統計學導論 1.1 節：統計學、機率與關鍵術語的定義

The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.

統計學（statistics）研究資料（data）的蒐集、分析、解讀與呈現。我們在日常生活中都會接觸並使用資料。

In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest halfhour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data: 5; 5.5; 6; 6; 6; 6.5; 6.5; 6.5; 6.5; 7; 7; 8; 8; 9 The dot plot for this data would be as follows:

請在課堂上試做這個練習。請班上同學寫下自己每晚的平均睡眠時間（以小時為單位，四捨五入至最接近的半小時）。老師會記錄這些資料。接著，將資料繪製成簡單的圖形（稱為點圖（dot plot））。點圖由一條數線及位於數線上方的點所組成。例如，請看以下資料：5; 5.5; 6; 6; 6; 6.5; 6.5; 6.5; 6.5; 7; 7; 8; 8; 9 這組資料的點圖如下：

Does your dot plot look the same as or different from the example? Why? If you did the same example in an English class with the same number of students, do you think the results would be the same? Why or why not? Where do your data appear to cluster? How might you interpret the clustering?

你的點圖看起來與範例相同還是不同？為什麼？如果在學生人數相同的英文課上做同樣的練習，你認為結果會相同嗎？為什麼會相同或不同？你的資料看起來集中在哪裡？你會如何解讀這種集中現象？

The questions above ask you to analyze and interpret your data. With this example, you have begun your study of statistics.

上述問題要求你分析並解讀自己的資料。透過這個範例，你已經開始學習統計學。

In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct.

在這門課中，你將學習如何整理與摘要資料。整理與摘要資料稱為敘述統計（descriptive statistics）。摘要資料的兩種方式是繪圖與使用數值（例如，求平均值）。學過機率（probability）與機率分配（probability distributions）之後，你將使用正式的方法，從「良好」的資料中得出結論。這些正式的方法稱為推論統計（inferential statistics）。統計推論（statistical inference）運用機率，判斷我們能多有把握認為結論正確。

Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.

要有效解讀資料（推論），必須以良好的資料產生程序及周詳的資料檢視為基礎。你將接觸到許多用來解讀資料的數學公式，可能會覺得公式多得過頭。統計學的目標不是利用公式進行大量計算，而是了解你的資料。計算可以交給計算機或電腦。理解則必須靠你自己。如果能徹底掌握統計學的基礎，你在生活中做決定時就能更有信心。

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 1/2 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996/2000 is equal to 0.498 which is very close to 0.5, the expected probability.

機率是研究隨機性（randomness）的數學工具。它探討事件（event）發生的可能性（機會）。例如，如果你擲一枚公正硬幣四次，結果不一定是兩次正面、兩次反面。然而，如果你擲同一枚硬幣 4,000 次，結果就會接近一半正面、一半反面。任何一次投擲出現正面的預期理論機率為 1/2 或 0.5。即使重複少數幾次的結果無法確定，重複許多次之後，結果就會呈現規律。英國統計學家 卡爾·皮爾森（Karl Pearson） 曾擲硬幣 24,000 次，結果出現 12,012 次正面，本書一位作者讀到這件事後，也擲了硬幣 2,000 次。結果出現 996 次正面。分數 996/2000 等於 0.498，非常接近預期機率 0.5。

The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.

機率理論起源於對撲克等機會遊戲的研究。預測以機率的形式呈現。我們利用機率來預測地震、下雨，或你在這門課拿到 A 的可能性。醫師利用機率，判斷接種疫苗引發其原本要預防之疾病的可能性。股票經紀人利用機率，判斷客戶投資的報酬率（rate of return）。你可能會利用機率來決定是否購買彩券。學習統計學時，你將透過機率計算，運用數學的力量來分析與解讀資料。

In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.

在統計學中，我們通常想研究的是母體（population）。你可以將母體想成研究對象中所有人、事物或物件的集合。為了研究母體，我們會選取一個樣本（sample）。抽樣（sampling）的概念是從較大的母體中選取一部分（或子集合（subset）），研究這一部分（樣本），以取得關於母體的資訊。資料是從母體抽樣所得到的結果。

Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.

由於檢視整個母體需要耗費大量時間與金錢，抽樣是一種非常實用的技術。如果你想計算全校學生的整體平均成績點數（grade point average），從就讀該校的學生中選取樣本是合理的做法。從樣本蒐集到的資料會是這些學生的平均成績點數。在總統選舉中，民意調查會抽取 1,000–2,000 人的樣本。民意調查應能代表全國人民的看法。罐裝碳酸飲料製造商會抽取樣本，確認容量為 16 盎司的罐子是否裝有 16 盎司的碳酸飲料。

From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.

我們可以從樣本資料計算出統計量（statistic）。統計量是代表樣本某項特性的數值。例如，如果將一個數學班視為所有數學班所構成之母體中的樣本，那麼該班學生在學期結束時的平均得分，就是統計量的一個例子。統計量是母體母數（population parameter）的估計值。母數（parameter）是整個母體的數值特徵，可以用統計量來估計。既然我們將所有數學班視為母體，那麼所有數學班學生的平均得分，就是母數的一個例子。

One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.

統計學領域主要關注的問題之一，是統計量估計母數的準確程度。準確程度取決於樣本能多充分地代表母體。樣本必須具備母體的特徵，才具有代表性。在推論統計中，我們同時關注樣本統計量與母體母數。在後面的章節中，我們將利用樣本統計量，檢驗既定母體母數的有效性。

A variable, usually notated by capital letters such as X and Y, is a characteristic or measurement that can be determined for each member of a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense).

