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首页CRC手册：概率与统计标准表格与公式全览

CRC手册：概率与统计标准表格与公式全览

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"《CRC标准概率与统计表与公式手册》是一本专为IT专业人士精心编纂的权威参考书籍，由Chapman & Hall/CRC在2000年出版。本书以其全面详尽的内容涵盖了概率与统计的基本理论、概念、定理和公式，适合于统计学家以及对相关领域感兴趣的读者。手册的特点在于其结构清晰，包括目录式的PDF书签，方便查阅。书中包含了大量的概率和统计表格，每个表格都配有详尽的文字描述，并通过实例加以解释，使抽象的概念变得具体易懂。例如，它可能包括概率分布表、假设检验统计量表、置信区间计算公式等，这些都是进行数据分析和决策的重要工具。此外，书中还探讨了许多高级统计话题，如贝叶斯统计、非参数统计、时间序列分析等，体现了现代统计学的深度和广度。对于学习者而言，本书不仅提供了扎实的基础知识，而且通过大量的实例和步骤解析，帮助读者理解并掌握统计方法的实际应用。即使是经验丰富的统计学家，也能从中找到新的视角和解决问题的新途径。欧洲数学学会在2001年的通讯中称赞这本书为鼓舞人心的统计问题指南，对统计工作者来说非常实用。然而，值得注意的是，尽管书中信息丰富，但并非所有内容都保证绝对准确，因此在使用过程中，读者需要根据实际需求和当前研究环境判断材料的有效性。版权方面，该书要求在复制或传播任何形式的内容时必须事先获得CRC Press LLC的书面许可，以尊重作者和出版商的权益。《CRC标准概率与统计表与公式手册》是IT行业中必备的一部参考文献，无论是学术研究还是实践应用，都能为读者提供强有力的支持和深入的理解。"

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CHAPTER 2

Summarizing Data

Contents

2.1Tabularandgraphicalprocedures

2.1.1Stem-and-leafplot

2.1.2Frequencydistribution

2.1.3Histogram

2.1.4Frequencypolygons

2.1.5Chernoﬀfaces

2.2Numericalsummarymeasures

2.2.1(Arithmetic)mean

2.2.2Weighted(arithmetic)mean

2.2.3Geometricmean

2.2.4Harmonicmean

2.2.5Mode

2.2.6Median

2.2.7p%trimmedmean

2.2.8Quartiles

2.2.9Deciles

2.2.10Percentiles

2.2.11Meandeviation

2.2.12Variance

2.2.13Standarddeviation

2.2.14Standarderrors

2.2.15Rootmeansquare

2.2.16Range

2.2.17Interquartilerange

2.2.18Quartiledeviation

2.2.19Boxplots

2.2.20Coeﬃcientofvariation

2.2.21Coeﬃcientofquartilevariation

2.2.22Zscore

2.2.23Moments

2.2.24Measuresofskewness

 2000 by Chapman & Hall/CRC

2.2.25Measuresofkurtosis

2.2.26Datatransformations

2.2.27Sheppard’scorrectionsforgrouping

Numerical descriptive statistics and graphical techniques may be used to sum-

marize information about central tendency and/or variability.

2.1 TABULAR AND GRAPHICAL PROCEDURES

2.1.1 Stem-and-leaf plot

A stem-and-leaf plot is a a graphical summary used to describe a set of ob-

servations (as symmetric, skewed, etc.). Each observation is displayed on the

graph and should have at least two digits. Split each observation (at the same

point) into a stem (one or more of the leading digit(s)) and a leaf (remaining

digits). Select the split point so that there are 5–20 total stems. List the

stems in a column to the left, and write each leaf in the corresponding stem

row.

Example 2.1 :

Construct a stem-and-leaf plot for the Ticket Data (page 2).

Solution:

Stem Leaf

4 39

11555689

02334445556777889

122344558

349

Stem = 10, Leaf = 1

Figure 2.1: Stem–and–leaf plot for Ticket Data.

2.1.2 Frequency distribution

A frequency distribution is a tabular method for summarizing continuous or

discrete numerical data or categorical data.

