curves will plot as straight lines on arithmetic normal probability paper without the use of a transformation. The constant is

computed from the equation

c=x\_{Cv/CV)-\\. (6)

The constant relative to and derived from the data of table 2 is

computed as follows :

i?=5.5992 [(0.70/0.15)-!] =20.53.

Figure 23 shows a plotting in terms of original measurement, also the

fitted curve as determined in table 2. The constant ({?=20.53) is added

to each observation and to the points of the fitted curve. Also shown in

figure 23 are the transformed data with a fitted straight line drawn

through them. Thus, it follows that the graphical fitting procedure

presented herein is based on the idea of transforming data to make them

more nearly log-normally distributed, fitting the pure log-normal distribution to the transformed data, and then retransforming back to the

initial values of measurement. The graphical procedure arrives at

the net result without the necessity of calculating the transforming

function.

FITTING EQUATIONS FOR KNOWN STATISTICAL

DISTRIBUTIONS

In some cases it may be found appropriate to use known statistical

distributions for fitting sets of hydrologie data. Thus, the hydrologie

analyst also needs procedures for fitting the more commonly used

statistical distributions.

NORMAL DISTRIBUTION

The best known of the frequency distributions is the normal. The

normal frequency curve is bell shaped and symmetrical about the mean.

Special graph paper has been developed on which the normal distribution curve will plot as a straight line. Figure 24 shows annual runoff

amounts plotted on this arithmetic probability paper.

For the normal distribution, the mean of the set of data {x) is
FITTING FREQUENCY DISTRIBUTIONS TO HYDROLOGIC DATA

computed as usual from ^x/n. The standard deviation (5) is estimated from

s-=-%{x-~xY/^xKy, (7)

where the values of Ky are read from a table of areas of the normal

distribution {S^^2)\ such values are also listed in table 3 for a range

of sample sizes. This fitting equation is used so as to be consistent

with the extreme-value fitting procedure.

The step-by-step procedure for fitting a normal distribution is given

below. A worked example is given in table 5 and shown in figure 24.

1. Rank the observed data in decreasing order.

2. Refer to table 3 for values of Ky corresponding to the size of

sample or the number of years of data {n).

3. Calculate the corrected sum of squares of the data, S(íP —a?)^=

4. Calculate the sum of cross products of the ranked x and their

respective Ky values, ^xKy,

5. Calculate 8 by equation 7 and the mean (a?), a?=2a?/n.

6. Multiply equation 5 through by ic to obtain

x=x-{-xCvK^ (8)

or, since Cv is defined as s/x^ ^

x^x-\-sK, (9)

The calculated x and s, together with the K values from the OV=0

row of table 4, are substituted into equation 9. The data can be plotted

on normal probability paper and the fitted straight line drawn in.

( See fig. 24. ) Plotting positions are given in table 3.

EXTREME-VALUE DISTRIBUTION

The extreme-value theory was first investigated by Fisher and Tippett (11) and applied to hydrologie data by Gumbel (13). The theory

states that the annual maximum values of n years of record approach

a definite frequency distribution when the number of observations

within each year becomes large. A description of the theory is given

by Gumbel (14-) - Figure 25 shows special graph paper on which the

extreme-value distribution plots as a straight line.

Chow (9) has shown that the extreme-value distribution is approximately the same as a log-normal distribution with Í7F=0.364. This

specifies the K values to be used from table 4. The mean of the set

of data (x) is computed as usual. The standard deviation of this

distribution (s) is calculated from

s=lM7^(x-xy/^x Ky, (10)

where the terms are the same as previously defined. This simplified

formulation of the fitting procedure was developed by Brakensiek (6).

The step-by-step procedures for fitting the extreme-value distribution are given below. A worked example is given in table 6 and shown

in figure 25.

1. Rank the observed data in decreasing order.

2. Refer to table 3 for values of Ky corresponding to the size of

sample (n).

3. Calculate the corrected sum of squares of the data,

^{x-xy=ix^- (^xy/n.

4. Calculate the sum of cross products of the ranked x and their

respective Ky values, 5a? Ky. _ _

5. Calculate s by equation 10 and the mean (x) hj x=:î,x/n,

6. Values of the frequency factor (K) for the extreme-value distribution correspond to the 6^F=0.364 row of table 4. These

values together with the calculated x and s values allow several

points on the fitted line to be calculated. If the original data

are plotted on extreme-value probability paper and utilize the

plotting positions of table 3, draw the fitted straight line.

The fit obtained by this simplified procedure was found by Brakensiek (5) to be close to the fit obtained by the method developed by

Gumbel (U).

LOG-NORMAL DISTRIBUTION

The procedure for fitting the log-normal distribution is identical

with that for the normal distribution with one exception : log x is used

instead of x. Therefore, a worked example is not given to illustrate

this case. The special paper designed to plot log-normal distributions

as a straight line has the same horizontal and vertical scale divisions

as figures 2 to 22.
8 AGRICULTURE HANDBOOK 2 59, U.S. DEPARTMENT OF AGRICULTURE

DiSCUSSION

Previous fitting procedures associated with the modified log-normal

distribution have had the disadvantage of requiring the calculation

of a coefficient of skew. As is well known {21)^ the estimation of this

statistic from small samples is unreliable. The procedure as presented

in this report for fitting modified log-normal distributions to data does

not depend on an explicit estimation of a coefficient of skew. An

advantage that may be derived from the use of the graphical procedure

is that approximate fits may be obtained much more easily and quickly

than with analytical methods.

The concept of rectifying skewed data by adding a constant to observed data is included in the procedures presented. A system for

rectifying skewed data was recommended by the subcommittee reviewing flood-frequency methods, American Society of Civil Engineers {2).

Determination of the constant, however, was accomplished by a trialand-error process. The procedures presented in this report make possible the direct determination of the rectifying constant.

The flexibility of utilizing the modified log-normal distribution is

quite apparent and was pointed out by Hazen {16). The range of

skewness that can be fitted by the modified log-normal distribution

covers most of the cases usually encountered in hydrologie analysis.

The graphical procedure permits more weight to be given to selected

parts of the data, an important consideration in some analyses where

engineering judgment must substitute for more theoretical statistical

considerations.

Recent work by Chow (P) has shown that the log-normal distribution

for C^F=0.364 corresponds quite closely to the extreme-value distribution. This fact adds considerable theoretical justification for using

modified log-normal distributions in hydrologie analysis. Since

modified log-normal distributions thus include the normal, log-normal,

and extreme-value distributions as special cases, the use of the modified

distribution would save the hydrologist from having to be familiar

with the theory of a number of statistical distributions.

