2409.26b,20
Page 1 of 14
FOREST SERVICE HANDBOOK
PORTLAND, OREGON
2409.26b - REFORESTATION HANDBOOK
R-6 Amendment No. 2409.26b-92-4
Effective February 21, 1992
POSTING NOTICE. Amendments to this handbook are numbered consecutively.
Check the last transmittal sheet received for this handbook to see that the above
amendment number is in sequence. If not, obtain intervening amendment(s) at
once from the Information Center. Do not post this supplement until the missing
one(s) is received and posted. After posting, place the transmittal at the front of the
title and retain until the first transmittal of the next calendar year is received.
The last Amendment to this handbook was 2409.26b-92-3 (!2409.26b
Contents).
Page Code
Chapter 10 (Entire Chapter)
Superseded New
(Number of Sheets)
20
Document Name
2409.26b,20
14
Digest:
Chapter 20 - This Chapter updates the procedures suggested for reforestaion
surveys. Current Chapter 10 to be rewritten and entitled Site Analysis and Plans.
JOHN F. BUTRUILLE
Regional Forester
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 2 of 14
FSH 2409.26b - REFORESTATION HANDBOOK
R-6 AMENDMENT 2409.26b-92-4
EFFECTIVE 2/21/92
CHAPTER 20 - REFORESTATION SURVEYS
20
REFORESTATION SURVEYS
21
ACREAGE SURVEYS
21.1
21.2
22
22.1
22.11
22.12
22.2
23
23.1
23.2
24
24.1
24.2
24.3
25
25.1
25.2
Standards
Determining Acreage from Aerial Photos
PREPLANTING SURVEYS
Procedures
Presale Reconnaissance Surveys
Planting Contract Preparation Surveys
Records
STOCKING SURVEYS
Standards
Procedures
SURVIVAL AND GROWTH SURVEYS
Standards
Procedures
Records
STATISTICAL BASIS
Sample Size
Reading References
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 3 of 14
20 - REFORESTATION SURVEYS. Surveys are done to determine the need for
planting or seeding, and to monitor the progress of new stands toward a certifiable
condition. A decision to replant may also result from such surveys. When surveys
reveal that a stand has progressed in age and stocking to the prescribed level, the
stand can be certified as restocked.
21 - ACREAGE SURVEYS. Acreage is used for many purposes, including plans,
estimates, contracts, payments, records, and reports. Needs are similar regardless
of treatment.
21.1 - Standards. The acreage in a project should be determined with reasonable
accuracy. This is essential for contract purposes, in fairness to the contractor and
the Government. In addition, accurate acreage determination is mandatory for
proper planning of project time, supplies, and materials.
The allowable error in determination of acreage for contract purposes is 5 percent.
The standard of accuracy for settling disputes will be the measurement of acreage
as determined by a traverse with a closing error of not more than one-half of 1
percent of the total traversed distance. This will be considered as 100 percent from
which the percentage variations of 5 percent as listed above are computed. A
traverse which meets the standard of one-half of 1 percent of the total traversed
distance is normally accomplished by using a Suunto (brand name) hand compass
(or a more accurate instrument), measuring tape, and a clinometer.
Example: A traverse around a section is 21,120 feet.
The determination of acreage in a contract area should be accomplished by the least
costly method to arrive at an acceptable contract acreage. It is not the intent here
to require all acreages to be determined by a traverse. As an example, the acreage
of a clearcut unit may have already been surveyed to the above standards so
another survey is not needed. In addition, there are numerous traverse computer
programs available for calculating acreage and preparing maps.
Notes and calculations used to determine acreages for contracts should be
maintained in Ranger District files until the contract is closed.
Acreages obtained from aerial photographs, using methods described in 21.2, will be
acceptable. These methods have an accuracy similar to a staff compass survey and
may be much cheaper. However, if there is a serious question or dispute as to the
true acreage, the traverse as defined above, should be used.
21.2 - Determining Acreage from Aerial Photos. Among the requirements for
successful aerial photo area measurements are accurate determination of photo
scale and clear definition of the boundaries of the measurement area on the aerial
photo. Examples of clearly defined boundaries may be sharp ridges, drainages,
roads, power lines, and timber cutting lines.