(1) Partition the measurement axis into 5–20 (usually equal) reasonable

subintervals called classes, or class intervals. Thus, each observation

falls into exactly one class.

(2) Record, or tally, the number of observations in each class, called the

frequency of each class.

(3) Compute the proportion of observations in each class, called the relative

frequency.

(4) Compute the proportion of observations in each class and all preceding

classes, called the cumulative relative frequency.

 2000 by Chapman & Hall/CRC

Example 2.2 : Construct a frequency distribution for the Ticket Data (page 2).

Solution:

(S1) Determine the classes. It seems reasonable to use 40 to less than 50, 50 to less

than 60, ..., 90 to less than 100.

Note: For continuous data, one end of each class must be open. This ensures

that each observation will fall into only one class. The open end of each class

may be either the left or right, but should be consistent.

(S2) Record the number of observations in each class.

(S3) Compute the relative frequency and cumulative relative frequency for each class.

(S4)TheresultingfrequencydistributionisinFigure2.2.

Cumulative

Relative relative

Class Frequency frequency frequency

[40, 50) 2 0.050 0.050

[50, 60) 8 0.200 0.250

[60, 70) 17 0.425 0.625

[70, 80) 9 0.225 0.900

[80, 90) 3 0.075 0.975

[90, 100) 1 0.025 1.000

Figure 2.2: Frequency distribution for Ticket Data.

2.1.3 Histogram

A histogram is a graphical representation of a frequency distribution. A (rela-

tive) frequency histogram is a plot of (relative) frequency versus class interval.

Rectangles are constructed over each class with height proportional (usually

equal) to the class (relative) frequency. A frequency and relative frequency

histogram have the same shape, but diﬀerent scales on the vertical axis.

Example 2.3 :

Construct a frequency histogram for the Ticket Data (page 2).

Solution:

(S1)UsingthefrequencydistributioninFigure2.2,constructrectanglesaboveeach

class, with height equal to class frequency.

(S2)TheresultinghistogramisinFigure2.3.

Note: A probability histogram is constructed so that the area of each rectangle

equals the relative frequency. If the class widths are unequal, this histogram

presents a more accurate description of the distribution.

2.1.4 Frequency polygons

A frequency polygon is a line plot of points with x coordinate being class

midpoint and y coordinate being class frequency. Often the graph extends to

 2000 by Chapman & Hall/CRC

Figure 2.3: Frequency histogram for Ticket Data.

an additional empty class on both ends. The relative frequency may be used

in place of frequency.

Example 2.4 :

Construct a frequency polygon for the Ticket Data (page 2).

Solution:

(S1)UsingthefrequencydistributioninFigure2.2,ploteachpointandconnectthe

graph.

(S2)TheresultingfrequencypolygonisinFigure2.4.

Figure 2.4: Frequency polygon for Ticket Data.

An ogive,orcumulative frequency polygon, is a plot of cumulative fre-

quencyversustheupperclasslimit.Figure2.5isanogivefortheTicketData

(page 2).

Another type of frequency polygon is a more-than cumulative frequency poly-

gon. For each class this plots the number of observations in that class and

every class above versus the lower class limit.

 2000 by Chapman & Hall/CRC

Figure 2.5: Ogive for Ticket Data.

A bar chart is often used to graphically summarize discrete or categorical

data. A rectangle is drawn over each bin with height proportional to frequency.

The chart may be drawn with horizontal rectangles, in three dimensions, and

maybeusedtocomparetwoormoresetsofobservations.Figure2.6isabar

chart for the Soda Pop Data (page 2).

Figure 2.6: Bar chart for Soda Pop Data.

A pie chart is used to illustrate parts of the total. A circle is divided into

slicesproportionaltothebinfrequency.Figure2.7isapiechartfortheSoda

Pop Data (page 2).

2.1.5 Chernoﬀ faces

Chernoﬀ faces are used to illustrate trends in multidimensional data. They

are eﬀective because people are used to diﬀerentiating between facial features.

Chernoﬀ faces have been used for cluster, discriminant, and time-series anal-

yses. Facial features that might be controllable by the data include:

(a) ear: level, radius

(b) eyebrow: height, slope, length

 2000 by Chapman & Hall/CRC

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