Probability Analysis 

Applied to a 

Water-Supply Problem

GEOLOGICAL SURVEY 

CIRCULAR 410
Probability Analysis Applied 

to a Water-Supply Problem

By Luna B. Leopold

GEOLOGICAL SURVEY CIRCULAR 410

Washington 1959
United States Department of the Interior

STEWART L. UDALL, Secretary

Geological Survey

William T. Pecora, Director

First printing 1959 

Second printing 1960

Third printing 1962 

Fourth printing 1967

Free on applkation to the U.S. Geological Survey, Washington, D.C. 20242
CONTENTS

Page 

Introduction................................................... 1

Variability.................................................... 2

Probability plotting ........................................ 4

Effect of persistence in hydrologic data ............. 8

Probable variation among means of

future samples .......................................... 11

Page 

Confidence in estimate of future variability ........ 11

Probable value of mean flow in next

61-year period .......................................... 13

Effect of storage on streamflow

variability................................................ 14

References ................................................... 15

ILLUSTRATIONS

Figure

Page

1. The normal distribution...................................................................................................... 3

2. The cumulative frequency distribution................................................................................... 5

3. Cumulative distribution curve, Colorado River at Lees Perry, 61 years 1896-1956 ..................... 7

4. Variability of means of records of various lengths in years ...................................................... 8

5. Effect of grouping tendency in streamflow data ....................................................................... 10

6. Probable variation of mean discharge values for periods of various lengths,

Colorado River at Lees Ferry.......................................................................................... 12

7. Variability of 61-year means............................................................................................... 13

8. Variability decreased by storage.......................................................................................... 16

9. Annual reservoir regulation in relation to mean flow................................................................ 17

10. Effect of various amounts of storage capacity on flow regulation.

Colorado River bksin...................................................................................................... 18

TABLES

Table 1. Reconstructed annual flows of Colorado River at Lees Ferry, Ariz................................................. 2

2. Computation of plotting position in probability analysis, annual flows of

Colorado River at Lees Ferry, Ariz..................................................................................... 6

3. Variability of group means of streamflow data............................................................................ 9

4. Capacity and regulation of some representative reservoirs........................................................... 15

III
PROBABILITY ANALYSIS APPLIED TO A WATERrSUPPLY PROBLEM

By Luna B. Leopold

INTRODUCTION

The literature on probability techniques applicable 

to problems in hydrology is ^abundant but scattered 

through scientific journals of ^x>th hydrology and statistics. Important administrative and judicial decisions presently face water-compact commissions, 

courts, and water-planning committees. These and 

other groups might find useful, a brief and simplified 

discussion of how statistical techniques can aid in 

analysing problems of water stipply. The interest expressed in this subject by various parties to the litigation concerning the Colorado RJiver prompts this publication of material, which was presented in August 

1958 before the Special Master of the Supreme Court 

hearing the proceedings of Arizona v. California et al. 

The examples presented here are the same as those 

used in testimony before the Special Master, but there 

are included the basic computations, which were too 

detailed to present in the actual testimony.

The specific example, which was analyzed in that 

testimony, was a 61-year series of annual discharge 

values of the Colorado River at Lees Ferry, 1896 to 

1956, inclusive. However, the methodology presented 

herein is generally applicable to many other streamflow records; and the specific data discussed should 

be viewed as exemplifying the types of information, 

which can be obtained from any streamflow record.

The series of data used are the 61 years of reconstructed record of annual discharge values representing the so-called virgin flow of the Colorado River at 

Lees Ferry.1 This particular series was used by 

witnesses from Arizona and California in the Colorado 

River litigation (California exhibit 2201 A). The series

*The division between the upper basin and lower 

basin of the Colorado River, as defined in the Colorado 

River compact of 1922, is a point on the Colorado 

River 1 mile downstream from the Paria River. This 

point is called Lee Ferry in the compact. Lees Ferry 

is the name of a nearby place where the Geological 

Survey makes river measurements.

was compiled from three sources. For the period 

1896 to 1947 the annual discharges were derived from 

the U. S. Bureau of Reclamation "Report on Colorado 

River Storage Project and Participating Projects," 

dated December 1950. The series from 1948 through 

1951 was obtained from the Bureau of Reclamation 

"Memorandum Supplement to Report on Water Supply 

of the Lower Colorado River Basin, Project Planning 

Report, November 1952," dated November 1953. The 

annual discharges for 1952 through 1956 were derived 

by the same methods used th the Bureau of Reclamation. 

The entire series from California exhibit 2201A is 

listed in columns 1 and 2 of table 1. The remaining 

columns are explained on page 4.

It is generally understood that the mean flow experienced during the period of record at a given gaging 

station will not necessarily be duplicated in future 

periods. However, it is not as well recognized that 

proper analysis can yield much information in addition 

to the mean flow for the period of record. Simple 

statistical techniques can be used to obtain values of 

the probability that any specific flow will be equaled 

or exceeded in the future, or that any particular value 

will not be reached in future periods. The objectively 

determined probabilities will not dictate a particular 

course of action of decision, but they do at least provide 

a framework within which decisions on water-supply 

problems can be made.

It should be emphasized that a statement of probability is not a forecast. Extensive studies of the variation of hydrologic phenomena clearly indicate that 

values of any hydrologic factor tend to vary with time, 

but these variations are not sufficiently regular to be 

deemed cyclic. For forecasting future hydrologic 

events there must be repetitive cyclical phenomena; 

cyclic phenomena imply that at some time in the future 

the experience of the past will tend to be duplicated. 

Repetitive cycles are, for all practical purposes, 

absent in hydrologic data. Therefore, the past record 

should be used as an indication only of the probability 

that certain events will occur in the future, not as a 

forecast.

1
PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

Table 1. Reconstructed annual flows of Colorado River at Lees Ferry, Ariz. 

(Flow represents discharge adjusted for upstream depletion]

(1)

Water 

year1

1896............

