生物统计分析第五版：关键值表概览

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《生物统计分析第五版》(J.H. Zar)是一本专注于生物统计学的权威教材。书中包含了丰富的统计表格和图形，这对于理解和应用各种统计方法至关重要。这些表格主要用于辅助计算在生物研究中常见的假设检验，如正态分布、卡方检验、t检验、F检验、q检验、Mann-Whitney U检验、Wilcoxon T检验、Kruskal-Wallis H检验、Friedman chi-square检验以及非参数多重比较测试等。 1. **统计表与图形：** 表格B.1至B.16提供了关键的临界值，例如Chi-squared（χ²）分布的临界值，这在评估数据的拟合度和显著性时极为有用。对于连续性和分组数据的正态分布比例、t分布、F分布、q分布（用于Tukey检验、Dunnett's Test）、Kolmogorov-Smirnov检验（包括单尾和双尾检验，以及修正后的版本）以及非参数检验的统计量如Mann-Whitney U、Wilcoxon T和Kruskal-Wallis H的临界值，都有详尽的数据支持。 2. **插值与估计：** 部分章节可能涉及数据缺失或不完全的情况，这些表格可以帮助研究者通过插值方法估算未知的统计参数，确保分析的准确性和完整性。 3. **实际应用：** 对于生物学家、医学研究者、临床试验分析师以及其他领域依赖统计分析的专业人员来说，这些表格是进行假设检验、确定显著性水平、解释结果并做出科学决策的重要工具。理解如何利用这些临界值可以提高研究的有效性和可靠性。 4. **非参数检验：** 非参数统计方法在数据分布不明确或者不符合特定假设时尤为实用，如Table B.15和B.16中的Q值，对于处理不同条件间的多个样本均值比较提供了非参数替代方案。《Biostatistical Analysis 5th》一书不仅包含了基础的统计理论，还提供了大量实用的数值资源，帮助读者在实际工作中进行精准的统计推断和决策。熟练掌握和运用这些表格，对于提升科学研究的质量和效率具有重要意义。

Acknowledgments

A book

this nature requires, and benefits from, the assistance

many.

For

the

preparation

the early editions, the outstanding library collections

the University

Illinois at

Urbana-Champaign

were invaluable, and for all editions I am greatly

indebted to the library materials and services of

Northern

Illinois University

and

its vast book- and journal-collection networks. I also gratefully acknowledge the

cooperation

the computer services at

Northern

lIIinois University. which assisted

in executing many

the

computer

programs 1 prepared to generate some

the

appendix tables.

For

the tables taken from previously published sources. thanks arc

given for the permission to reprint them.

and

acknowledgment

each source

given

immediately following the appearance

the reprinted material. Additionally, I am

pleased

recognize the editorial and production staff

Pearson Education and

Laserwords for their valued professional assistance

transforming my manuscript

into the published product in hand.

Over

many years, teachers, students,

and

colleagues have aided in leading me to

important biostatistical questions and to

the

material presented in this volume. Special

recognition must he made

Charles Kendeigh (1904-1986)

the University

Illinois at

Urbana-Champaign.

who. through considerate mentorship. first made

aware

the value

quantitative analysis

biological data, leading me to produce

the first edition; Edward Batschelet

(1914-1979).

the University

Zurich, who

provided me with kind encouragement

and

inspiration on statistical matters during

the preparation

much

the first two editions;

Arthur

Ghent

(1927-2001),

University

Illinois at

Urbana-Champaign.

who was constantly supportive through

the first four editions. offering statistical

and

biological commentary both stimulating

and challenging; and Carol J. Feltz

(1956-2001). Northern lllinois University. who

provided substantive consultation

some

major new material for the fourth edition.

Prior to publication, the book drew

upon

the expertise and insights

reviewers,

including Raid Amin. University

West Florida; Franklin R. Ampy, Howard

University: Sulekha Anand. San Jose

State

University: Roxanne Barrows, Hocking

College; Thomas Beitinger, University

North Texas; Mark Butler. Old Dominion

University: William Caire, University

Central Oklahoma;

Gary

Cobbs, University

of Louisville; Loveday

L. Conquest, University

Washington;

Todd

A. Crowl,

Utah

State University; Matthew Edwards. University

California, Santa Cruz;

Todd

Fearer, University

Arkansas-Monticello; Avshalom Gamliel, Wavemetrics, Inc.;

