OpenIntro Statistics 3rd版：概率与数据分析入门

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《开放入门统计学第三版》是一本由杜克大学的 Mine Çetinkaya-Rundel 教授、Christopher D. Barr 和 David M. Diez 协同编写的教材，专为Coursera上统计学与R专项的第一门课程设计。该书以概率和数据分析为核心，旨在为学习者提供一个全面且易于理解的统计学习平台。在本书的开篇，作者通过案例研究引入主题，比如使用支架预防中风，展示了统计在实际问题中的应用。章节1.2介绍了数据的基本概念，包括数据类型（如定量和定性数据）、数据收集原则，以及观测研究和抽样策略的重要性。读者能够在此部分学习到如何有效地理解和处理不同类型的数据。接下来，章节1.5探讨了实验设计，强调了实验在科学推理中的作用，以及如何设计和分析实验数据以测试假设。这部分内容对于理解因果关系和控制变量至关重要。在第1.6节，作者详细讲解了数值数据的分析方法，涉及描述性统计、中心趋势度量和分散程度衡量等。对于分类数据，章节1.7提供了深入解析，涵盖了频率、百分比和概率的计算，以及如何用图表来呈现和理解类别数据。特殊话题的案例研究——性别歧视，出现在1.8章，讨论了如何运用统计方法识别和评估潜在的偏见或歧视现象。概率作为一门核心学科，被单独列为2.1节，介绍基本的概率概念，如样本空间、事件、概率的定义和性质，以及条件概率和独立事件的概念。这为后续的统计推断和假设检验打下坚实基础。全书还包括丰富的练习题，旨在帮助读者巩固所学知识，并逐步提升他们的统计技能。每章末尾的习题涵盖了理论知识和实践应用，适合不同水平的学习者进行自我检测和提高。总体来说，《开放入门统计学第三版》是一本实用的统计学教材，它以清晰的语言和实例驱动的教学方式，使学生能够系统地掌握概率与数据的基本理论和实践技巧，是学习者从基础到进阶统计学的理想入门资源。版权信息表明，这本书遵循创意共享许可，用户可以通过访问openintro.org获取免费PDF版或源代码，了解更多的许可证信息和相关资源。

16 CHAPTER 1. INTRODUCTION TO DATA

Figure 1.10: In February 2010, some

media pundits cite d one large snow

storm as valid evidence against global

warming. As comedian Jon S t ewart

pointed out, “It’s one storm, in one

region, of one country.”

Anecdotal evidence

Be careful of dat a collected in a haphazard fash i on. Such evidence may be true

and veriﬁable, but it may only represent extraordinary cases.

Anecdotal evidence typically is composed of u nusual cases th at 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 t h an the six others who graduated in four year s.

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 collec t in g a sample of students. All gr adu at es in the last 5 years repr ese nt the

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

In general, we always seek to randomly select a sample from a population. 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 st u dy. What kin d of students do you think she

might c ol le ct ? Do you think her sample would be representative of all graduates?

Perhaps she would pick a di sp roportionate number of graduates f rom health-related

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

selecting samples by hand, we run the r i sk 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 gr adu at es are randomly selected from th e

population to be included in the sample.

all graduates

sample

graduates from

health−related fields

Figure 1.12: Instead of sampling from all gr ad uates equally, a nutrition

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

proportionately often.

If someone was permitted to pick and choose exactly which graduates were i nc l ud ed

in the sample, it is entirely possible that the sample could be skewe d to that person’s int e r-

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

randomly helps resolve this problem. The most basic random sampl e is calle d a simple

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

each case in the populat i on has an equal chance of being included and there 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, c aut i on

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 i t is unclear whether the results are

representative of the entire populati on. This non-response bias can skew results.

Another common downfall is a convenience sample, where individuals who are easily

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

is done by stopping people walking in the Bronx, this will not represent al l of Ne w York

City. It is often diﬃcult to discern what sub-population a convenience sample repr ese nts.

Guided Practice 1.9 We can easily acc es s ratings for products, sellers, and com-

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

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

you thi nk t h i s means that 50% of buyers are dissat i sﬁ ed wi t h the product?

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

products that fell b elow 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 studies may

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

times impossible, to completely ﬁx this problem.

1.3.4 Explanatory and response variables

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

(1) Is federal spending, on average, higher or lower in cou nties 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 con si d er 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 s us pected 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 variable 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 likely be above or below the 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 primary types of data col l ect i on : observational studi e s and experiments.

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

does not directly inte rf er e with how the data arise. For instance, researchers may collect

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

similar individuals 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 naturall y occurring associat ion between variables, but they cannot by

themselves show a caus al conne ct i on.

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

duct an experiment. Usually there will be bot h an explanat ory and a response variab l e.

For instanc e, we may suspect admini s t er in g a dr u g will reduce mortality in heart attack

patients over the foll owing year. To che ck if there really is a causal connection between

the explanat ory variable and the response, researchers wil l collect a sample of individuals

and split them into groups. The 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 i n 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 ex am pl e 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 r and omi z ed experiment.

1.4 Observational studies and sampling strategies

1.4.1 Observational studies

Generally, data in observational studies are collected only by monitoring what occurs,

while experiments require the primary explanatory variable in a st ud y be assigned for each

subject by the researchers.

Making causal conclusions based on experiments is often reasonable. However, making

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

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

Guided Practice 1.10 Suppose an observat i onal study tracked sunscreen use and

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

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

Some previous research tells us that using sunscreen actually reduces sk i n cancer risk,

so maybe there is another variable that can ex p l ain this hypothetical association between

sunscreen usage and skin cancer. On e important piece of information that is absent is sun

exposure. If someone is out in the sun all day, she is more likely to use sunscreen and m ore

likely to get skin cancer. Ex posure 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 th at

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

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 me asu re d.

In the same way, the county data set is an observational study with confou nd i ng

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

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

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

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

ables. Suggest one or more other variables that mi ght explain the relations hi p visible

in Figure 1.9.

Observational studies come in two forms: prospecti ve and retrospective studies. A

prospective stu d y identiﬁes individu als and collects informat i on as events unfold. For

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

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 nur se s and then collects data from them usin g

questionnaires. Retrospective studies collect data after events have taken place, e.g.

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

may c ontain both prospectively- and retrospectively-collected variables. Local governments

prospectively coll ec t some variables as events unfolded (e.g. retails sales) while the federal

gover n ment retrospectively collected others during the 2010 census (e.g. county population

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 ran d om framework from a population, these stati st i cal

methods – the estimates and errors associated wi t h the estimates – are not reliable. He re

we c ons id er four random sampling techniques: simple, stratiﬁed, cluster, and multistage

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

member of one of the league’s 30 team s . To take a simple random sample of 120 baseball

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

Also called a lurking variable, confounding factor,oraconfounder.

Answers will vary. Population density may be important. 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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