97............

98............

99............

1900............

01............

02............

03............

04............

1905............

0.6............

07............

08............

09............

1910............

11............

12............

13............

14............

1915............

16............

17............

18............

19............

1920............

21............

22............

23............

24............

1925............

OC

27............

(2) 

Annual flow 

(in thousands 

of acre-feet)

10,089 

18,009 

13.815 

15,874

13,228 

13,582 

9,393 

14,807 

15,645

16,027 

19,121 

23,402 

12,856 

23,275

14,248 

16,028 

20,520 

14,473 

21,222

14,027 

19,201 

24,037 

15,364 

12,462

21,951 

23,015 

18,305 

18,269 

14,201

13,033 

15,853 

18,616

(3) . 

Deviation 

from mean 

(in thousands 

of acre -feet)

-5,091 

+2,829 

-1,365 

+694

-1,952 

-1.598 

-5,787 

-373 

+465

+847 

+3,941 

+8,222 

-2,324 

+8,095

-932 

+848 

+5,340 

-707 

+6,042

-1.153 

+4,021 

+8.857 

+184 

-2,718

+6,771 

+7,835 

+3,125 

+3.089 

-979

-2,147 

+673 

+3.436

(4)

Square of 

deviation 

from mean 

(nx 10">)

2,590 

801 

186 

48

381 

255 

3,340 

13 

22

72 

1.550 

6,840 

540 

6,550

87 

72 

2,860 

50 

3.650

133 

1,615 

7,840 

3 

735

4,590 

6,060 

974 

954 

94

461 

45 

1,180

(1)

Water 

year1

1928..........

1929..........

1930..........

31..........

32..........

33..........

34..........

1935..........

36..........

37..........

38..........

39..........

1940..........

41..........

42..........

43..........

44..........

1945..........

46..........

47..........

48..........

49..........

1950..........

51..........

52..........

53..........

54..........

1955..........

56..........

(2) 

Annual flow 

(in thousands 

of acre -feet)

17,279 

21,428

14,885 

7,769 

17,243 

11,356 

5,640

11,549 

13,800 

13,740 

17,545 

11,075

8,601 

18,148 

19,125 

13,103 

15,154

13,410 

10,426 

15,473 

15,613 

16.376

12,894 

11,647 

20,290 

10,670 

7,900

9,150 

10,720

925,957

(3) 

Deviation 

from mean 

(in thousands 

of acre -feet)

+2,099 

+6,248

-295 

-7,411 

+2,063 

-3,824 

-9,540

-3,631 

-1,380 

-1,440 

+2.365 

-4,105

-6,579 

+2,968 

+3,945 

-2,077 

-26

-1,770 

-4,754 

+293 

+433 

+1,196

-2,286 

-3,533 

+5,110 

-4,510 

-7,280

-6,030 

-4,460

(4) 

Square of 

deviation 

from mean 

(n x 1010 )

438 

3,900

9 

5,480 

425 

1,461 

9,100

1,318 

190 

207 

560 

1,685

4,330 

884 

1,555 

430 

0

314 

2,260 

9 

19 

143

522 

1,250 

2.610 

2,035 

5,300

3,640 

1,990

106,655

1 12 months ending September 30 of year shown. 

VARIABILITY

Probability analysis is in essence an analysis of the 

variability of a sample. A streamflow record represents a time sample out of an indefinitely long time 

period. Therefore, more information can be obtained 

if the data from that record are treated as other 

sampling data.

To determine the characteristics of any large population by taking a sample, the most obvious parameter 

indicated by the sample is the mean value. Of equal 

significance is the spread or dispersion of individual 

values about the arithmetic mean. In streamflow, for 

example, the annual discharge values include a few 

exceptionally large ones, a few small ones, and a preponderance of discharges centered around some 

central value. The distribution of annual discharges 

is shown graphically by a histogram that shows the

number of years in which the discharge falls into different categories of size.

When the distribution of sizes or quantities in a population follows the so-called normal law, the histogram 

will present the shape of the normal distribution. A 

smooth curve drawn through the points on the histogram will result in a bell-shaped curve, such as is 

shown in figure 1. A normally distributed population 

is; by definition, one whose histogram can be approximated by the bell-shaped curve illustrated.

Normal distributions have certain standard characteristics. One is that the arithmetic mean of the values 

should be identical with the mode, which is that category having the largest number of cases. This characteristic will be present only when the bell-shaped 

curve is symmetrical. The symmetry of the bellshaped curve is direct evidence that 50 percent of the
VARIABILITY

cases have values higher than the mean, and 50 percent have values lower than the mean.

By the same reasoning, a characteristic of a 

normal distribution is that a spread of values may 

be defined within which 50 percent of the individual 

values fall. Such a spread is defined as one probable deviation on each side of the mean. This 

spread is is indicated on the graph of figure 1.

For the same reason a given percentage of the total 

cases will fall within the limits defined by two probable deviations on either side of the mean. It is a 

characteristic of normally distributed populations 

that 82 percent of the total number of cases will lie 

within 2 probable deviations on either side of the 

mean, and that 96 percent of the cases will lie 

within the limits of 3 probable deviations on either 

side of the mean.

SIZE OR QUANTITY

o z.

LU

z> 

o

HI

o: 

u_

' p.d.- -p. d.- -p.d. - -p.d.- -p.d- -p.d.-

NOTE: _

p. d. = probable deviation

 50 ' 

percent

82 percent 

96 percent

p. d. - probable deviation

Arithmetic average (mean)* size class having largest number of cases (mode)

Figure 1. The normal distribution: The frequency of occurrence of various sizes or quantities.
PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

The bell-shaped curve of a normal distribution is 

asymtotic to the abscissa or horizontal coordinate; 

that is, the two tails of the bell-shaped curve gradually approach, but do not reach, the horizontal 

line of the graph.

The probable deviation may be computed for a 

normally distributed population in a simple manner. 