Peter

Homann, Western Washington University;

Robert

Huber,

Bowling

Green

State

University; James W. Kirchner. University

California, Berkeley;

Gary

A. Lamberti,

University

Notre Dame: David Larsen, University

Missouri, Columbia: David

R. McConville, Saint Mary's University

Minnesota; J. Kelly McCoy, Angelo State

University; David J. Moriarty. California

State

Polytechnic University; Mark Rizzardi,

Humboldt State University; Michael

A. Romano, Western Illinois University; and

Thomas

A. Wolosz, the State University

New York, Plattsburgh. I also acknowledge

my wife. Carol, for

her

prolonged

and

consistent patience during the thirty-five years

production

the five editions

this

book.

Jerrold H.

Zar

DeKalb, Illinois

xiii

CHAPTER

Data: Types and Presentation

1.1

TYPES

BIOLOGICAL

DATA

1.2

ACCURACY

AND

SIGNIFICANT

FIGURES

1.3

FREQUENCY

DISTRIBUTIONS

1.4

CUMULATIVE

FREQUENCY

DISTRIBUTIONS

Scientific study involves the systematic collection, organization. analysis. and presen-

tation

knowledge. Many investigations

the hiological sciences are quantitative.

where knowledge

the form

numerical ohservations called data.

(One

numerical

observation

a dawl11.*)

order

for the presentation and analysis

data to be

valid and useful,

must use methods appropriate to the type

data obtained. to

the design

the data collection. and

the questions asked

the data: and the

limitations

the data.

the data collection. and

the data analysis should be

appreciated when formulating conclusions. This chapter. and those that follow.

will

introduce many concepts relevant to this goal.

The

word statistics

derived from the Latin for

"state."

indicating the historical

importance

governmental data gathering, which related principally

demographic

information (including census data and "vital statistics") and often to their use

military recruitment and tax collecting.

The

term statistics

often encountered as a synonym for

data:

One

hears

college

enrollment statistics (such as the numbers

newly admitted students. numbers

senior students, numbers

students from various geographic locations). statistics

a basketball game (such as how many points were scored by each player. how many

fouls were committed). lahor statistics (such as numbers

workers unemployed.

numbers employed

various occupations). and so on. Hereafter. this usc of the word

statistics

will

not

appear

in this hook. Instead,

will be used

its

other

common

manner: to refer to the orderly collection, analysis.

and

interpretatiol1

(~f

data with

a view

to objective evaluatioll

conclusions based

Oil

the data. (Section 2.4

will

introduce another fundamentally important use

the term statistic.)

Statistics applied

biological problems

simply called biostatistics or, sometimes.

biometry* (the latter term literally meaning "biological measurement"). Although

*The IeI'm

dllfll

sometimes seen as a singular noun meaning

"numcrical

information." This

hook

rdrains

from that usc.

t Peters

(llJ~7:

79) and Walker (llJ29: 32) attrihute the first use

the term Sflllistin to a

German

professor. Gottfried Achenwall (1719-1772). who uscd the

German

word Swti.wik

74lJ.

and the

lirst puhlished use

the English word to

John

Sinclair

(1754-1~35)

1791.

*The word

biollletry. which literally means "biological

measurement."

had. since

the

nineteenth

century. hcen found

sl.!veral contexts (such as demographics and. later. yuantitative genetics:

Armitage. 1

9~S:

Stigler.

2(K)O).

but using

mean the application

statistical methods

niological

information apparently was conceived hetween

IX92

and

1901

by Karl Pearson. along with the name

Biollletrika for the still·important English journal he helped found: and

was first

publi~hed

the

inaugural

is!\uc

this journ,,1

IlJO}

(Sncdccor. 1954).

The

Biometrics Section

the Amcric,m

4 Chapter 1 Data: Types

and

Presentation

their magnitudes relative to each other:

success

learning to run a maze may be

recorded as

A. B.

often true that biological data expressed on the ordinal scale could have been

expressed on the interval

ratio scale had exact measurements been obtained (or

obtainable). Sometimes data that were originally on interval

ratio scales will be

changed to ranks: for example, examination grades of99. 85. 73. and 66% (ratio scale)

might be recorded as A,

C, and D (ordinal scale), respectively.