An analytical fitting procedure for the modified log-normal distribution has been given by Brakensiek {5). This technique may be used

to refine the graphical fit if desired. The procedure is not illustrated

with an example in this report.

SELECTED REFERENCES

(1) AiTCHisoN, J., and BROWN, J. A. O.

1Ö56. THE LOG-NORMAL DISTRIBUTION. Cambridge Univ., Dept. Applied

Econ. Monog. 5, 176 pp.

(2) AMERICAN SOCIETY OF CIVIL ENGINEERS.

1953. REVIEW OF FLOOD FREQUENCY METHODS, FINAL REPORT OF THE SUBCOMMITTEE OF THE JOINT DIVISION COMMITTEE ON FLOODS. Amer. SOC.

Civ. Engin. Trans. 118: 1220-1230.

(3) ARKIN, H., and COLTON, R. R.

1950. TABLES FOR STATISTICIANS. 152 pp. New York. (College Outline

Series, No. 75.)

(4) BEARD, L. R.

1954. ESTIMATION OF FLOOD PROBABILITIES. Amer. Soc. Civ. Engin. Proc, v.

80, Separate 438.

(5) BRAKENSIEK, D. L.

1958. FITTING A GENERALIZED LOG-NORMAL DISTRIBUTION TO HYDROLOGIC DATA.

Amer. Geophys. Union Trans. 39: 469-473.

(6) CHOW, V. T.

1951. DISCUSSION OF ANNUAL FLOODS AND THE PARTIAL-DURATION SERIES.

Amer. Geophys. Union Trans. 31: 939-941.

(7)

1951. A GENERAL FORMULA FOR HYDROLOGIC FREQUENCY ANALYSIS. Amer.

Geophys. Union Trans. 32 : 231-237.

(8)

1953. FREQUENCY ANALYSIS OF HYDROLOGIC DATA WITH SPECIAL APPLICATION

TO RAINFALL INTENSITIES. 111. Engin. Expt. Sta. Bul. 414, 80 pp.

(9)

(10)

1954. THE LOG-PROBABILITY LAW AND ITS ENGINEERING APPLICATIONS. Amer.

Soc. Civ. Engin. Proc, v. 80, Separate 536.

1959. DETERMINATION OF HYDROLOGIC FREQUENCY FACTOR. Amer. Soc. CiV.

Engin. Proc, v. 85, Separate 2084.

(11) FISHER, R. A., and TIPPETT, L. H. C.

1928. LIMITING FORMS OF THE FREQUENCY DISTRIBUTION OF THE SMALLEST

AND LARGEST MEMBER OF A SAMPLE. Cambridge Phil. Soc. Proc. 24:

180-190.

(12) GOODRICH, R. D.

1927. STRAIGHT LINE PLOTTING OF SKEW FREQUENCY DATA. Amer. SoC. CiV.

Engin. Trans. 91: 1.

(13) GUMBEL, E. J.

1941. THE RETURN PERIOD OF FLOOD FLOWS. Ann. Math. Stat. 12: 163-190.

(14)

1954. STATISTICAL THEORY OF EXTREME VALUES AND SOME PRACTICAL APPOCATioNS. [U.S.I Nati. Bur. Standards Applied Math. Series 33,

51 pp.
FITTING FREQUENCY DISTRIBUTIONS TO HYDROLOGIC DATA 9

(15) GuMBEL, B. J., and SCHELLING, H. VON.

1950. THE DISTRIBUTION OF THE NUMBER OF EXCEEDANOES. Ann. Math. Stat.

21: 247-262.

(16) HAZEN, A.

1930. FLOOD FLOWS, A STUDY OF FREQUENCIES AND MAGNITUDES. 199 pp.

John Wiley & Sons, Inc., New York.

(17) IZZARD, C. F.

1954. PEAK DISCHARGE FOR HIGHWAY DRAINAGE DESIGN. Amer. SOC. CiV.

Engin. Trans. 119: 1005-1015.

(18) MANN, H. B., and WHITNEY, D. R.

1947. ON A TEST OF WHETHER ONE OF TWO RANDOM VARIABLES IS STOCHASTICALLY LARGER THAN THE OTHER. Ann. Math. Stat. 18: 50-60.

(19) MocKUS, V.

1960. SELECTING A FLOOD-FREQUENCY METHOD. Amer. Soc. AgT. Engin.

Trans. 3: 48-51, 54.

(20) POWELL, R. W.

1943. A SIMPLE METHOD OF ESTIMATING FLOOD FREQUENCIES. CiV. Engin.

13: 105-106.

(21) SLADE, J. J., JR.

1936. THE RELIABILITY OF STATISTICAL METHODS IN THE DETERMINATION OF

FLOOD FREQUENCIES. U.S. Geol. Survey Water-Supply Paper 771,

pp. 421-432.

(22) SNEDECOR, G. W.

1956. STATISTICAL METHODS. 5th ed., 534 pp. Ames, Iowa. (Iowa State

Col.)

[U.S.]

(23) [UNITED STATES] NATIONAL BUREAU OF STANDARDS.

1953. PROBABILITY TABLES FOR THE ANALYSIS OF EXTREME-VALUE DATA.

Nati. Bur. Standards Applied Math. Series 22, 82 pp.

(24) UNITED STATES SOIL CONSERVATION SERVICE.

1957. NATIONAL ENGINEERING HANDBOOK, SECTION 4-A, THE HYDROLOGY GUIDE.

U.S. Dept. Agr. Soil Conserv. Serv. [n.p.]

SYMBOLS

Cv —CoefRcient of variability of a set of x variâtes.

CV—Coefficient of variability of a set of transformed variâtes.

Fn —Plotting position.

K —Frequency factor.

Ky—Standardized normal deviate; equals K when CV=0.

jfi —Order number of variate when set is ranked in decreasing order.

n —Number of observed variâtes in set.

P —Precipitation.

P —Mean precipitation.

Q —Runofif.

Q —Mean runoff.

s —Standard deviation; the sample estimate of <T.

t —"Students" t statistic.

T —Sum of order numbers of a subset of variâtes.

U —Nonparametric statistic for testing whether or not two samples are drawn

_^ from the same population.

U —Mean of the U statistic.

X —Observed variate.

X —Mean of a set of variâtes.

a —Standard deviation of the population; square root of the population variance.