One probable deviation equals 0.6745 times a quantity 

called the standard deviation. The standard deviations 

may be computed by the following formula:

Standard deviation*

where 3t is the difference between the value of an 

individual measurement and the mean of all the 

measurements in a sample, and n is the number 

of measurements in the sample.

An example of the computation of the standard 

deviation is given in table 1. Column 2 in table 1 

shows the annual flow in acre-feet at Lees Ferry 

for each water year. The arithmetic mean of 

column 2 is computed to be 15,180,000 acre-feet. 

Deviations from the mean shown in column 3 are 

merely the difference between the individual annual 

discharge values and the above-mentioned mean. 

Column 4 is a tabulation of the square of the deviations or the square of each value in column 3. 

The probable deviation is indicated as the product 

of the factor 0.6745 times the standard deviation, or

Probable deviation = 0.6745 x 4.22 = 2.84 million 

acre-feet

By definition, therefore, one probable deviation 

on either side of the mean would include 50 percent of all the values listed in column 2. That is, 

50 percent of the values in column 2 lie within the 

limits of 18.02 and 12.34 million acre-feet. The data 

in column 2 will verify this statement approximately.

2The sums (table 1) of the annual flow, in thousands 

of acre-feet, and the square of deviation from the 

mean are 925,957 and 106,655 x 10 10 , respectively. The 

mean is derived by dividing the sum of the annual flow 

by the total number of years from 1896 1956, thus

Mean- 925,957. 15.130 million acre-feet. 

61

The computation of the standard deviation is as 

follows:

Standard deviation ;

by substituting values for symbols, the equation then 

simplifies to

106.655x1018- A777xlOio ; 

61-1

therefore.

Standard deviation- 4.22 million acre-feet.

Probability analysis is greatly simlified where the 

data in the sample are distributed in a normal manner.3 

That the data shown in table 1 are normally distributed 

will be shown, after a discussion of the use of probability paper for plotting.

PROBABILITY PLOTTING

For ease of analysis, the bell-shaped graph of a 

normal distribution can be plotted in a somewhat different manner by accumulating progressively the 

number of cases equal to, or less than, any particular 

value. Such a cumulative curve is shown in figure 2A. 

For the vertical scale, probable deviations on either 

side of the mean are used. Note that, as indicated 

above, in defining a normal distribution 50 percent of 

the cases are greater than the mean and^therefore, 

the value of 50 percent on the abscissa corresponds to 

0 on the ordinate scale. One probable deviation below 

the mean should also correspond with 25 percent of 

the cases, inasmuch as the spread between the mean 

and one probable deviation below the mean must consist of 25 percent of the total values. Thus, the 25- 

percent point on the abscissa corresponds to -1 probable deviation on the ordinate; and, similarly, 75 percent on the abscissa corresponds to one probable 

deviation above the mean on the ordinate. It was also 

stated that one probable deviation on either side of the 

mean will include 50 percent of the total cases; on 

figure 2 the difference between 75 percent and 25 percent is 50 percent of the total. Also note that in figure 

2A the cumulative distribution approaches but does 

not reach abscissa values of 0 and 100.

In order to further simplify the cumulative frequency 

distribution curve, the abscissa scale can be changed 

by spreading out the values in such a way that the Sshaped line of figure 2A becomes a straight line, as 

shown in figure 2B. The expanded horizontal scale 

should be such that the characteristics of a normal 

distribution are fulfilled. As an example, 82 percent 

of the cases should lie within two probable deviations 

on either side of the mean. Thus, the value of -2 on 

the ordinate scale will appear at an abscissa value of 

9 percent, and +2 on the ordinate scale will appear 

opposite 91 percent on the abscissa.

The abscissa scale so constructed that a normal 

cumulative frequency distribution will plot as a straight 

line is, by definition, a probability scale. Graph paper 

in which the probability scale is printed as the abscissa 

is widely used and can be purchased at most 

engineering-supply stores.

To test whether a series of data is normally distributed, the values may be arranged in order of magnitude 

and plotted on probability paper. Table 2 lists the 

annual flows of the Colorado River in order of magnitude, beginning with the largest. If such data aline

3The annual discharges of all rivers are not normally 

distributed. In general, however, the annual flow of the 

larger streams, and those in humid regions tends toward normality.
100

PERCENTAGE OF CASES OR PROBABILITY

NOTE:

By adjusting the scale, a normal 

distribution plots as a straight line

If measurement data such as acrefeet are plotted on ordinafe, then 

the slope of the line is a measure 

of the variability

Figure 2. The cumulative frequency distribution of a normal population.
6 PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

themselves approximately in a straight line on probability paper, the values in the sample may be considered to be normally distributed. In making such a graph, 

the abscissa position of any individual value is obtained 

by using the formula

Plotting position i

in which m is the rank of the individual number in the 

array and n is the total number of cases in the sample. 

The plotting positions for data in table 2 are computed 

by this formula and shown in column 3. In table 2, 1 

year of 61 years constitutes 1.6 percent of the total 

sample. Having arranged the 61 values in order of 

magnitude, as shown in column 2, the computed plotting 

position places each point in abscissa positions 1.6 

percent apart.

The data in table 2, plotted on arithmetic probability 

paper, are shown in figure 3, where the 61 points aline 

themselves in a reasonable approximation to a straight 

line, as illustrated by a line drawn to conform with 

the points.

As indicated previously (p. 4), the probable deviation 

is computed to be 2.84 million acre-feet; 2.84 million 

acre-feet on either side of the mean (fig. 3) should include 50 percent of the total values in table 2. The 

graph in figure 3 confirms this computation. The 

ordinate value of 12.34 corresponds to an abscissa 

value of 25 percent and 18.02 corresponds to an abscissa value of 75 percent.

The value of one probable deviation can be obtained 

by reading the difference between the ordinates corresponding to 50 percent and 25 percent on the plotted 

graph, without going through the numerical computation 

shown on page 4.