1. Most Ranger Districts have ortho photos. These photos are distortion free
and may be used for acreage determination with no adjustment for error.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 4 of 14
2. Electronic planimeters, if available, are the preferred system for measuring
acreage. They are very fast and accurate.
3. Acreage of an area may be measured on an aerial photo by use of a dot grid.
The 64-dot per square inch is satisfactory. The acres per dot for the specific scale of
the aerial photo will have to be calculated. The formula for determining acres per
dot with 64 dots per square inch grid is:
Area Per Dot = (Photo Scale Reciprocal/10,000)
4
For example: Photo Scale = 1:13,200
AD = (1.32)2 = 0.436
4
If a more intensive grid is wanted, the 64-dot grid may be repeatedly placed on the
area in a random fashion until the desired number of dots are counted. Four
successive counts with a 64-dot grid would be the equivalent of using a 256-dot grid.
Where areas are measured directly on photos, there may be some errors due to
topographic displacement unless the topography is relatively flat. Photographs
should be selected on which the area measured lies closest to the center of the
photo. This practice helps to lower the error.
When an area has a topographical displacement with slopes above 30 percent, photo
area should be transferred to a 4-inch to the mile base map with a radial line plotter
before the area is measured. This procedure should also be used when photo scale
cannot be accurately determined.
22 - PREPLANTING SURVEYS. Preplanting surveys should predict plantability to
a reasonable accuracy. In practice, preplanting survey information is used to fit
contract specifications to ground conditions. They may also be used to predict
plantability before existing commercial timber is cut. Since contract specifications
are necessarily written to cover ground conditions in a broad-brush fashion, a great
deal of statistical accuracy is not required of preplanting surveys. While a
preplanting survey of some type is recommended for all units to be reforested, a
Regional standard is not specified.
22.1 - Procedures.
22.11 - Presale Reconnaissance Surveys. Presale reconnaissance surveys are used
to prescribe the regeneration system to be used. These may be part of a stand
exam. The following is a partial list of the type of information needed:
1. Soil depth, texture, and type.
2. Vegetative type.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 5 of 14
3. Slope and aspect.
4. Depth of duff.
5. Animal damage potential.
6. Frost damage potential.
7. Physical plantability. The tool which is to be used for planting is sometimes
used to check this.
Many of these variables may be rated individually using numerical values. The
numerical value may then be combined into an overall index which indicates the
effort necessary to assure reforestation success. For example, soil depth, soil
texture, percent rock, percent slope, and exposure are physical variables that may
be rated using the following scales:
Variable 1 - % Slope
0 - 10
1
11 - 20
2
21 - 30
3
31 - 40
4
41 - 50
5
51 - 60
6
61 - 70
7
71 - 80
8
81 - 90
9
91 -100 10
Variable 3 - Soil Depth
37" - 40" 1
33" - 36" 2
29" - 32" 3
25" - 28" 4
21" - 24" 5
17" - 20" 6
13" - 16" 7
9" - 12"
8
5" - 8"
9
0" - 4"
10
Variable 2 - % Rock
0 - 10
1
11 - 20
2
21 - 30
3
31 - 40
4
41 - 50
5
51 - 60
6
61 - 70
7
71 - 80
8
81 - 90
9
91 -100
10
Variable 4 - Soil Texture
Sandy Loam 1
Loam
2
Silt Loam
3
Silt
4
Sandy Clay
5
Silty Clay
6
Clay Loam
7
Clay
8
Sand
9
Gravel
10
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
Variable 5
N
Flat
NE
NW
E
W
SE
S
SW
2409.26b,20
Page 6 of 14
Exposure
1
2
3
4
5
6
7
8
10
The individual scores may then be combined into an overall index with predicted
management constraints similar to the table below:
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 7 of 14
Plantability Index
Index
Numerical Scale
Management Constraints
Good
5 – 15
Planting may proceed with no
difficulties expected.
Fair
16 – 25
Minor problems
failure not likely.