473-008 O - 72 - 2
APPENDIX

TABLES

TABI>E 1.—Test of representativeness of anm/aal precipitation between

1939 and 1969 at Coshocton^ Ohio

RANKED YEARLY PRECIPITATION FOR 1909-59 PERIOD

TABLE 2.—Fitting a frequency curve to annual runoff amounts {Q) hy

the modified log-normal method

Rank No. Annual

(m) Year precipitation

Inches

10 1909 46.96

34 1910 38. 42

9 1911 47.62

29 1912 39. 99

4 1913 49.36

22 1914 42.45

18 1915 45.32

27 1916 40.00

28 1917 40.00

40 1918 37.32

13 1919 46.29

24 1920 40.98

2 1921 49.46

44 1922 36.05

36 1923 38.24

39 1924 37.78

47 1925 34.48

Rank No. Annual

(m) Year precipitation

Inches

1 1926 52. 60

14 1927 46. 20

46 1928 34. 49

12 1929 46.77

51 1930 22.03

38 1931 37.94

42 1932 36.30

31 1933 39. 13

49 1934 30.48

11 1935 46. 86

25 1936 40.55

3 1937 49. 43

21 1938 42.53

17 1939 45.53

5 1940 49.23

37 1941 38.21

35 1942 38.31

Rank No.

(m) Year

30 1943

50 1944

7 1945

20 1946

32 1947

26 1948

23 1949

15 1950

16 1951

41 1952

48 1953

43_.____ 1954

33 1955

6 1956

19 1957

45 1958

8 1959

= 21

-305

= 52.2

Annual

precipitation

Inches

39.22

30.28

48.08

42.56

38.98

40.02

42.32

46. 18

46. 10

36. 51

30. 74

36. 16

38.89

49.01

43.23

35.26

47.82

CALCULATION OF U AND TEST OF REPRESENTATIVENESS BY t

T=556 6 = 30

f7=21(30)+ 21(21 +1)/2-556 =

i; = 21(30)/2=315

<r t;2=21(30) (30+21+l)/12=2730;<rc/ =

¿= (305^315)/52.2=-0.19

The probability of a ¿-value equal to or greater than ± 0.19 is approximateîy 85 percent.

Rank No. (m)

Ranked

runoff

(Q (=x))

Plotting

position xlx Computations

1 _

Inches

11.76

9.82

8.82

7.23

5.83

5.79

4.74

4.35

3.72

3.20

.97

.96

Percent

7.7

15.4

23. 1

30.8

38.5

46.2

53.8

61. 5

69.2

76.9

84.6

92.3

2. 100

1.754

1.575

1.291

1.041

1.034

.847

.777

.664

.572

. 173

.171

lQ = 67.19/12

2 = 5.5992

3

4

5 _. _ .

6

7 _-.-__

»Graphical estimates:

C,=0.7

8 CV=0.15

9

10

11

12 . _ -

Total 67. 19

If the value of C, is substituted into equation 5, the fitted frequency equation

becomes

a:/x=l + 0.7jK:

Since the frequency line is curved, more than two points are needed to orient the

line. Several points lying on the fitted frequency hne are found as follows:

Probability A" value 2 j/i=l+C. K Q=Q(I+C.liO

Percent

99.0

95.0

80.0

50.0

20.0

5.0

1.0

. 1

.01

-2.01

-1.51

-.86

-.08

.81

1.76

2.66

3.80

4.81

0.398

.944

1.567

2.232

2.862

3. 660

4.367

2.23

5.29

8.77

12.50

16.02

20.49

24.45

1 Read from the n=12 column of table 3.

2 For CF=0.15, see table 4.

10
FITING FREQUENCY DISTRIBUTIONS TO HYDROLOGIC DATA 11

TABLE 3.—Plotting positions (Fn) and standardized normal deviates

{Ky) for a range of sample sizes or nvmber of years of record (n)

Rank

No.

n=9 71 = 10 n=ll n = 12 n = lS

(m)

Fn K, Fn K, Fn K, Fn K, Fn K,

1.--

2.__

3.--

4__-

5.--

6.-_

7_-_

8_--

9__-

10

Percent

10.0

20. 0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

1.28

.84

. 52

.25

.00

-. 25

-.52

-.84

-1. 28

Percent

9. 1

18. 2

27. 3

36.4

45. 5

54.5

63.6

72. 7

81. 8

90.9

1.34

.91

.60

.35

. 11

-.11

-.35

-.60

-. 91

-1.34

Percent

8.3

16.7

25.0

33.3

41.7

50.0

58.3

66. 7

75.0

83.3

91. 7

1.38

.97

.68

. 43

. 21

.00

-.21

-.43

-.68

-.97

-1.38

Percent

7.7

15.4

23. 1

30. 8

38.5

46. 2

53.8

61.5

69.2

76. 9

84.6

92.3

1.43

1.02

. 74

.50

.29

. 10

-. 10

-.29

-. 50

-. 74

-1.02

-1.43

Percent

7. 1

14.3

21.4

28.6

35. 7

42. 9

50.0

57. 1

64. 3

71. 4

78.6

85.7

92.9

1. 46

1.07

. 79

. 57

.37

. 18

.00

-. 18

-. 37

- 57

11 — 79

12 — 1 07

13 — 1 46

TABLE 3.—Plotting positions (Fn) and standardized normal deviates

{Ky) for a range of sample sizes or number of years of record

(n)—Continued

Rank n = 14 n = 15 n = 16 n = 17 n == 18

No.

(m)