Not all hydrologic data are normally distributed. 

Such data should be transformed to use a statistical 

term to provide a series of values that are normally 

distributed. In some cases, though the individual values 

are not normally distributed, the logarithms of those 

values will be normally distributed. The handling of 

log-normal distributions is beyond the scope of this 

report.

Table 2. Computation of plotting position in probability analysis, annual flows of Colorado River at Lees Ferry

(1)

Serial

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 ...........

(2) 

Annual flow in order 

of magnitude 

(thousands of acre -feet)

24,037

23,402

23,275

23.015

21.951

21.428

21.222

20.520

20,290

19,201

19.125

19.121

18.616

18.305

18.269

18.148

18,009

17,545

17,279

17,243

16,376

16.028

16.027

15.874

15.853

15,645

15.613

15,473

15,364

15,154

14.885

(3) 

Plotting position 

(probability)

98.4

96.1

95.2

93.6

92.0

90.4

88.8

017 *

85.5

83.8

82.2

80.6

79.0

77.4

75.8

74 2

72.5

70.9

69.3

67.7

66.1

64.5

62.9

61.3

59.7

58.1

56.4

54.8

53.2

51.6

50.0

(1) 

Serial

32 ...........

33 ...........

34 ...........

35 ...........

36 ...........

^7

38 ...........

QQ

40 ...........

41 ...........

42 ...........

43 ...........

44 ...........

45 ...........

46 ...........

47 ...........

48 ...........

49 ...........

50 ...........

51 ...........

52 ...........

53 ...........

54 ...........

55 ...........

56 ...........

57 ...........

58 ...........

59 ...........

60 ...........

,61 ...........

(2) 

Annual flow in order 

of magnitude 

(thousands of acre-feet)

14.807

14.473

14.248

14,201

14,027

13.815

13,800

19 740

13,582

13.410

13.228

13.103

13.033

12.894

12.856

12.462

11.647

11,549

11,356

11,075

10,720

10,670

10.426

10.089

9,393

9,150

8,601

7,900

7.769

5,640

(3) 

Plotting position 

(probability)

48.4

46.8

45.2

43.6

42.0

40.4

38.7

37.1

35.4

33.8

32.2

30.6

29.0

27.4

25.8

24.2

22.5

21.0

19.3

17.7

16.2

14.4

12.8

11.2

9.6

8.0

6.4

4.8

3.2

1.6
PROBABILITY PLOTTING

25

hLU

UJ

UJ

a: 

o 20

z: 

o

15

10

ID 

Z 

Z

i

I 10 25 50 75 90 99 

PROBABILITY, IN PERCENTAGE, OF FLOWS 

LESS THAN INDICATED

Figure 3. Cumulative distribution curve. Colorado River at Lees Ferry, 61 years '1896 1956.

In data obtained by sampling, where the individual 

values in the sample are normally distributed, the 

means of groups of data in the sample will also be 

normally distributed. For example, in a streamflow 

record the means of 10-year periods may be computed, and the values of 10-year means may be 

treated as items in another sample. In a 61 -year 

record only 6 independent 10 -year means may be 

computed a relatively small sample. However, the 

characteristics of the whole population may be 

approximated by the characteristics of a sample 

because the "standard error of the mean" of the 

sample is a close approximation to the standard 

deviation of the means of other samples from the 

same population. The standard error of the mean 

is

§

"x Standard deviation M» *^ ~ "^ * ^ * ~ ~ f

where 11 is the number of items comprising the 

sample.

The standard error of the 61-year mean is computed 

to be

S y- *- 22. = 0.54 million acre-feet. ~" V6T 

The probable error is

0.6745 x 0.54 a 0.364 million acre-feet.

This figure can be interpreted in two ways: (a) as a 

50-percent chance that the 61-year mean of record lies 

within 0.364 million acre-feet of the true mean of an 

indefinite series of years or (b)as the probable deviation 

of an indefinitely large number of 61-year means.

However, this calculation is based on an assumption 

that the data occur in random order a requirement 

not actually met in hydrologic data. That is, neither 

the individual annual values nor the means of other 

successive periods occur in random order. This lack 

of random sequence is explained below and a method 

of correcting for it is discussed.
PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

EFFECT OF PERSISTENCE IN HYDROLOGIC DATA

Experience in our daily lives verifies the fact that 

rainy days occur together and dry days occur together. 

For similar meteorological reasons wet years tend to 

occur in groups and dry years similarly occur together. 

This tendency for grouping is called persistence. It is 

clear that if the means of groups that included a nonrandom assortment of individuals were computed, the 

spread or deviation among the means would be greater 

than if the groups consisted of a random selection of 

individuals. As an example, imagine measurements of 

the mean height of 10-men groups on a college campus 

Suppose the groups were made up by a random process 

so that each individual group of 10 consisted of some 

short men and some tall ones. In contrast, suppose 

that one of the 10-man groups consisted of the football 

team, another the basketball team, a third the coxswains of the crew, and so forth. The mean height of 

the basketball players would be larger than the mean 

of 10 randomly selected individuals. The mean of the 

coxswains would be smaller than the mean of a randomly selected group. Thus, the variation among the 

means of 10-men groups would be larger in the team 

groups the nonrandomly selected groups than in the 

randomly selected ones. Likewise, the spread of 

means of groups consisting of wet years and those 

consisting of dry years would be greater than if the 

groups consisted of randomly ordered individual years.

That the variability of groups of streamflows in 

their natural order of occurrence is actually larger 

than if the same flow values occurred in random sequence was sharply brought to the attention of the 

engineering profession by a distinguished British

engineer, H. E. Hurst (1950). By working with the longest record of river stage in the world, the 1,050 years 

of recorded stage of the Nile at the Roda gage, Hurst 

obtained evidence that the tendency for wet years to 

occur together and dry years together increased variability of means of various periods. Other scientists 

confirmed this tendency with independent data.

