OpenIntro Statistics 3e：免费教材，涵盖概率与数据分析

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《OpenIntro Statistics 3e》是一本专为初学者设计的开源统计学教材，由David M Diez、Christopher D Barr和Mine Çetinkaya-Rundel三位作者共同编写。本书第三版于2015年9月29日更新，遵循Creative Commons许可协议，旨在提供免费的教育资源。全书结构清晰，涵盖了数据分析的基本概念和实践技巧。章节一"Introduction to Data"是整个课程的入门部分，首先通过一个实际案例（使用支架预防中风）引入统计学的应用，帮助读者理解数据在实际问题中的重要性。随后章节分别探讨了数据的基本要素，如数据收集的原则，观察性研究和抽样策略，实验设计，以及数值和分类数据的分析方法。性别歧视是一个专门讨论的主题，强调了统计学在社会科学研究中的公正性和敏感性处理。章节二"Probability (special topic)"深入探讨概率理论，它是统计学的基础，涉及如何定义和量化随机事件发生的可能性。这一部分的内容包括概率的计算、条件概率、独立事件和互斥事件等概念，对于理解和应用统计推断至关重要。在整个学习过程中，每章都配有丰富的案例研究和练习题，鼓励读者通过实际操作来掌握理论知识。如果你对这本书的内容印象深刻并希望支持作者，可以通过访问openintro.org获取免费PDF版本，或下载源代码文件，了解更多关于版权许可的信息。《OpenIntro Statistics 3e》以其简洁易懂的语言和实用的指导，适合对统计学感兴趣的初学者，无论是学术背景还是职业发展，都能从中受益匪浅。通过学习这本书，读者将建立起扎实的统计基础，为日后在数据分析、科研或商业决策等领域应用所必需的技能打下坚实的基础。

16 CHAPTER 1. INTRODUCTION TO DATA

Figure 1.10: In Februar y 2010, some

media pundits cited one large snow

storm as valid evidence agai nst global

warming. As comedian Jon Stewart

pointed out, “It’s one storm, in one

region, of one country.”

Anecdotal evidence

Be careful of data collected in a haphazard fashion. Such evidence may be t ru e

and veriﬁable, but it may only represent extraordinary cases.

Anecdotal evidence typically is composed of unusual cases that we recall based on

their striking characteristics. For instance, we are more likely to remember the two people

we met who took 7 years to graduate than the six others who graduated in four years.

Instead of looking at the most unusual cases, we should examine a sample of many cases

that represent the population.

1.3.3 Sampling from a population

We might try to estimate the time to graduation for Duke undergraduates in the last 5

years by collectin g a sample of student s. All graduates in the last 5 years represent the

population , and graduates who are selected for review are collectively called the sample.

In general, we always seek to randomly select a sample from a populat i on. The most

basic type of random selection is equivalent to how ra✏es are conducted. For example, in

selecting graduates, we could write each graduate’s name on a ra✏e ticket and draw 100

tickets. The selected names would represent a random sample of 100 graduates.

Why pick a sample randomly? Why not just pick a sample by hand? Consider the

following scenario.

Example 1.8 Suppose we ask a student who happens to be majoring in nutrition

to select several graduates for the study. What kind of students do you think she

might collect? Do you think her sample would be representative of all graduates?

Perhaps she would pick a d i spr opor t i onat e number of graduates from health-related

ﬁelds. Or perhaps her selection would be well-representative of the population. When

selecting samples by hand, we run the risk of picking a biased sample, even if that

bias is unintentional or diﬃcult to discern.

1.3. OVERVIEW OF DATA COLLECTION PRINCIPLES 17

all graduates

sample

Figure 1.11: In this graphic, ﬁve graduates are randomly selected from the

population to be included in the sample.

all graduates

sample

graduates from

health−related fields

Figure 1.12: Instead of sampling from all graduates equally, a nutrition

major might in ad vertently pick graduates with health-related majors dis-

proportionately often.