Fn Ky Fn K, Fn Ks Fn Ky Fn Ky

Percent Percent Percent Percent Percent

1___ 6. 7 1. 50 6. 2 1. 53 5.9 1.57 5.6 1. 60 5.3 1.62

2___ 13.3 1. 11 12. 5 1. 15 11.8 1. 19 11. 1 1. 22 10. 5 1. 25

3___ 20.0 . 84 18. 8 .89 17. 6 .93 16.7 .97 15. 8 1.00

4___ 26. 7 . 62 25.0 .67 23. 5 .72 22.2 . 76 21. 1 .80

5.__ 33.3 .43 31. 2 . 49 29. 4 .54 27. 8 .59 26.3 .63

6_._ 40.0 .25 37.5 .32 35.3 .38 33.3 .43 31. 6 .48

7.-- 46. 7 .08 43. 8 . 16 41.2 .22 38.9 .29 36.8 .33

8_-- 53. 3 -.08 50. 0 . 00 47. 1 .07 44. 4 . 14 42. 1 . 20

9_-_ 60.0 -. 25 56. 2 -. 16 52. 9 -. 07 50. 0 . 00 47. 4 .07

10__ 66.7 -.43 62. 5 -.32 58. 8 -. 22 55. 6 -. 14 52.6 -.07

11__ 73.3 -.62 68.8 -.49 64. 7 -.38 61. 1 -.29 57. 9 -.20

12__ 80.0 -.84 75.0 -. 67 70.6 -. 54 66. 7 -. 43 63. 2 -.33

13__ 86. 7 -1. 11 81. 2 -. 89 76. 5 -. 72 72. 2 -. 59 68. 4 -. 48

14__ 93.3 -1. 50 87. 5 -1. 15 82. 4 -.93 77.8 -.76 73.7 -.63

15 93. 8 -1. 53 88. 2

94. 1

-1. 19

-1. 57

83.3

88. 9

94. 4

-.97

-1. 22

-1.60

79. 0

84.2

89.5

94. 7

-. 80

16 -1. 00

17 -1.25

18 -1.62
12 AGRICULTURE HANDBOOK 259, U.S. DEPARTMENT OF AGRICULTURE

TABLE 3.—Plotting positions (Fn) and standardized normal deviates

{Ky) for a range of sample sizes or number of years of record

(n)—Continued

TABLE 3.—Plotting positions (Fn) and standardized normal deviates

{Ky) for a range of sample sizes or number of years of record

(n)—Continued

Rank

No.

n = 19 n =20 n =21 n =22 n ==23

(m)

Fn K, Fn Ky Fn K, Fn Ky Fn Ky

PercerU Percent Percent Percent Percent !___ 5.0 1. 64 4. 8 1.67 4.5 1. 69 4.3 1. 71 4.2 1.73

2__. 10. 0 1.28 9.5 L31 9. 1 1.34 8.7 1. 36 8. 3 1.38

3--- 15. 0 1. 04 14.3 L07 13.6 1. 10 13. 0 L 12 12.5 1. 15

4___ 20. 0 . 84 19.0 .87 18.2 .91 17. 4 .94 16.7 . 97

5_-- 25.0 . 67 23.8 .71 22. 7 . 75 21.7 .78 20.8 .81

6.-- 30. 0 .52 28.6 .57 27. 3 .61 26. 1 . 64 25. 0 . 67

7.__ 35. 0 .39 33.3 .43 31. 8 . 47 30.4 .51 29. 2 .55

8__- 40.0 .25 38. 1 .30 36.4 .35 34.8 .39 33.3 .43

9___ 45. 0 . 13 42. 9 . 18 40.9 .23 39. 1 .27 37.5 .32

lü__ 50. 0 . 00 47.6 .06 45.5 . 12 43.5 . 16 41.7 .21

11__ 55. 0 -. 13 52. 4 -.06 50. 0 .00 47.8 .05 45.8 . 10

12-- 60.0 -.25 57. 1 -. 18 54.5 -. 12 52. 2 -.05 50. 0 . 00

13-- 65. 0 -.39 61.9 -.30 59. 1 -.23 56.5 -. 16 54. 2 -. 10

14-- 70. 0 -.52 66.7 -.43 63.6 -.35 60.9 -.27 58.3 -.21

15-- 75.0 -. 67 71.4 -.57 68.2 -.47 65.2 -.39 62.5 -.32

16.- 80.0 -. 84 76.2 -. 71 72. 7 -. 61 69. 6 -.51 66.7 -. 43

17-- 85. 0 -1. 04 81. 0 -.87 77.3 -. 75 73.9 -. 64 70. 8 -. 55

18_- 90. 0 -1.28 85. 7 -1.07 81.8 -.91 78.3 -.78 75. 0 -. 67

19-- 95.0 -1. 64 90.5 -1.31 86. 4 -1. 10 82. 6 -.94 79. 2 -.81

20 95 2 — 1 67 90 9 — 1 34 87. 0

91 3

-1. 12

-1.36

— 1 71

83. 3

87.5

91.7

95.8

-. 97

-1. 15

-1.38

-1. 73

21 95 5 — 1 69

22-- 95 7

23--

Rank

im)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

Fn

Percent

4. 0

8.0

12. 0

16.0

20. 0

24. 0

28. 0

32. 0

36. 0

40. 0

44. 0

48.0

52. 0

56.0

60. 0

64. 0

68. 0

72.0

76.0

80.0

84.0

88.0

92.0

96.0

Ky

1.75

1. 40

1. 18

.99

.84

. 71

. 58

. 47

.36

. 25

. 15

.05

-.05

-. 15

-. 25

-.36

-.47

-.58

-.71

-.84

-. 99

-1. 18

-1.40

-1.75

n=25

Fn

Percent

3.8

7.7

11. 5

15. 4

19. 2

23. 1

26. 9

30. 8

34. 6

38.5

42. 3

46. 2

50.0

53. 8

57. 7

61. 5

65. 4

69. 2

73. 1

76. 9

80.8

84. 6

88.5

92. 3

96. 2

Ky

1.77

1.43

1. 20

1.02

.87

. 74

. 61

. 50

. 40

.29

. 19

. 10

.00

-. 10

-. 19

-. 29

-.40

-.50

-. 61

-. 74

-.87

-1. 02

-1. 20

-1.43

-1. 77

n=26

Percent

3. 7

7.4

11. 1

14. 8

18. 5

22. 2

25. 9

29.6

33.3

37. 0

40. 7

44. 4

48. 1

51. 9

55. 6

59. 3

63.0

66. 7

70.4

74. 1

77.8

81. 5

85. 1

88.9

92. 6

96.3

Ky

1.79

1. 45

1. 23

1.04

.90

. 77

. 65

.54

.43

.33

. 24

. 14

.05

-.05

-. 14

-. 24

-.33

-.43

-. 54

-.65

-.77

-.90

-1.04

-1. 23

-1.45

-1. 79

n=27

Percent

3.8

7. 1

10. 7

14.3

17. 9

21. 4

25. 0

28. 6

32. 1

35. 7

39.3

42. 9

46. 4

50.0

53. 6

57. 1

60. 7

64. 3

67. 9

71.4

75. 0

78.6

82. 1

85. 7

89.3

92. 9

96.4

Ky

1.80

1.47

1. 24

1.07

.92

. 79

.67

. 56

. 46

.37

. 27

. 18

.09

. 00

-.09

-. 18

-. 27

-.37

-.46

-. 56

.67

.79

. 92

.07

. 24

w=28

-1. 47

-1. 80

Percent

3.4

6.9

10.3

13. 8

17. 2

20.7

24. 1

27.6

31. 0

34. 5

37.9

41. 4

44. 8

48. 3

51. 7

55. 2

58. 6

62. 1

65.5

69.0

72. 4

75.9

79.3

82.8

86. 2

89. 7

93. 1

96.6

Ky

1.82

1.48

1. 26

1. 09

.95

.82

.70

.59

. 50

.40

.31

.22

. 13

.04

-.04

-. 13

-. 22

-. 31

-.40

-. 50

-.59

-.70

-.82

-.95

-1. 09

-1.26

-L48

-1.82
FITTESTG FREQUENCY DISTRIBUTIONS TO HYDROLOGIC DATA 13

TABLE 3.—Plotting positions (Fn) and standardised normal deviates

(Ky) for a range of sample sises or number of years of record

{n)—Continued

TABLE 3.—Plotting positions (Fn) and standardized normal deviates

{Ky) for a range of sample sizes or number of years of record

(n)—Continued

Rank n =29 n-=30 n =31 n =32 n==33

No.