Poor
26 – 40
Energetic quality control is
needed.
Seed source may be
desirable.
Very Poor
41 – 50
Potential failure due to planting
difficulty is probable, seed source
is highly recommended.
possible,
but
As originally conceived, this index included a rating based on plant communities
and species to be planted. While not applicable Region-wide, it is recommended
that Forests develop a scale for plant communities or associations to be combined
into this type index. It will allow for the differing silvical characteristics of the
seedling species.
22.12 - Planting Contract Preparation Surveys. Contract preparation surveys are
becoming increasingly important due to changes in site preparation, logging
methods, and silvicultural prescriptions. These items change the specifications
needed for planting to accomplish the objective of establishing the desired number
of trees per acre. Some of the problems occurring during the administration of a
planting contract stem from an inadequate preplanting survey.
The primary purpose of a preplanting survey is to obtain enough information about
an area to specify contract clauses that meet ground conditions. The survey should
be made when ground conditions are similar to those expected at the time work is to
be done. For instance, when trees are to be planted on broadcast-burned or
mechanically-terraced areas, the survey should be made after the burning or
terracing. Sometimes ground conditions from one area to another are very similar.
Examples are recently burned clearcuts or wildfires, or mechanically prepared
areas. In these cases a survey of a representative area may be all that is required.
The survey should be made before preparing either force account work plans or
contracts. Sufficient lead time is essential in both instances, especially when
contracting to ensure the appropriate specifications are incorporated in the
contract.
Occasionally, actual work may be delayed following an initial survey. During this
lapse of time, ground cover conditions may change to the extent that specifications
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 8 of 14
used in the first survey will not result in satisfactory work. A second survey using
revised specifications should be performed if necessary.
Preplanting surveys should be made on preconceived specifications for the specific
project, or they should be designed to inventory the elements necessary to obtain
the correct specifications for contracting or force account planting. The preplanting
survey should obtain information on the following:
- Species of trees to plant.
- Seedling specifications.
- Number of planting spots per plot for the average spacing or percent
plantable area.
- Depth of duff.
- Size of slash to be moved.
- Depth of slash to be moved.
- Species of vegetation.
- Density of vegetation.
- Soil depth.
- Soil texture.
- Percent slope.
- Percent rock.
- Recommended planting method.
- Potential frost pockets.
- Potential for animal damage.
- Residual stocking distribution.
A large portion of this information should be determined during the presale
examination and recorded on the Reforestation Activity Record (R6-2400-91).
However, when it is not, or if there is a need to verify the information, the data may
be divided between what should be determined by the plot and what should be
determined for the unit or a portion of the unit. The above items may be divided as
follows:
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 9 of 14
Item Numbers
Plot
Portion of Unit
Unit
3, 4, 5, 6, 12
1, 2, 7, 8, 11, 14, 15
1, 2, 7, 8, 9, 10, 11, 13, 14, 15
This is not a complete list of items to be inventoried on a preplanting survey. A
specific note system for preplanting surveys has not been designed.
A
recommended method of data collection using the Reforestation Stocking Survey
form, much of which should have been gathered in the NEPA documentation and
silvicultural prescription, is this. Survey the following:
- Species to plant.
- Seedling specifications.
- Vegetation type.
- Soil depth and textures.
- Slope and aspect.
- Planting method.
- Animal damage potential.
- Frost potential.
Next, prepare a list of items to be surveyed on each plot. An example of the list may
be as follows:
- Number of trees on plot.
- Percent plantable or number of planting spots.
- Number of crop trees.
- Depth of duff.
- Depth of slash.
- Animals present (gopher mounds, rabbit pellets, mountain beaver evidence,
big game trails).
- Vegetation present or scalping required.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 10 of 14
Last, prepare a list for the variables and change the column headings on the
reforestation survey form. Record the coded information for each plot as the survey
is taken.
22.2 - Records. Field notes from the presale survey will be used in making the
silvicultural prescription. Site data that is not recorded in the written silvicultural
prescription should be recorded on the Reforestation Activity Record.