Ordinal-scale data contain and convey less information than ratio

interval data,

for only relative magnitudes are known. Consequently. quantitative comparisons are

impossible (e.g., we cannot speak

a grade

C being half as good as a grade

the difference between cell sizes I and 2 being the same as the difference

between sizes 3 and

4). However. we

will

see that many useful statistical procedures

are,

fact. applicable to ordinal data.

(d)

Data

in Nominal Categories. Sometimes the variable being studied

classified

some qualitative measure it possesses rather than by a numerical measurement.

In such cases the variable may be called an attribute, and we are said to be dealing

with nominal,

categorical. data. Genetic phenotypes are commonly encountered

biological attributes: The possible manifestations

an animal's eye color might be

brown

blue: and if human hair color were the attribute

interest,

might

record black. brown. blond,

red. As other examples

nominal data (nominal

from the Latin word for

"name"),

people might be classified as male

female,

right-handed

left-handed. Or, plants might be classified as dead

alive,

as with

without fertilizer application. Taxonomic categories also form a nominal

classification scheme (for example, plants

a study might be classified as pine, spruce.

fir).

Sometimes. data that might have been expressed on an ordinal. interval,

ratio

scale

measurement may be recorded

nominal categories. For example. heights

might be recorded as tall

short.

performance on an examination as pass

fail.

where there

an arbitrary cut-off point on the measurement scale to separate tall

from short and pass from fail.

As will be seen. statistical methods useful with ratio. interval.

ordinal data

generally are not applicable to nominal data. and

must. therefore. be able to

identify such situations when they occur.

(e) Continuous

and

Discrete Data. When

spoke previously

plant heights.

were dealing with a variable that could be any conceivable value within any observed

range; this

referred to as a continuous variable. That

is.

measure a height

and a height

36 cm, an infinite number of heights

possible in the range

from

to 36

cm:

a plant might be 35.07

tall

35.988 cm tall,

35.3263 cm tall,

and so on. although,

course,

not have devices sensitive enough to detect this

infinity

heights. A continuous variable

one for which there

a possible value

between any

other

two values.

However, when speaking

the number

leaves on a plant,

are dealing

with a variable that can take on only certain values. It might be possible to observe

leaves.

28 leaves, but 27.43 leaves and 27.9 leaves are values

the variable

that are impossible to obtain. Such a variable

termed a discrete

discontinuous

variable (also known as a meristic variable). The number

white blood cells in 1

blood. the number

giraffes visiting a water hole. and the number

eggs laid

by a grasshopper are all discrete variables.

The

possible values

a discrete variable

generally are consecutive integers. but this is not necessarily so. If the leaves on

our

plants are always formed

pairs, then only even integers are possible values

the

variable.

And

the ratio

number

wings to number

legs

insects

a discrete

variable that may only have the value

0,0.3333

...•

0.6666

...

(i.e.,

respectively). *

Ratio-, interval-, and ordinal-scale data may be either continuous

discrete.

Nominal-scale data by their nature are discrete.

1.2

ACCURACY

AND

SIGNIFICANT

FIGURES

Accuracy

the nearness

a measurement to the true value

the variable being

measured.

Precision

not a synonymous term but refers to the closeness to each

other

repeated measurements

the same quantity. Figure

1.1

illustrates the difference

between accuracy and precision

measurements .

•

••

•••

2 3 4

(a)

•

••

•••

·r··,

(c)

5 6 kg 0

•

••••••••

I I I

234

(b)

•

•••••••

I I I

2 3 4 5

(d)

6 kg

FIGURE

1.1: Accuracy and precision

measurements. A 3-kilogram animal

weighed

10 times. The 10

measurements shown in sample (a) are relatively accurate and precise; those in sample (b) are relatively

accurate

but

not

precise; those

sample

(c)

are relatively precise

but

not

accurate; and those

sample

Cd)

are relatively inaccurate and imprecise.

Human

error

may exist

the recording

data.

For

example. a person may

miscount the number

birds in a tract

land

misread the numbers

a heart-

rate monitor.

Or,

a person might obtain correct data

but

record them in such a way

(perhaps with poor handwriting) that a subsequent

data

analyst makes an

error

reading them. We shall assume that such errors have not occurred, but there are

other

aspects

accuracy that should he considered.

Accuracy

measurement can be expressed in numerical reporting.

we report

that the hind leg

a frog

8 cm long, we are stating the

number

8 (a value

continuous variable) as an estimate

the frog's true leg length. This estimate was

made using some sort

a measuring device.