(m)

Fn Ky Fn K, Fn K, Fn K, Fn K,

Percent Percent Percent Percent Percent

1--- 3.3 1.84 3.2 1.85 3.1 1.87 3.0 1.88 2.9 L90

2... 6.7 1.50 6.5 1. 51 6.2 1.54 6. 1 L55 5.9 L56

3___ 10.0 1.28 9.7 1.30 9.4 L32 9. 1 1.33 8.8 L35

4_-- 13. 3 1. 11 12.9 1. 13 12.5 1. 15 12. 1 L 17 1L8 L18

5._. 16.7 .97 16. 1 .99 15.6 1.01 15.2 L03 14.7 1.05

6_.- 20.0 . 84 19.4 .86 18.8 .88 18. 2 . 91 17.6 .93

7_._ 23.3 . 73 22.6 .75 21. 9 .78 21.2 .80 20.6 .82

8-_- 26. 7 .62 25.8 .65 25.0 . 67 24. 2 .70 23. 5 .72

9___ 30.0 .52 29.0 . 55 28. 1 .58 27.3 .60 26.5 .63

10. _ 33.3 .43 32.3 .46 31.2 .49 30.3 .52 29. 4 .54

11__ 36. 7 . 34 35.5 .37 34.4 .40 33.3 .43 32.4 .46

12_- 40.0 .25 38. 7 . 29 37.5 .32 36.4 .35 35.3 .38

13_- 43. 3 . 17 41. 9 .20 40.6 .24 39.4 .27 38.2 .30

14__ 46. 7 .08 45.2 . 12 43.8 .16 42.4 . 19 4L 2 . 22

15__ 50.0 .00 48. 4 .04 46.9 .08 45.5 . 11 44. 1 . 15

16__ 53.3 -.08 51.6 -.04 50.0 .00 48. 5 .04 47. 1 .07

17_- 56.7 -. 17 54.8 -. 12 53. 1 -.08 51.5 -.04 50.0 .00

LS-- 60.0 -.25 58. 1 -.20 56. 2 -. 16 54.5 -. 11 53.0 -.07

19_- 63.3 -.34 61.3 -. 29 59.4 -.24 57. 6 -. 19 55. 9 -. 15

20_- 66.7 -.43 64.5 -.37 62. 5 -.32 60.6 -.27 58.8 -. 22

21__ 70.0 -. 52 67.7 -. 46 65.6 -.40 63. 6 -.35 6L 8 -.30

22.- 73.3 -.62 71. 0 -.55 68.8 -.49 66.7 -.43 64.7 -.38

23__ 76.7 -.73 74.2 -.65 71.9 -.58 69.7 -.52 67.6 -.46

24.. 80.0 -.84 77.4 -.75 75.0 -.67 72. 7 -.60 70.6 -.54

25-- 83.3 -.97 80.6 -.86 78. 1 -.78 75. 8 -.70 73.5 -.63

26_- 86.7 -1. 11 83.9 -.99 81.2 -.88 78.8 -.80 76.5 -. 72

27_- 90. 0 -1.28 87. 1 -1. 13 84. 4 -LOI 81.8 -.91 79.4 -.82

28__ 93.3 -1.50 90.3 -1.30 87.5 -L 15 84.8 -L03 82.4 -. 93

29__ .96.7 -1.84 93.5 -1.51 90.6 -1.32 87.9 -L 17 85.3 -L05 HO 96 8 — 1 85 93.8

96 9

-L54

— 1 87

90.9

93 9

-L33

-L55

-1.88

88.2

9L2

94. 1

-L 18

-L35

— 1. 56

31

32-. 97,0

33. - 97. 1 -L90

Rank

No.

n =34 n =35 n =36 n =37 n==38

(m)