Field notes from the contract preparation survey will supplement those from the
presale survey on the Reforestation Activity Record. They should be tied into an
existing database, such as ORACLE or the District GIS system.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 11 of 14
23 - STOCKING SURVEYS.
23.1 - Standards. Surveys are done to determine the need for planting or seeding,
and to monitor the progress of new stands toward a certifiable condition. A decision
to replant may also result from such a survey. When surveys reveal that a stand
has progressed in age and stocking to a point where silviculturists can agree that no
further reforestation work will be necessary, and the stand complies with the
reforestation prescription and Forest stocking standards, the stand should be
certified as reforested (reference NFMA). Stands will henceforth be described
as "certified" rather than as "established." This usually occurs after the 3rd
year stocking survey exam. Extensive or "walk through" surveys may be used at
local discretion, but certification must be based on a survey done to an acceptable
standard, similar to the survival and growth surveys.
23.2 - Procedures. Determining the proportion of the total area that is stocked is
the primary objective of stocking surveys. Survey results are judged against
previously stated or prescribed standards to determine the need for further
treatment, or for certification.
There are several methods of taking stocking surveys. The following three will be
discussed in this Chapter. They are the staked point, the plot count, and the
stocked quadrate. As they are all discussed adequately in existing literature, they
will not be discussed in detail here.
The staked point method, offers the advantage of insight gained from repeated
examination of the same tree or plot. It is a good method of evaluating the results
of a reforestation practice. Staking trees and repeated examination require costly
effort and without supplemental sampling, staked point methods evaluate only the
results for marked trees. (This method is discussed further in section 24, Survival
and Growth Surveys.)
The plot count method, involves making a total count of trees on a series of
sample plots, determining the average number of trees per plot, and expanding this
average to an acre basis. If an adequate sample is taken, a reliable figure for trees
per acre may be obtained. This is meaningful information in an older stand where
competition has produced some uniformity of distribution. In young plantations,
however, the number of trees per acre tells little about spatial distribution.
The stocked quadrat method, is the most widely used method in the western
United States. While the word quadrat implies a square shaped plot, the use of
circular plots is also acceptable and is easier to measure in the field. The system is
based on the assumption that if an area is divided into plots of such a size that one
seedling will fully stock the plot at prescribed stocking levels, then the percentage of
stocked plots indicates the portion of land being utilized by tree growth. For proper
application, it is important to recognize that the system was designed to
automatically compare actual stocking with a prescribed stocking level, and that
the sample plot size must be related to the prescribed stocking level.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 12 of 14
If the plot size is not consistent with the prescribed stocking level, one of two
estimating errors may occur. If too large a plot is used, the estimate of stocking is
likely to be high since a large plot is more likely to be stocked. A plot is classified as
stocked if one or more countable trees are found, and nonstocked if no countable
trees are found. A single stocking percentage is calculated from this data that
represents the percentage of the unit that is stocked. For example, a series of
stocked 1/200-acre plots indicates 100 percent stocking and 200 trees per acre. If
400 trees per acre are prescribed, the area is only 50 percent stocked. If too small a
plot is used the number of stocked plots may be adequate but tree distribution may
still be in doubt. For example, 50 percent stocking of 1/1000-acre plots assures 500
trees per acre, but large gaps may occur in their distribution if only one plot in two
has stocking present.
The arrangement of plots and sample lines involves a compromise between
efficiency and sampling accuracy. The random sample is recommended by
statisticians. Systematic samples on a preplanned grid are the most prevalent
system used by foresters. The more nearly square the grid, the more accurate the
sample. In general, practice plot lines may be two to five chains apart and the plots
equally spaced at intervals along the lines (section 25, Statistical Basis). The two to
five chain distance is a workable distance within which most of the unsampled
detail of the unit may be observed as the surveyor progresses along the sample grid,
and is determined on a site-by-site basis.
In practice, more information than tree stocking is desirable. Supplemental data
may be efficiently gathered as the survey is being taken. (For example, tree species
growth, trees per plot, vegetative species and density, animal impacts, and diseases
present may be systematically gathered and recorded along with the stocking
information.) Form R6-FS-2400-109 (Selected Tree Register) is a versatile form
that can be used unchanged or adapted to record stocking survey information for
most situations. A Forest designated form may also be used. An aerial photo, or a
sketch map, should be utilized as the survey progresses to help in relocating
problem areas in the future.