Some records which show this effect are presented 

in table 3. These samples include some of the longest 

stream-discharge records in existence. Column 5 

shows the standard deviation of the means of annual 

discharges for natural 5-year groups, for example, 

1901 05, 1906 10, and so forth. Column 7 shows 

standard deviation of 10-year means in the same records, such as 1901 10, 1911-20, etc. Similarly, 

columns 9 and 11 are for 15- and 20-year groups.

Column 3 shows the standard deviation of annual 

discharge values. The standard deviation of annual 

values is unaffected by sequence; and, thus, the variability of annual values, expressed by the standard 

deviation, can be used as a yardstick against which the 

variability of means of 5-year, 10-year, and other 

periods may be compared. For this reason the ratio 

of variability of 5-year means to 1-year values is a 

factor that is independent of length of record, and 

various streams can be compared by their ratios as 

shown in column 6. Similarly, the ratio of standard 

deviation of 10-year means to that of annual values 

appears in columns 8, 10, and 12.

The data in table 3 were plotted to derive the dashed 

curve of figure 4. Ordinate values of individual points 

through which the dashed curve was, drawn are not

LJ

2

100

80

.± 60

2 u- 

< O

5 H 40 2 5

d < 20

Is

> o

Weight of evidence of 

all long-term records 

of stream flow, United 

States and Europe

Colorado River data

I Random sequence

I 2 4 5 10 20 50 

LENGTH OF RECORD, IN YEARS

Figure 4. Variability of mean values of streamflow for records of various lengths.

100 200
PROBABILITY PLOTTING

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10 PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

shown; the ordinates are, for example, values shown 

in column 6, table 3, and a corresponding abscissa of 

5 years. For Niagara River at Niagara Falls, for 

example, 5-year means (abscissa value of 5) have a 

ratio of 0.74 or 74 percent of the variability of the 

annual values in the same record. The dashed curve 

was extended beyond the 20-year abscissa to follow a 

smooth logarithmic line with slope of -0.35.

The actual values of the Colorado River at Lees 

Ferry (table 3) are plotted as circles in figure 4. For 

example, in 61 years of record, the 12 five-year means 

had a standard deviation, which was 64 percent of the 

standard deviation of the 61 annual values. Thus the 

cross at abscissa value of 5 has an ordinate position 

of 64 percent.

The circles representing Colorado River data plot 

consistently with the dashed line representing the data 

for the other rivers in table 2. The above results can 

be compared with similar results from a group of data 

that are not only normally or randomly distributed about 

their mean, but are randomly ordered as well. If group 

means are comprised of randomly chosen individuals, 

then the variability of these group means would decrease 

as the square root of the number of individuals making 

up the groups. Thus, if annual values of streamflow 

were randomly ordered occurred in random sequence 

variability of means of groups would decrease inversely 

as the square root of the number of years comprising 

the group. Thus, the means of 100-year groups would

be -JL or as variable as 1-year values. 

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25 50 75 100 125 150

LENGTH OF PERIOD, IN YEARS

175

NOTE:

This graph shows that the variability of lOO-year means 

of streamflow Is the same as that of 25-year means if 

the flows occurred in random sequence.

Figure 5. The effect of grouping tendency in streamflow data.
CONFIDENCE IN ESTIMATE OF FUTURE VARIABILITY 11

Randomly ordered data are described by the solid 

line in figure 4. Relative to the variability of 1-year 

values, means of 100-year records would be VlO as 

variable. Thus, the solid line goes through the ordinate 

value of 10 percent for an abscissa value of 100 years.

The difference between the variability of naturally 

occurring groups of streamflow data and the same data 

randomly ordered is defined by the difference between 

the dashed line and solid line of figure 4. For periods 

of equal length (for example, 5-year group averages) 

the variability (ordinate value) is larger in naturally 

ordered data than the same values randomly ordered.

The same idea can be presented in another way, as 

shown on figure 5. An actual record of 100 years has 

about the same variability as means randomly ordered 

in a 25-year record. Similarly, the means of 200 years 

of actual records have about the same variability as 

the means of 40 years if these discharges occurred in 

random order rather than in their natural order. A 

100-year record is, therefore, required to estimate 

the mean flow with the same confidence as could be 

estimated from 25 years of record were the flows to 

occur in random order. After a record exceeds 100 

years in length (fig. 5), the estimate of the long-term 

mean improves at a decreasing rate that is, the effectiveness of a record of 200 years is but 20 percent 

better than that of ah estimate based on TOO years of 

record.

To summarize, the tendency toward persistence in 

meteorologic and climatic events leads to a larger 

variability of mean values than if the same events 

occurred in random order. A group of values, such as 

annual discharge quantities, may be normally distributed 

about their mean value but occur in nonrandom sequence. The greater variability caused by this linkage 

is similar in data from rivers of widely different size 

and type; and as shown below, the similarity between 

rivers provides a means of correcting for this linkage in 

making probability analyses of any given streamflow 

record.

PROBABLE VARIATION AMONG MEANS OF 

FUTURE SAMPLES

Variability is defined as the dispersion or spread of 

values about their mean. Thus, in figure 3 the slope of 

the line on the graph is a measure of the variability in 

the sample data. Figure 4 defines the variability of 

means of groups of various sizes (periods of years) 

in terms of the variability of annual streamflow 

quantities. This ratio, read as percentage on the 

ordinate of figure 4, can be used to express the slope 

of a line representing the probable dispersion of means 

of future samples. The percentage is merely multiplied by the slope of the line showing distribution of 

annual values to obtain the slope of a line representing 

the dispersion of means of 5-year, 10-year, or some 

other period.

The dispersion of 10-year means and 61-year means 

is indicated by appropriate lines on figure 6 as examples. For comparison, the dispersion of the annual 

values is also shown. The position and slope of the 

line representing annual values is the same as that 

shown in figure 3. It will be recalled that the slope of 

that line can be expressed by two points. One point is

the mean plotted at 50-percent probability. The second 

point is the mean, 15.18 million acre-feet, plus the 

probable deviation, 2.84 million acre-feet, plotted at 

75-percent probability.