Fn K, Fn K, Fn Ky Fn Ky Fn Ky

Percent Percent Percent Percent Percent

1 2.9 L90 2.8 L91 2.7 1.93 2.6 L94 2.6 1.94

2 5.7 1.58 5.6 1. 59 5.4 L61 5.3 1. 62 5. 1 1. 64

3 8. 6 1.37 8.3 1.39 8. 1 L 40 7.9 1.41 7.7 1.43

4 11.4 1. 21 ILl 1.22 10.8 L24 10. 5 1. 25 10. 3 1. 26

5 14.3 1. 07 13.9 L08 13. 5 L 10 13.2 L 12 12.8 L 13

6 17. 1 .95 16. 7 .97 16.2 .99 15.8 1. 00 15.4 1. 02

7 20. 0 .84 19. 4 .86 18.9 .88 18.4 .90 17.9 .92

8 22.9 . 74 22.2 .77 2L 6 .79 2L 1 .80 20.5 . 82

9 25.7 .65 25. 0 . 67 24.3 .70 23.7 '. 72 23. 1 . 74

10 28.6 .56 27.8 .59 27. 0 .61 26.3 .63 25.6 .66

11 31.4 .48 30.6 .51 29.7 .53 28.9 .56 28.2 .58

12 34.3 .40 33.3 .43 32.4 .46 3L 6 .48 30.8 .50

13 37. 1 .33 36. 1 .36 35. 1 .38 34. 2 . 41 33. 3 . 43

14 40.0 .25 38.9 .28 37.8 .31 36.8 .34 35.9 .3Q

15 42.9 . 18 4L 7 .21 40.5 .24 39.5 .27 38.5 .29

16 45.7 . 11 44. 4 . 14 43.2 . 17 42. 1 .20 41. 0 .23

17 48. 6 .04 47.2 .07 45.9 . 10 44. 7 . 13 43.6 . 16

18 51.4 -.04 50.0 .00 48.6 .04 47. 4 .07 46. 2 . 10

19 54.3 -. 11 52.8 -.07 5L4 -.04 50. 0 .00 48.7 .03

20 57. 1 -. 18 55. 6 -. 14 54. 1 -.10 52.6 -.07 5L3 -.03

21 60.0 -.25 58.3 -.21 56.8 -. 17 55.3 -. 13 53.8 -. 10

22 62.9 ^.33 61. 1 -.28 59.5 -.24 57.9 -,20 56.4 -. 16

23 65.7 -.40 63.9 -.36 62.2 -.31 60.5 -. 27 59.0 -.23

24 68.6 -.48 66.7 -.43 64.9 -.38 63.2 -.34 6L5 -.29

25 7L4 -.56 69.4 -.51 67. 6 -.46 65.8 -.41 64. 1 -.36

26 74.3 -.65 72.2 -.59 70.3 -.53 68.4 -.48 66. 7 -.43

27 77. 1 -.74 75.0 -.67 73.0 -.61 71. 1 -.56 69.2 -.50

28 80.0 -.84 77.8 -.77 75.7 -.70 73.7 -.63 71.8 -.58

29 82.9 -.95 80.6 -.86 78.4 -.79 76.3 -.72 74.4 -. 66

30 85. 7 -L 07 83.3 -.97 81. 1 -.88 78.9 -.80 76.9 -. 74

31 88.6 -1.21 86. 1 -1.08 83.8 -.99 81.6 -.90 79.5 -.82

32 91.4 -L37 88.9 -1. 22 86.5 -L 10 84.2 -1.00 82. 1 -.92

33 94.3 -1.58 9L7 -1.39 89.2 -1.24 86.8 -1. 12 84.6 -1. 02

34 97. 1 -1.90 94. 4 -L59 9L9 -1. 40 89.5 -1.25 87.2 -L 13

35 97. 2 -1.91 94. 6 — 1. 61 92. 1 — 1. 41 89.7 -1.26

36 97.3 -1.93 94.7

97. 4

-1.62

-1.94

92.3

94.9

97.4

— 1. 43

37 — 1. 64

38 — L94
14 AGRICULTURE HANDBOOK 2 59, U.S. DEPARTMENT OF AGRICULTURE

TABLE 3.—Plotting positions {Fn) and standardized normal deviates

(Ky) far a range of sample sizes or number of years of record

(n)—Continued

TABLE 4.—Theoretical log-prohability frequency factors

Rank

No.

im)

1_-

2._

3--

4__

5--

6__

7-_

8-_

9__

10.

11.

12-

13-

14.

15-

16.

17_

18-

19-

20-

21.

22-

23-

24-

25-

26-

27-

28-

29-.

30-

31-.

32-.

33-.

34-.

35-.

36-.

37-.

38-.

39-.

40-.

41-

42-

43.

Percent

2.5

5.0

7.5

10. 0

12. 5

15.0

17.5

20.0

22. 5

25.0

27.5

30.0

32.5

35.0

37.5

40.0

42.5

45.0

47.5

50.0

52.5

55. 0

57.5

60.0

62.5

65. 0

67. 5

70.0

72.5

75.0

77.5

80. 0

82.5

85.0

87.5

90.0

92.5

95.0

97.5

Ky

1. 96

1. 64

1. 44

1. 28

L 15

1.04

.93

.84

. 76

. 67

. 60

.52

. 45

.39

.32

.25

. 19

. 13

.06

.00

-.06

-.13

-. 19

-.25

-.32

-.39

-.45

-.52

-. 60

-. 67

-. 76

-.84

-.93

-1. 04

-1. 15

-1.28

-1. 44

-1. 64

-1.96

Fr,

Percent

2. 4

4.9

7.3

9.8

12.2"