If someone was permitted to pick and choose exactly whi ch graduates were included

in the sample, it is entirely possible that the sample could be skewed to that person’s inter-

ests, which may be entirely unintentional. This introduces bias into a sample. Sampling

randomly helps resolve this problem. The most basic random sample is called a simple

random samp le, and which is equivalent to using a ra✏e to select cases. This means that

each case in the popul at ion has an equal chance of being included and ther e is no implied

connection between the cases in the sample.

The act of taking a simple random sample helps minimize bias, however, bias can

crop up in other ways. Even when people are picked at random, e.g. for surveys, caution

must be exercised if the non-response is high. For instance, if only 30% of the people

randomly sampled for a survey actually respond, then it is unclear whether the results are

representative of the entire popu lat i on . This non-res ponse bias can skew results.

Another common downfall is a convenience sample, where indivi d uals who are easily

accessible are more likely to be included in the sample. For instance, if a political survey

is done by stopping people walking in the Bronx, this will not represent all of New York

City. It is often diﬃcult to discern what sub-population a convenience sampl e repr esents.

Guided Practice 1.9 We can easily access ratings for products, seller s, and com-

panies through websites. These ratings are based only on t hose people who go out of

their way to provide a rating. If 50% of online reviews for a product are negative, do

you th i nk this means that 50% of buyers are dissatisﬁed with the product?

Answers will vary. From our own anecdotal experiences, we believe people tend to rant more about

products that fell below expectations than rave about those that perform as expected. For this reason, we

suspect there is a negative bias in product ratings on sites like Amazon. However, since our experiences

may not be representative, we also keep an open mind.

18 CHAPTER 1. INTRODUCTION TO DATA

population of interest

sample

population actually

sampled

Figure 1.13: Due to the possibility of non-response, surveys studi es may

only reach a certai n group within the population. It is diﬃcult, and often

times impossible, to completely ﬁx this problem.

1.3.4 Explana tory and response variables

Consider the following question from page 13 for the county data set:

(1) Is feder al spending, on average, higher or lower in counties with high rates of poverty?

If we suspect poverty might a↵ect spending in a county, then poverty is the explanatory

variable and federal spending is the response variable in the relationship.

If there are

many variables, it may be possible to consider a number of them as explanatory variables.

TIP: Explanatory and response variables

To identify the explanatory variable in a pair of variables, identify which of the two

is suspected of a↵ecting the other and plan an appropriate analysis.

might affect

explanatory

variable

response

variable

Caution: association does not imply causation

Labeling variables as explanatory and response does not guarantee the relationship

between the two is actually causal, even if there is an association identiﬁed between

the two variables. We use these labels only to keep track of which vari abl e we

suspect a↵ects the other.

In some cases, there is no explanatory or response variable. Consider the following

question from page 13:

(2) If homeownership is lower than the national average in one county, will the percent

of multi-unit structures in that county l i kely be above or below th e national average?

It is diﬃcult to decide which of these variables should be considered the explanatory and

response variable, i.e. the direction is ambiguous, so no explanatory or response labels are

suggested here.

Sometimes the explanatory variable is called the independent variable and the response variable

is called the dependent variable. However, this becomes confusing since a pair of variables might be

independent or dependent, so we avoid this language.

1.4. OBSERVATIONAL STUDIES AND SAMPLING STRATEGIES 19

1.3.5 Introducing observational studies and experiments

There are two pri mary types of data collection: observational studies and experi ments.

Researchers perform an obser vational study when they collect data in a way that

does not directly inter fer e with how the data arise. For instance, researchers may collect

information via surveys, review medical or company records, or follow a cohort of many

similar in d ivi d ual s to study why certain diseases might develop. In each of these situations,

researchers merely observe the data that arise. In general, observational studies can pro-

vide evidence of a naturally occurring association between variables, but they cannot by

themselves show a causal connection.

When researchers want to investigate the possibility of a causal connection, they con-

duct an experiment. Usually there wil l be both an explanatory and a response variable.