To obtain the slope of the line representing 10-year 

means, the 10= year abscissa in figure 4 is read from 

the dashed curve representing actually ordered or 

linked hydrologic data; a value of 44 percent is indicated. The slope would be determined as follows: 

0.44 x 2.84 - 1.25 million acre-feet

(probable deviation)

Thus, if the mean for the 61 years is the true mean, 

75 percent of 10-year means would be expected to be 

equal to, or less than,

15.18 + 1.25s 16.43 million acre-feet. 

In figure 6, the 10-year line has an ordinate value of 

16.43 at an abscissa value of 75 percent.

Similarly, the line representing the dispersion of 

means of 61-year periods will have an ordinate value 

of

15.18 + (0.24 x 2.84) = 15.86

corresponding to an abscissa value of 75 percent. This 

can be interpreted as follows: Assuming the 61-year 

mean to be the true mean, three-fourths of the discharge 

values representing means of 61-year periods will be 

equal to or less than 15.86 million acre-feet. By the 

same token, one-fourth of the means of future 61-year 

periods will be equal to or less than

15.18 - (0.24 x 2.84) . 14.5 million acre-feet. 

Throughout this discussion it will be understood that 

the computations are for the reconstructed record of 

virgin flow-at Lees Ferry and the actual runoff will 

be less than these figures owing to upstream depletion.

CONFIDENCE IN ESTIMATE OF FUTURE 

VARIABILITY

In figure 6 shows the most probable distribution of 

means of future 10-year and 61-year periods, assuming 

the 61-year mean to be the true mean. The phrase 

"most probable" implies that the true variation, which 

actually will be experienced, may be somewhat different. This is reasonable because an estimate made 

from a sample would be unusual if it were a perfect 

expression of the whole population. Probability theory 

allows an objective estimate of the expected deviation 

of any sample or group of samples from the true 

attributes of the whole population. In the previous 

analysis, the sample mean was assumed to be the same 

as the true mean. The probability that the sample 

mean may differ from the true mean is considered 

below.

The objective estimate of sampling differences is 

called the confidence limit or confidence band. Confidence limits are derived from the characteristic of 

normal distributions already employed, so that the 

standard error of the mean of a sample is an approximation of the standard deviation of the means of many 

samples of equal size.

The mean value of the 61-year sample of annual discharge values is 15.18 million acre-feet. The probable 

error (p.e.) of this mean is equal to

0.24 x 2.84 0.68 million acre-feet. 

Stated more simply, there is a 50 percent chance that 

the true mean of the whole population (indefinitely long
12 PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

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Probability of future /neons, taking 

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of hydro logic events

25 50 75 95 99

PROBABILITY, IN PERCENTAGE, OF FLOWS 

LESS THAN INDICATED

Figure 6. Probable distribution of mean discharge values for periods of various lengths, Colorado River

at Lees Ferry.

period of time) lies within one probable deviation on 

either side of the mean of the sample, or

15.18+ 0.68s 14.50-15.86 million acre-feet. 

Thus, the line representing the most probable distribution of means of future 61-year periods drawn on 

figure 6 could lie in a slightly upward or downward 

position on the chart. There is a 50 percent chance 

that its true position is within a spread upward or 

downward of the mean by an amount equal to 0.24 probable deviation or 0.68 million acre-feet.

As defined earlier, there is an 82-percent chance 

that a limit of twice this value above or below the 

sample mean would include the true mean of the whole 

population. In such a manner bands expressing any

desired confidence could be defined. The present 

discussion deals with only one of these various possible confidence limits that represents a 50-percent 

chance.

The 50-percent confidence band has been drawn on 

figure 7 as a pair of parallel lines lying 0.68 million 

acre-feet above and below the mean at the abscissa 

value of 50 percent. The slope of the parallel lines 

was determined previously; that is, the ordinate value 

at 75-percent probability is 0.68 million acre-feet 

higher than the ordinate at 50-percent probability. 

Thus, the confidence-limit lines are parallel to the 

line on figure 6 that represented the most probable 

dispersion of means of future 61-year periods.
"PROBABLE VALUE OF MEAN FLOW IN NEXT 61-YEAR PERIOD 13

O

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One-year 

flows,

5O~percent chance that 

true distribution of 

61-year means lies 

within these limits

25 50 75 95 99

PROBABILITY, IN PERCENTAGE,OF FLOWS 

LESS THAN INDICATED

Figure 7. Variability of 61-year means, Colorado River at Lees Ferry.

A more elegant but only slightly different construction of confidence limits yields curved lines rather 

than the parallel straight lines of figure 7.

Because a sample yields only an estimate of value 

of the true mean of the whole population and only an 

estimate of the dispersion of the means of other 

samples drawn from the same population, there is a 

50-percent chance that the line representing the actual 

distribution of means of future samples may be anywhere within the band defined by the confidence limits. 

Two possible positions of the line representing such a 

distribution are shown by the dashed lines within the 

confidence band shown on figure 7. An infinite number 

of such possible positions exist, having various slopes 

within the limit of the confidence band and having 

various vertical positions within that band. Any of

these possible positions are equally probable; therefore, 

according to the assumptions used in figure 7, there 

is also a 50-percent chance that the distribution of 

future 61-year means will lie outside of the confidence 

band drawn.

PROBABLE VALUE OF MEAN FLOW IN NEXT 

61-YEAR PERIOD

When the probable value of the mean of some prospective period in the future, such as the next time 

period, is discussed, then it must be considered that 

the period will also be a sample with the same probability of variation from the true mean as was the 

sample already available. The variation from the true 

value of the mean of the available sample must be
14 PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

coupled with the variation in the next sample to obtain 

the total possible variation from the true, but unknown, 

value applicable to the whole population.