Had

the device been capable

accuracy, we might have declared that the leg was 8.3 em long,

perhaps 8.32 em

long. When recording values

continuous variables. it

important to designate the

accuracy with which the measurements have been made.

convention, the value

8 denotes a measurement

the range

7.50000

...

to 8.49999

...

, the value 8.3

designates a range

8.25000

...

to 8.34999

...•

and the value 8.32 implies that the

true value lies within the range

8.31500

...

to 8.32499

....

That

is,

the reported

value

the midpoint

the implied range. and the size

this range

designated

by the last decimal place

the measurement.

The

value

8 cm implies an ability to

*The ellipsis marks (

...

) may be read as

"and

so on." Here. they indicate that

and

are

repeating decimal fractions. which could just as well have been written as 0.3333333333333

...

and

0.6666666666666

....

respectively.

6 Chapter 1 Data: Types and Presentation

determine length within a range

1 cm. 8.3 cm implies a range

0.1

cm. and 8.32 cm

implies a range

0.01

cm. Thus. to record a value

8.0 implies

greater

accuracy

measurement

than

does the recording

a value

8, for in the first instance the

true value

said to lie between 7.95000

...

and 8.049999

...

(i.e., within a range

0.1

cm). whereas 8 implies a value between 7.50000

...

and

8.49999

...

(i.e

.•

within a

range

I cm).

state

8.00 cm implies a

measurement

that ascertains the frog's limb

length

be between 7.99500

...

and

8.00499

...

cm (i.e

within a range

0.01

cm).

Those

digits in a

number

that

denote

the accuracy

the measurement are referred

as significant figures. Thus. 8 has

one

significant figure, 8.0

and

8.3 each have two

significant figures, and 8.00 and 8.32 each have three.

In working with exact values

discrete variables. the preceding considerations

not apply.

That

is. it

sufficient to state that

our

frog has four limbs

that its

left lung contains thirteen flukes.

The

use

4.0

13.00 would

inappropriate. for

as the numbers involved are exactly 4

and

13.

there

question

accuracy

significant figures.

But there are instances where significant figures

and

implied accuracy come

into

play with discrete data.

entomologist may report

that

there

are

72,000 moths in

a particular forest area. In doing so. it is probably not being claimed

that

this is the

exact

number

but an estimate

the exact number.

perhaps

accurate

two significant

figures. In such a case. 72,000 would imply a range

accuracy

1000. so that the true

value might lie anywhere from 71,500 to 72,500.

the

entomologist wished to convey

the fact that this estimate is believed

be accurate to the nearest 100 (i.e ..

three

significant figures),

rather

than

to the nearest 1000, it would be

better

to present the

data

in the form

scientific l1otation,* as follows: If the

number

7.2 x

(

= 72.000)

written, a range

accuracy

0.1 x

1000)

implied.

and

the true value

is assumed to lie between 71,500 and 72,500. But if 7.20

1(}4

were written. a range

accuracy

0.01

(

= 100) would be implied,

and

the true value would be

assumed to be in the range

71,950 to 72,050. Thus. the accuracy

large values

(and

this applies

continuous as well as discrete variables) can be expressed succinctly

using scientific notation.

Calculators

and

computers typically yield results with more significant figures than

are justified by the data. However. it

good

practice-to

avoid rounding

error-to

retain many significant figures until the last

step

in a sequence

calculations.

and

attaining the result

the

final

step

round

off to the

appropriate

number

figures.

A suggestion for the

number

figures

report

given at the

end

Section 6.2.

1.3

FREQUENCY

DISTRIBUTIONS

When collecting and summarizing large

amounts

data, it

often helpful to record

the

data

in the form

a frequency table. Such a table simply involves a listing

all

the

observed values

the variable being studied

and

how many times each value is

observed. Consider the tabulation

the frequency

occurrence

sparrow nests

in each

several different locations. This is illustrated in Example

l.l,

where the

observed kinds

nest sites are listed, and for each kind

the

number

nests observed

recorded.

The

distribution

the total

number

observations

among

the various

categories is termed a

frequency distribution. Example

1.1

is a frequency table for

nominal data. and these

data

may also be presented graphically by means

a bar

graph (Figure 1.2). where the height

each

bar

is proportional to the frequency

in the class represented.

The

widths

all

bars

in a

bar

graph

should be equal so

*The use

scientific

notation-by

physicists-can be traced back to at least the

18605

(Miller.

2004b).

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