變數（variable）通常以 X、Y 等大寫字母表示，是母體中每個成員皆可確定的一項特徵或測量值。變數可以是數值變數（numerical variables），也可以是類別變數（categorical variables）。數值變數的取值具有相等的單位，例如以磅為單位的重量，以及以小時為單位的時間。類別變數則將人或事物歸入某個類別。如果令 X 等於一名數學課學生在學期結束時的得分，那麼 X 就是數值變數。如果令 Y 為一個人的政黨歸屬，那麼 Y 的例子包括 共和黨（Republican）、民主黨（Democrat） 及無黨籍。Y 是類別變數。我們可以對 X 的值進行一些數學運算（例如，計算平均得分），但對 Y 的值進行數學運算並沒有意義（計算平均政黨歸屬並沒有意義）。

Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.

資料是變數的實際取值。它們可以是數字，也可以是文字。單筆資料（datum）是一個單一的值。

Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is 22/40 and the proportion of women students is 18/40. Mean and proportion are discussed in more detail in later chapters.

統計學中經常出現的兩個詞是平均數（mean）與比例（proportion）。如果你在數學課參加三次考試，分別得到 86、75 和 92 分，將三次考試的分數相加，再除以三，就能算出平均分數（取到小數點後一位時，平均分數為 84.3）。如果你的數學班有 40 名學生，其中 22 名是男生、18 名是女生，那麼男學生的比例為 22/40，女學生的比例為 18/40。平均數與比例將在後面的章節中更詳細地討論。

The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean," and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."

「平均數」與「平均值（average）」這兩個詞經常互換使用。以其中一個詞替代另一個詞，是常見的做法。正式術語是「算術平均數（arithmetic mean）」，而嚴格來說，「平均值」是一種中心位置。然而，在非統計學家的一般使用情境中，普遍接受以「平均值」表示「算術平均數」。

Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively.

判斷下列研究中的關鍵術語各指什麼。我們想知道 ABC 學院（ABC 學院） 一年級學生購買學用品的平均花費（平均數），其中不包括書籍。我們隨機調查了該校 100 名一年級學生。其中三名學生分別花了 $150、$200 和 $225。

Solution 1.1 The population is all first year students attending ABC College this term.

解答 1.1 母體是本學期就讀 ABC 學院 的所有一年級學生。

The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population).

樣本可以是 ABC 學院 某一班統計學入門課程的所有修課學生（不過，這個樣本可能無法代表整個母體）。

The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term.

母數是本學期 ABC 學院 一年級學生的平均花費（平均數），其中不包括書籍。

The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample.

統計量是樣本中一年級學生的平均花費（平均數），其中不包括書籍。

The variable could be the amount of money spent (excluding books) by one first year student. Let X = the amount of money spent (excluding books) by one first year student attending ABC College.

變數可以是一名一年級學生的花費金額（不包括書籍）。令 X = 就讀 ABC 學院 的一名一年級學生的花費金額（不包括書籍）。

The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225.

資料是這些一年級學生花費的美元金額。資料的例子包括 $150、$200 和 $225。

Determine what the key terms refer to in the following study.

判斷下列研究中的關鍵術語各指什麼。

As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies. Here is the criterion they used:

在一項旨在測試汽車安全性的研究中，美國國家運輸安全委員會（National Transportation Safety Board） 蒐集並檢視了汽車碰撞對測試假人影響的資料。以下是他們採用的準則：

Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars.

將假人放在汽車前座，讓汽車以每小時 35 英里的速度撞向牆壁。我們想知道，駕駛座上的假人如果是真正的駕駛人，頭部會受傷的比例是多少。我們先抽取由 75 輛汽車組成的簡單隨機樣本（simple random sample）。

Solution 1.3 The population is all cars containing dummies in the front seat.

解答 1.3 母體是所有前座放有假人的汽車。

The sample is the 75 cars, selected by a simple random sample.

樣本是以簡單隨機抽樣選出的 75 輛汽車。

The parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population.

母數是母體中駕駛座假人（如果是真人）頭部會受傷的比例。

The statistic is proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample.

統計量是樣本中駕駛座假人（如果是真人）頭部會受傷的比例。

The variable X = the number of driver dummies (if they had been real people) who would have suffered head injuries.

變數 X = 駕駛座假人（如果是真人）頭部會受傷的數量。

The data are either: yes, had head injury, or no, did not.

資料為以下兩者之一：是，頭部有受傷；或否，頭部未受傷。

Determine what the key terms refer to in the following study.

判斷下列研究中的關鍵術語各指什麼。

An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.

一家保險公司想知道，所有醫師中曾涉及一次或多次醫療疏失訴訟（malpractice lawsuits）的比例。該公司從專業名錄中隨機選取 500 名醫師，確認樣本中曾涉及醫療疏失訴訟的人數。

Solution 1.4 The population is all medical doctors listed in the professional directory.

解答 1.4 母體是專業名錄中列出的所有醫師。

The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population.

母數是母體中曾涉及一次或多次醫療疏失訴訟的醫師比例。

The sample is the 500 doctors selected at random from the professional directory.

樣本是從專業名錄中隨機選出的 500 名醫師。

The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample.

統計量是樣本中曾涉及一次或多次醫療疏失訴訟的醫師比例。

The variable X = the number of medical doctors who have been involved in one or more malpractice suits. The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not.

變數 X = 曾涉及一次或多次醫療疏失訴訟的醫師人數。資料為以下兩者之一：是，曾涉及一次或多次醫療疏失訴訟；或否，未曾涉及。