14. 6

17. 1

19. 5

22. 0

24. 4

26.8

29.3

31. 7

34. 1

36. 6

39.0

41. 5

43. 9

46. 3

48.8

51.2

53. 7

56. 1

58.5

61. 0

53. 4

65.9

68. 3

70. 7

73. 2

75. 6

78.0

80.5

82.9

85. 4

87.8

90. 2

92.7

95. 1

97. 6

Ky

1.98

1. 66

1. 45

1. 29

1. 16

1.05

. 95

. 86

. 77

. 69

. 62

.54

.48

.41

.34

.28

. 21

. 15

.09

.03

-.03

-.09

-. 15

-. 21

-.28

-. 34

-. 41

-.48

-.54

-.62

-. 69

-. 77

-.86

-.95

-1.05

-1. 16

-1. 29

-1. 45

-1. 66

-1.98

Fn

Percent

2. 4

4.8

7. 1

9.5

11.9

14.3

16. 7

19. 0

21.4

23.8

26.2

28. 6

31.0

33.3

35.7

38. 1

40.5

42. 9

45. 2

47.6

50. 0

52. 4

54. 8

57. 1

59.5

61.9

64.3

66. 7

69.0

71.4

73.8

76. 2

78. 6

81. 0

83.3

85.7

88. 1

90.5

92.9

95. 2

97. 6

Ky

1.98

1. 67

1. 47

1.31

L 18

L07

.97

.88

. 79

. 71

. 64

.57

.50

.43

.37

.30

. 24

. 18

. 12

.06

.00

-.06

-. 12

-. 18

-. 24

-.30

-.37

-.43

-.50

-.57

-. 64

-. 71

-.79

-.88

-.97

-1.07

-1. 18

-1.31

-1.47

-1. 67

-1.98

71=42

Percent

2.3

4. 7

7.0

9.3

11.6

14. 0

16. 3

18. 6

20.9

23.3

25. 6

27.9

30.2

32.6

34.9

37.2

39.5

41. 9

44. 2

46.5

48. 8

51. 1

53. 5

55.8

58. 1

60. 5

62.8

65. 1

67. 4

69.8

72. 1

74. 4

76.7

79. 1

81. 4

83.7

86. 0

88. 4

90. 7

93. 0

95. 3

97.7

Ky

«=43

1.99

1. 68

1. 48

1. 32

L 19

1. 08

.98

.89

.81

. 73

.66

.59

.52

.45

.39

.33

.27

. 21

. 15

.09

.03

-.03

-.09

-. 15

-. 21

-.27

-.33

-.39

-. 45

-.52

-.59

-.66

-. 73

-.81

-.89

-.98

-1.08

-1. 19

-1. 32

-1. 48

-1.68

1. 99

Fn

Percent

2.3

4.5

6.8

9. 1

11. 4

13. 6

15. 9

18. 2

20. 5

22. 7

25.0

27.3

29.5

31.8

34. 1

36.4

38. 6

40. 9

43. 2

45.5

47. 7

50. 0

52.3

54.5

56.8

59. 1

61. 4

63. 6

65. 9

68.2

Ky

81.8

84. 1

86.4

88. 6

90.9

93. 2

95.5

97. 7

2.00

1. 69

1. 49

1.34

1. 21

1. 10

1. 00

. 91

.83

.75

. 67

. 60

.54

. 47

. 41

.35

.29

. 23

. 17

. 11

.06

.00

-.06

-. 11

-. 17

-.23

-.29

-.35

-. 41

-.47

70.5 -.54

72. 7 -. 60

75. 0 -.67

77.3 -. 75

79.5 -.83

-.91

-1. 00

-1. 10

-1. 21

-1. 34

-1. 49

-1. 69

-2. 00

cv

0.000

. 010

.020

. 030

.040

.050

. 060

.070

.080

. 090

. 100

. 125

. 150

. 175

.200

. 225

. 250

. 275

. 300

.325

.350

. 364

.375

.400

.425

.450

.475

. 500

. 525

.550

.575

.600

.625

.650

.675

. 700

. 725

. 750

.775

.800

. 825

.850

.875

.900

.925

.950

.975

1. 000

K values for probability in percentage greater than the given variate

2.33

2.31

2. 29

2. 27

2. 24

2. 22

2. 20

2. 18

2. 16

2. 13

2. 11

2.06

2.01

1. 96

1.91

1.86

1.81

1.77

1.72

1.68

1.63

1.61

1.59

1. 55

1. 51

1.47

1.43

1.40

1.37

1.34

1.31

1.28

1.24

1.22

1. 19

1. 16

1. 13

1. 11

1.08

1.06

1.03

1.01

.99

.97

.95

.93

. 92

.90

1. 64

1. 64

1. 63

1. 63

1. 62

1.61

1. 60

1. 59

1. 58

1.57

1. 56

1. 54

1.51

1. 49

1.46

1.44

1.41

1.39

1.36

1.34

1.32

1.30

1. 29

1.27

1.24

1.22

1.20

1. 17

1. 15

1. 13

1. 11

1.09

1.07

1. 05

1. 03

1.01

.99

.98

. 96

.94

.92

.91

.89

.88

.86

.85

. 83

. 82

0.84

.85

.85

.85

.85

.85

.85

.85

.85

. 85

.85

.85

.86

.85

.85

.85

.84

.84

.84

.83

.83

.82

.82

. 82

.81

.81

.80

.80

.79

. 78

. 78

.77

.76

. 76

.75

. 74

.73

. 73

. 72

. 71

. 70

. 70

.69

.68

.67

.67

. 66

.65

0.00

. 01

.02

.02

.03

.03

.04

.04

.05

. 05

.06

.07

.08

.09

. 10

. 11

. 12

. 13

. 14

. 15

. 16

. 16

. 17

. 18

. 19

.20

.20

. 21

. 22

. 22

.23

.24

.24

.25

.26

.26

.27

.27

.27

.28

. 28

. 28

. 29

.29

. 29

.29

. 29

. 29

20

+

0.84

.84

.84

.84

.84

.83

.83

.83

.82

.82

.82

. 81

.81

.80

.79

.78

. 77

. 76

.75

. 74

. 73

. 73

.72

.71

. 70

.69

.68

.66

.65

. 64

.63

.61

.60

. 59

. 58

.57

. 56

. 54

. 53

. 52

. 51

. 50

.48

.47

.46

.45

.43

.42

5

1. 64

1. 65

1. 66

1.67

1.68

1. 69

1. 69

1. 70

1. 71

1. 72

1. 72

1. 74

1. 76

1. 78

1. 79

1.81

1.82

1.83

1. 84

1.85

1.86

1.87

1.87

1.87

1.88

1.88

1.89

1. 89

1.89

1. 89

1. 89

1. 89

1.89

1.89

1. 89

1.88

1. 88

1. 87

1. 87

1.86

1. 85

1.85

1. 84

1. 83

1. 82

1.81

1. 79

1.78

1

2.33

2.35

2.38

2.40

2.42

2.44

2. 46

2. 49

2.51

2. 53

2.55

2.61

2.66

2. 72

2. 78

2.83

2.88

2.94

2.99

3.04

3. 10

3. 12

3. 15

3. 20

3. 25

3. 30

3.34

3.38

3.42

3.46

3. 50

3.53

3.57

3. 60

3.63

3.66

3.69

3. 71

3. 74

3. 76

3. 78

3.80

3.82

3.84

3. 86

3.88

3.90

3.91

0.1

-f3.09

3. 13

3. 17

3.22

3.26

3.30

3.35

3.40

3. 45

3. 50

3. 56

3.67

3.80

3. 92

4.05

4. 18

4.32

4.46

4.60

4.74

4.88

4.94

5.02

5. 16

5.30

5.44

5. 58

5. 71

5.86

6.00

6. 14

6.27

6.41

6.55

6. 70

6.83

6. 97

7. 10

7.24

7. 37

7.50

7.62

7.74

7.86

7.98

8.09

8. 20

8. 30

^ Adapted from Chow {9). Each column of Chow's table 2 was plotted against

CV. Values in this table were then read from these graphs. Additional values

may be found by a graphical method described by Chow {lO).
FITTING FREQUENCY DISTRIBUTIONS TO HYDROLOGIC DATA 15

TABLE 5.—Fitting a normal distribution to annual runoff amounts (Q) TABLE 6-

Rank No, (m)

Ranked

runoff

(Q (=x))

Plotting

position

standard

normal

deviate

Computations

1

In.jYr.

23.34

19.37

18.46

17.65

16.72

14. 10

13.75

12. 92

12.71

12.70

12.43

n. 52

11.24

10.39

9.93

8.28

6.72

5.69

4.37

Percent

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

55.0

60.0

65 0

70.0

75.0

80.0

85.0

90.0

95.0

L64

1.28

L04

.84

.67

.52

.39

.25

. 13

.00

-.13

-.25

-.39

-.52

-.67

-.84

-1.04

-L28

-1.64

^ 202 = 3514 7377

2

3 (20)2/19 =5055». 7076

4

5 2(0-0)2=425.0301

6 ___ _ 20 X^=76.5454

7 .

8 0= 242. 29/19

9

10 — 12 7521

11 By equation 7:

12

13

14 s = 425. 0301/76. 5454

15_._

16

= 5. 5527

17

18

19

Total 242. 29

-Fitting an extreme value distribution to ^i-hour maximum

precipitation amounts (P)

^ Read from the n=19 column of table 3.

If the values of s and 0 are substituted into equation 9, the fitted frequency equation

becomes

0=12. 75 + 5. 55K.

Two points lying on the fitted frequency curve are found as follows:

(1) From the Cy=0 row of table 4, read K= 1.64 corresponding to a probability

of 5 percent. Then the value of 0 for this probability level is calculated to

be

0=12.75+ 5.55(1.64) =21.9.