24 - SURVIVAL AND GROWTH SURVEYS.
24.1 - Standards. Regional consistency in making these surveys is critical because
the data will be used in performance analysis of Forest Supervisors. Also, survival
and "first time success" data will be used by the Chief to evaluate the reforestation
performance of the Regional Forester. Survival sampling error must be + 10
percent at the 95 percent confidence level as determined by staked trees (FSM
2472.4). Each major seedling species on each Ranger District will be sampled.
24.2 - Procedures. Survival and growth surveys are an essential part of any
reforestation program. They are used to gather information concerning the
effectiveness of techniques used in reforestation. Therefore, it is important to take
a critical look at the need for information that a survey should produce. After
clearly identifying the objective and determining what is already known about the
site, the type of survey may be determined. The sample should truly represent the
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 13 of 14
population being sampled. A survey to determine plantation survival must include
samples from all variables represented that affect survival. For example, if only
southerly aspects are sampled the results are likely to be biased.
The following type of survey is commonly used to measure survival and growth.
Staked tree. Staked trees are the preferred system used to compare the survival
and growth of trees. A system for measuring survival and growth of seedlings is
described below:
1. During, or very soon after planting, 50 trees are staked in a unit using
consecutively numbered, highly visible, and durable stakes.
2. A reasonable effort is made to locate this staked row so that it represents
the planting unit in which it is located. Because each stake must be relocated in the
future a sketch map showing the starting point and direction of travel may be
helpful. It is expected that the staked row should traverse across a site to represent
the entire range of elevations and aspects on the site.
3. Enough staked rows or staked trees are established to proportionately
represent the major variables affecting plantation survival and growth. (See section
25, Statistical Basis.)
4. At the end of the first growing season, the staked row is ready for survival
and growth. The tree is tallied as dead or alive; if living, the terminal leader is
measured. If all, or part, of the terminal is missing, the causative agent is noted
and the length is not measured.
5. For each staked row a survival rate may be determined, which roughly
represents the unit in which the row is located. The same is true for average leader
growth. If the data shows high survival and can be verified by a cursory exam of
the rest of the plantation, it may not be necessary to conduct any additional surveys
of the unit for stocking. However, this type of information has no statistical basis
for individual units.
6. Using staked rows as paired plots may be very useful in assessing the
effectiveness of special techniques. By staking trees in treated areas and in
untreated areas, the effectiveness of animal repellants, for instance, may be tested.
Good principles of experimental design must be used if this information is to be
useful.
24.3 - Records. Survival and growth data should be summarized by major planting
species for each Ranger District and reported yearly. Survival will be reported by
expressing the number of live trees as a percentage of the total trees sampled.
Growth will be expressed as the average growth per tree to the nearest centimeter.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 14 of 14
25 - STATISTICAL BASIS.
25.1 - Sample Size. The size of a sample for a specified level of precision is
dependent upon the inherent variation in the population. The more homogeneous
the population, the smaller the sample needed to determine population estimates
for a given level of precision.
1. Plot count sampling method. The variance of the stocking level within a
unit is a measure of the dispersion of individual plot stocking levels around the
mean plot stocking levels. The estimate of this variance is given by:
_
S2 = E(Xi - X)2 = EX2- (EX)2
n-1
n
n-1
Where: S2 = Sample estimate of the population variance.
Xi = The value of the ith unit in the sample.
X = The arithmetic mean of the sample.
n = The number of observations taken.
This formula may be illustrated by a sample calculation. Suppose we observe 7
trees, 8 trees, and 12 trees on three separate plots. The estimate of the variance
would then be:
S2 = (72 + 82 + 122) - (27)2
3
= 257 - 243 = 7
2
2
The Standard Deviation of this sample would be the square root of the variance
or 7 = 2.6458.