These errors used in a statistical sense are not 

added together but combined as the square root of the 

sums of their respective squares.

To present a practical example, in a water-supply 

problem the engineer is interested in the probability 

of the mean discharge of flae next period of time being 

higher or lower than the mean value during the period 

of available record. He is particularly interested in 

estimating the probability of a lower value of streamflow in the next period than he observed in the last one, 

because in water-supply problems, deficient flow can 

be critical. In the case of the 61-year period of the 

Colorado River at Lees Ferry, compute the lowest 

value that the mean of the next 61 years is likely to be 

in 1 chance out of 4. Inasmuch as the variability of 

means of particular future 61-year periods relative to 

the variability of annual data from a 61-year sample 

equals the statistical sum of the variability of the 

sample and the future periods,

\/(0.24) 2 + (0.24)2 = 0.34.

The width of the 50-percent-confidence band is, thus,

0.34 x 2.84 = 0.96 million acre-feet 

above and below the mean already experienced in the 

available sample. Therefore, there is a 50-percent 

chance that the next 61-year mean lies between 

15.1810.96 million acre-feet or between 16.14 and 

14.22 million acre-feet. Thus, it can be said that 

there is a 25-percent chance, or 1 chance in 4, that 

the mean of the next 61 years would be less than 14.22 

million acre-feet.

By a similar procedure the variability means of 

future 10-year periods can be computed and the lower 

limit for the probable mean value of the next 10-year 

period can be found. In this case, the combined 

variability would be

\/(0.24)2 + (0.44)2 = 0.50.

There is a 50-percent chance that the mean flow 

during a specific 10-year period, such as the next 

10-year period, will be

15.18t (0.5 x.2.84) = 16.60-13.76

million acre-feet.

There is also an 82-percent chance that the next 

10-year mean will lie within two probable deviations 

from the sample mean or

15.18t (2 x 0.5 x 2.84) = 18.02-12.34

million acre-feet.

Therefore, it can be stated that there is a 9-percent 

chance (1 out of 11) that the next 10-year mean will be 

less than 12.34 million acre-feet.

With regard to the record of the Colorado River at 

Lees Ferry, an inquiry could be made as to whether 

the means actually experienced during the driest 

10-year period of record greatly exceeded reasonable 

expectations. The lowest 10-year period was the 

decade 1931-40, with a mean of 11.83 million acrefeet. This discharge was only slightly below that for 

the 9-percent probability. Thus, it may be concluded 

that the lowest 10-year mean in the 61-year period at 

Lees Ferry might have been expected in 1 chance out 

of 11 a reasonable probability of occurrence.

EFFECT OF STORAGE ON STREAMFLOW 

VARIABILITY

Variability of discharge is an inherent characteristic 

of rivers. Storage reservoirs are devised by man to 

make variable river flows match his needs for water; 

that is, water-supply reservoirs are built to hold over 

water from wet periods in order that it may be discharged during dry ones. Reservoir storage, therefore, 

is merely a feasible way for man to reduce the natural 

variability of stream discharge.

In figure 8, the line having the greatest slope is the 

most probable distribution of 10-year means and is 

identical with the 10-year line of figure 6; it may be 

considered to represent what nature has provided. The 

line having the lesser slope (fig. 8) is arbitrarily 

drawn on the graph to indicate the lower variability, 

which man desires to achieve by reservoir storage. 

The hatched area between the two lines is a quantitative 

measure of the regulation achieved by storage. Absolutely even flow would be represented in figure 8 by 

the horizontal dashed line, but it is theoretically and 

physically impossible to achieve a uniform outflow 

because, as indicated on figure 4, even long periods, 

say 200 years, have means which are likely to vary by 

a considerable amount. Thus, if successive 200-year 

periods have considerably different mean values of 

flow, a relatively large amount of reservoir storage 

would have to be built to hold over water, which 

occurred as high flows in one 200-year period, to 

supply low periods in a succeeding 200 -year period of 

relative dryness.

By the same reasoning, each additional increment of 

storage capacity yields a smaller and smaller increment of actual flow regulation; that this is the actual 

experience with reservoirs built and operated in the 

United States is shown in figure 9 (Langbein, 1959). 

The ordinate of this chart represents the present regulation, in which regulation is defined as the average of 

the total increments to storage that occurred on an 

annual basis. Regulation on this graph is expressed as 

a percentage of the mean annual flow. It would be 

physically impossible to add, on the average, more 

water to storage than the average annual flow of the 

stream. For this reason, ordinate values cannot 

possibly exceed 100 percent of the mean annual flow.

The abscissa scale represents the storage capacity, 

expressing capacity as a ratio to mean annual flow. 

Each point represents reservoir data tabulated in 

table 4. The reservoirs listed in table 4 were chosen 

to represent a reasonable sample of different kinds of 

reservoirs and have a variety of storage capacities in 

terms of the annual flow of the respective streams. 

Reservoirs built exclusively for flood control are not 

included. The point representing Lake Mead, for 

example, is plotted at an abscissa value of about 2 inasmuch as the 29 million acre-feet of usable storage 