For instance, we may suspect administeri n g a dr u g will reduce mortality in heart attack

patients over the following year. To check if there really is a causal connection between

the explanat or y variable and the response, researchers will collect a sample of individuals

and split them into groups. Th e individuals in each group are assigned a treatment. When

individuals are randomly assigned to a group, the experiment is called a randomized ex-

periment. For example, each heart attack patient in the drug trial could be randomly

assigned, perhaps by ﬂipping a coin, into one of two groups: the ﬁrst group receives a

placebo (fake treatment) and the second group receives the drug. See the case study in

Section 1.1 for another example of an experiment, though that study did not employ a

placebo.

TIP: association 6= causation

In general, association does not imply causation, and causation can only be inferred

from a randomized experiment.

1.4 Observational studies and sampling strategies

1.4.1 Obs ervational studies

Generally, data in observational studies are collected only by moni tor i n g what occurs,

while experiments require the primary explanatory variable in a study be assigned for each

subject by the researchers.

Making causal conclusions based on experiments is often reason abl e . However, making

the same causal conclusions based on observational data can b e treacherous and is not rec-

ommended. Thus, observational studies are generally only suﬃcient to show associations.

Guided Practice 1.10 Suppose an observational study tracked sunscreen use and

skin cancer, and it was found that the more sunscreen someone used, the more likely

the person was to have skin cancer. Does this mean sunscreen causes skin cancer?

Some previous research tells us that using sunscr een actually reduces skin cancer risk,

so maybe there is another variable that can explain thi s hypotheti cal association between

sunscreen usage and skin canc er . On e important piece of informati on that is absent is sun

exposure. If someone is out in the su n all day, she is more likely to use sunscreen and more

likely to get skin cancer. Exposure to the sun is unaccounted for in the simple investigation.

No. See the paragraph following the exercise for an explanation.

20 CHAPTER 1. INTRODUCTION TO DATA

sun exposure

use sunscreen

skin cancer

Sun exposure is what is called a confounding variable,

which is a variable that

is correlated with both the explanatory and response variables. While one method t o

justify making causal conclusions from observational studies is to exhaust the search for

confounding variables, there is no guarantee that all confounding variables can be examined

or measured.

In the same way, the county data set is an observational study with confoundi n g

variables, and its data cannot easily be used to make causal conclusions.

Guided Practice 1.11 Figure 1.9 shows a negative association between the home-

ownership r ate and the percentage of multi-unit structures in a county. However, it

is unreasonable to conclude that there is a causal relationshi p between the two vari-

ables. Suggest one or more other variables that might explain the relationship visible

in Figure 1.9.

Observational studies come in two forms: prospective and retrospective studies. A

prospective stud y identiﬁes individual s and collects information as events unfold. For

instance, medical researchers may identify and follow a group of similar individuals over

many years to assess the possible inﬂuences of behavior on cancer risk. One example

of such a study is The Nurses’ Health Study, started in 1976 and expanded in 1989.

This prospective study recruits registered nurses and then collects dat a from them using

questionnaires. Retrospective studies collect data after events have taken p l ace, e.g.

researchers may review past events in medical records. Some data sets, such as county,

may contain both prospectively- and retrospectively-collected variables. Local governments

prospectively collect some variables as events unfolded (e.g. ret ai l s sales) while the federal

govern ment retrospectively collected others during the 2010 census (e.g. county pop u lat i on

counts).

1.4.2 Four sampling methods (special topic)

Almost all statistical methods are based on the notion of implied randomness. If observa-

tional data are not collected in a rand om framework from a population, these statisti cal

methods – the estimates and errors associated with the estimates – are not reliable. Here

we consi d er four random sampling techniques: simple, stratiﬁed, cluster, and multi stage

sampling. Figures 1.14 and 1.15 provide graphical representations of these techniques.

Simple random sampling is probably the most intuitive form of random sampling.

Consider the salaries of Major League Baseball (MLB) players, where each player is a

memb er of one of the league’s 30 teams. To take a simple r and om sample of 120 b asebal l

players and their salaries from the 2010 season, we coul d write the names of that season’s

Also called a lurking variable , confounding factor,oraconfounder.

Answers will vary. Population density may be imp ortant. If a county is very dense, then this may

require a larger fraction of residents to live in multi-unit structures. Additionally, the high density may

contribute to increases in property value, making homeownership infeasible for many residents.

www.channing.harvard.edu/nhs

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