(2) From the CV=0 row of table 4, read K= —1.64 corresponding to a probability of 95 percent. Then the value of 0 for this probability level is

calculated to be

0= 12.75 + 5.55(-1.64) =3.6.

Columns 2 and 3 of the above data are plotted on normal probability paper, figure

24. The fitted Hne is oriented by drawing it through the above calculated points,

i.e., 5 percent=21.9 and 95 percent= 3.6.

Rank No. (m)

Ranked

precipitation

(Pi=x))

Plotting

position

{Fn)i

standard

normal

deviate

Computations

1 .

Inches

3.49

3.37

3. 14

2.97

2.70

2.63

2.59

2.54

2.47

2.21

2.06

1.90

1.77

1. 75

1.75

1.73

1.66

Percent

5.6

11. 1

16.7

22.2

27.8

33.3

38.9

44.4

50.0

55.6

61. 1

66.7

72.2

77.8

83.3

88.9

94.4

1.60

1.22

.97

.76

.59

.43

.29

. 14

.00

-. 14

-.29

-.43

-.59

-.76

-.97

-1.22

-1.60

] 2P2= 103.3595

2

3 (XP)^I17 =974884

4 - -

5 . --- --. - 2(P-P)2=5.8711

6 2PK„=8.2862

7

8 _ P=40.71/17

9

10— =2.3947

11 _ _ .- _- -

12 By equation 10:

13

14 - -. s=1.067(5.8711)/8.2862

15 _ --.

16

=0.7560

17

Total-- 40.71

1 Read from the n=17 column of table 3.

If the values of s and P are substituted into equation 9, the fitted frequency equation becomes

P=2.39+ 0.76E:.

Two points lying on the fitted frequency curve are found as follows:

(1) From the CF=0.364 row of table 4, read K=l.S7 corresponding to a

probability of 5 percent. Then the value of P for this probability level is

calculated to be

P=2.39 + 0.76(1.87) = 3.81.

(2) From the CF=0.364 row of table 4, read K=-1.W corresponding to a

probability of 95 percent. Then the value of P for this probabihty level is

calculated to be

P=2.39 + 0.76(-1.30) = 1.40.

Columns 2 and 3 of the above data are plotted on extreme-value probability paper,

figure 25. The fitted line is oriented by drawing it through the above calculated

points, i.e., 5 percent=3.81 and 95 percent=1.40.
FIGURES

473-008 0-72-3
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 2 1 0.5 0.2 0.1 0.05 0.01

5^

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18 FIGURE 1.—^Modified log-normal frequency cun e fitted graphically to data of table 2.
PROBABILITY OF A LARGER VALUE (percent)

10

99.99 99.9 99.8 99.5 99 98 05 0.01

IX

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<

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FIGURE 2.—Modiflea log-normal curves for Cv=0.1. 19
99.99

101

PROBABILITY OF A LARGER VALUE (percent)

99.9 99.8 99.5 99 98 95 90 80. 70 60 50 40 30 20 10 5 2

1 0.5 0.2 0.1 0.05 0.01

IX

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<

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20 FIGURE 3.r—Modified log-normal curves for C„=0.2.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 2 1 0.5 0.2 0.1 0.05 0.01

10

IX

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FIGURE 4.—Modified log-normal curves for (7^=0.3. 21
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 1 0.5 0.2 0.1 0.05 0.01

10

IX

UJ

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Q:

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22 FiGUEE 5.—Modified log-normal curves for O«=0.4.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10

10

1 0.5 0.2 0.1 0.05 0.01

IX

X

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FiGUSE 6.—Modified log-normal curves for Cv=0.5. 23
99.99

PROBABILITY OF A LARGER VALUE (percent)

99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

FIGURE 7.—Modified log-normal curves for (7«=0.6.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

10

X

<

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FiGUBE 8.—Modified log-normal curves for C»=0.7. 25
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

10

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26 FIGURE 9.—Modified log-normal curves for Cv=0.8.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 2 1 0.5 0.2 0.1 0.05 0.01

10

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FIGURE 10.—Modified log-normal curves for C«=0.9. 27
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 0.01

FIGURE 11.—Modified log-nonnal curves for (7^=1.0.
PROBABILITY OF A LARGER VALUE (percent)

99.99

10

IX

X

<

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99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

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30 FIGURE 13.—Modified log-normal curves for CV=0.1.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 1 0.5 0.2 0.1 0.05 0.01

10

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FIGURE 14.—Modified log-normal curves for (7F=0.2. 31
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 1 0.5 0.2 0.1 0.05 O.Ot

10

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32 FiGUBE 15.—McKiified log-normal curves for (7F=0.3.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 1 0.5 0.2 0.1 0.05

1 r . - l! I ' T ' ' " ! I ' ' ! t ^ ' ' ' i I 1 ' ' I i I I ' ' I ' ' 1 ? M "' 1 ^ ''-' '' i • r i 1 I I ^ r"i i —^ r— 1 'I I ' —r—r0.01

FIGURE 16.—Modified log-normal curves for CV=OA. 33
PROBABILITY OF A LARGER VALUE (percent)

0.2 0.1 0.05 0.01

FIGURE 17.—Modified log-normal curves for CF=0.5.
99.99

O

PROBABILITY OF A LARGER VALUE (percent)

99.9 99.8 99.5 99 98 95 90 80^ 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

FIGURE 18.—Modified log-normal curves for (77=0.6. 35
PROBABILITY OF A LARGER VALUE (percent)

99.99

10

99.9 99.8 99.5 99

1^

0.2 0 1 0.05 0.01

FiGUBE 19.—Modified log-normal curves for CV=0.7.
99

10

PROBABILITY OF A LARGER VALUE (percent)

99 99.9 99.8 99.5 99 ^8 95 90 80 70 60 50 40 30 ^ 10 5 2

1 0.5 0.2 0.1 0.05 0.01

FIGURE 20.—Modified log-norinal curves for CV=0.8.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

10

X

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FIGURE 21.—Modified log-normal curves for Cy=0.9.
PROBABILITY OF A LARGER VALUE (percent)

99.99 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20 10 5 2 1 0.5 0.2 0.1 0.05 0.01

10^

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ANNUAL RUNOFF (in.)

FiGUBE 23.—Modified log-normal distribution fitted graphically to data of table 2.

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PROBABILITY OF A LARGER VALUE (percent)

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42

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FIGURE 25.—Extreme-value distribution fitted to data of table 6.

U. S. GOVERNMENT PRINTING OFFICE : 1972 O - 473-008