in that reservoir represents approximately 2 times 

the annual mean discharge of the river. The ordinate 

value for Lake Mead represents the average annual 

increment to storage actually experienced since the 

reservoir was built. It will be noted that the operating 

experience at Lake Mead, represented by a point on 

figure 9, fits reasonably well with the points for other 

reservoirs plotted on this chart.
REFERENCES 

Table 4. Capacity and regulation of some representative reservoirs

15

Reservoir and State

Usable capacity

Acre-feet 2

Detention

period 

(years) 3

Mean annual regulation 1

Acre-feet 

per year

Ratio to 

capacity

Ratio to

mean annual

flow

Piney, Pa............................................ 13,000

Great Falls, Tenn................................. 49,400

Ocoee No. 1, Tenn................................ 33;000

Claytor, Va......................................... 100,000

Mascoma Lake, N. H............................. 7,744

Franklin D. Roosevelt Lake, Wash........... 5,072,000

West Fork Bitterroot, Mont.................... 31,700

Hiwassee, N. C.................................... 1,376,000

Green Mountain, Colo............................ 146,900

Gibraltar, Calif.................................... 7,731

Stillwater, N. Y.................................... 106,000

Sacandaga, N. Y................................... 762,300

First and Second Connecticut Lakes, N. H. 88,106

Norris, Tenn....................................... 2,281,000

Shasta Lake, Calif................................ 4,377,000

Lake Alamanor, Calif............................ 649,800

Salmon River Canal Co., Idaho................ 182,650

Henrys Lake, Idaho .............................. 79,351

Lake Mead, Ariz.-Nev........................... 27,207,000

Lake Mead plus Lake Mohave................. 29,000,000

Fort Phantom Hill, Tex......................... 69,500

Lake Kickapoo, Tex.............................. 106,000

Elephant Butte, N. Mex.......................... 2,185,000

Elephant Butte plus Caballo, N. Mex........ 2,526,000

Quabbin, Mass..................................... 1,279,000

San Carlos, Ariz.................................. 1,205,000

Lake Henshaw, Calif............................. 194,320

0.011

.021

.035

.04

.05

.07

.14

.265

.34

.39

.40

.51

.60

.72

.80

1.0

1.6

2.1

2.1

2.3

2.3

2.6

2.6

3.0

6.0

6.7

21.8

70,000

204,000

204,000

150,000

22,000

4,800,000

26.000

330,000

111,000

4,300

91,000

560,000

62.000

1,070,000

1,530,000

250,000

57,500

20,600

45,750,000

46,500,000

11,900

20,200

4 375,000

4 470,000

119,000

4 117,000

8,170

5.4

4.1

3.6

1.5

2.8

.95

.82

.90

.75

.55

.86

.75

.70

.47

.35

.38

.31

.26

.21

.22

.175

.19

.17

.185

.09

.097

.042

0.06

.089

.13

.06

.14

.07

.12

.24

.26

.21

.34

.38

.42

.34

.28

.38

.54

.54

.44

.51

.40

.49

.45

.55

.56

.65

.91

1 For reservoirs with monthly detention period greater than 0.1 year, regulation was computed from monthly 

changes in reservoir contents. Daily data were used for smaller reservoirs. 

2Thomas and Harbeck, 1956. 

3Ratio of usable capacity to mean annual flow. 

Including evaporation losses.

The smooth line representing the general experience 

of reservoir operation has been drawn through the 

points; it approaches but does not reach the ordinate 

value of 100 percent, as has been explained above. 

Thus, both theoretically and from actual operating 

experience, complete regulation yielding the mean 

annual flow of the stream is impossible to achieve. 

The smooth line in figure 9 confirms that successive 

increments of reservoir capacity add increasingly 

smaller increments to regulation.

In plotting figure 9, evaporation losses from the 

reservoirs have been computed and included in the 

annual regulation. If the regulation less evaporation 

losses were computed, the smooth curve drawn in 

figure 9 would lie at a lower value and become asymptotic or even fall away from the horizontal line where 

large values of reservoir capacity are shown.

By applying the generalized experience indicated by 

the smooth curve in figure 9 to the Colorado River, 

ordinate values have been read off the smooth curve 

and used to compute the increments of regulation that 

would be attained by different assumed reservoir 

capacities constructed in the upper Colorado River 

basin. The results of this computation, again including

evaporation losses as a part of the annual regulation, 

define the solid line in figure 10. Evaporation losses 

subtracted from ordinate values yield net regulation 

as defined by the dashed line. It can be seen from this 

dashed line that total reservoir capacity in excess of 

about 40,000,000 acre-feet would achieve practically 

no additional water regulation if evaporation loss is 

subtracted from annual regulation. Thus, generalized 

experience with representative reservoirs in the 

United States indicates that if reservoirs with capacity 

beyond an additional 10 million to 15 million acre-feet 

are constructed in the upper Colorado River basin, 

evaporation loss will thereafter offset the hydrologic 

benefit of the regulation so achieved.

REFERENCES

Hurst, H. E., 1950, Long-term storage capacity of 

reservoirs: Am. Soc. Civil Engrs. Proc., v. 76, 

Separate 11.

Langbein, W. B., 1959, Water yield and reservoir storage in the United States: U. S-. Geol. Survey Circ. 409.

Thomas, N. O., and Harbeck, G. E. Jr., 1956,

Reservoirs in the United States: U. S. Geol. Survey 

Water-Supply Paper 1360-A.
16 PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

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Absolute even flow 

cannot be achieved*

nature 

provides

What man 

desires

'Crosshatched area 

denotes regulation 

by storage

'/0-year means, Colorado 

River at Lees Ferry

I_____I _____I

15 25 50 75 95 99 

PROBABILITY, IN PERCENTAGE, OF 10-YEAR MEANS

Figure 8. The decrease of variability by storage.
PROBABIUTY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM 17

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18 PROBABILITY ANALYSIS APPLIED TO A WATER-SUPPLY PROBLEM

1.00

TOTAL RESERVOIR CAPACITY,

IN MILLIONS OF ACRE-FEET

20 30 40 50 60 70

.80

zf 

g

33

le

z _

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.20

I 1 I 1 1 1

Present 

main-stem 

development

» Annual regulation without 

evaporation loss

Evaporation loss

regulation

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TOTAL RESERVOIR CAPACITY, 

IN RATIO TO MEAN FLOW

NOTE: For computation, historic mean flow taken as 13 million acre-feet, 

representing depleted flow at Lees Ferry> evaporation /oss 

assumed 27,OOO acre-feet per year per million acre-feet of 

reservoir capacity.

Figure 10. The effect of various amounts of storage capacity on flow regulation, Colorado River basin.

* U.S. GOVERNMENT PRINTING OFFICE : 1967 O 272-010