There is a method for estimating sample size based on the coefficient of variation,
S/X and the relative accuracy with which the mean is to be estimated. How many
observations does one need to be reasonably sure that the true mean will be found
within a given distance of the sample mean? The equation to use is:
n = Nt2 + C2
_____________
NA + t2C2
Where:
n = Required number of plots.
C = Coefficient of variation, that is, the ratio of the
standard deviation to the mean (S/X) expressed as a
percent.
t = Tabular "Student's t" value of the desired confidence
limit.
A = Allowable error percent.
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
2409.26b,20
Page 15 of 14
N = Maximum number of plots. (Acres/plot size in acres.)
In theory, the t value is dependent on the sample size (that is, n-1 degrees of
freedom), which is what we are trying to find. In practice it is acceptable to use t =
1.0 for a 67 percent confidence limit, t = 2.0 for a 95 percent confidence limit and t =
3.0 for a 99 percent confidence limit. If the value of C is not known, as is often the
case, here are three acceptable ways of estimating C.
a. Use the value of C as obtained from other analysis of similar
materials.
b. Use S (sample standard deviation) from previous analyses, estimate
X (sample mean) expected in the analysis, and calculate C where C =
(S/X)100.
c. Estimate the highest (H) and lowest (L) observation which might be
expected. Then compute C where C = ((H - L)/(2 (H + L))100.
Example: Using the data from the example in 25.1a variability, the coefficient of
variation is calculated to be C = (S/X)100 = (2.6458/9)100 = 29 percent. How many
additional plots should be taken to estimate the number of trees per acre in the
same area to within 10 percent of it's true mean with a 95 percent confidence limit
if the plot size is 1/50 acre and the unit is 30 acres in size.
Answer:
n = Nt2C2 = (1500)(2)2 (29)2
= 5046000 = 32.9
2
2
2
2
2
2
NA + t C
(1500)(10) + (2) (29)
153364
To obtain the desired accuracy with the desired degree of confidence would require
33 plots. Since we already have three we need an additional 30 plots.
2. Staked row sampling method. If each staked row is considered to be a
plot, the same methods as used for the count plot sample size may be used. When
using the staked row sampling method to estimate average growth, the growth per
tree is used in the formula for variance instead of tree counts per plot. The answer
will indicate how many staked rows are necessary to estimate average growth per
tree.
If the staked row is used to measure survival, the previous method will call for more
samples than are necessary for a given reliability.
A better formula to use for staked row survival or for the stocked quadrate, where
each tree is the plot and is dead or alive is:
n = t2 pq
A2
R-6 AMENDMENT 2409.26-92-4
EFFECTIVE 2/21/92
Where:
t
limit.
p
q
A
2409.26b,20
Page 16 of 14
= Tabular "Students t" value of the desired confidence
= An advanced estimate of survival percent.
= 100 minus p.
= Allowable error percent.
The term "t" and "A" are the same as previously defined.
Example: How many staked trees (or quadrats) must be observed to estimate the
percent survival of a plantation that is estimated to have a survival of 75 percent?
The sample must estimate survival within 15 percent to a 95 percent confidence
interval.
Answer: n = t2 pq = (2)2(75) (25)
A2
152
= 33.3
There is no adjustment for the size of the tract in this formula. The finite
population correction factor (N-n) is so close to 1 that it is
N-1
essentially meaningless and omitted. For tracts greater than 30 to 40 acres in size
it may be advisable to sample them as two or more smaller blocks.
25.2 - Reading References.
Freeze, Frank. 1967. Elementary Statistical Methods for Foresters.
Agriculture Handbook 317, U.S.D.A. Forest Service. 87pp.
Freeze, Frank. 1962. Elementary Forest Sampling. Agriculture
Handbook 232, U.S.D.A. Forest Service. 91 pp.
Cleary, Brian D., Greaves, Robert D., and Herman, Richard K. 1978.
Regenerating Oregon's Forests. Oregon State University Extension Service.
282 pp.
Freund, John E. 1984. Modern Elementary Statistics. Prentice Hall,
In., Sixth Edition, EngleWood Cliffs, New Jersey. 561 pp.