경제통계학

계량경제학 기초자료 1 - 기술통계학

통계학의 정의

Statistics is concerned with scientific methods for collecting, organizing, summarizing, presenting and analyzing data as well as with drawing valid conclusions and making reasonable decisions on the basis of such analysis.

cf. Status(=state): 중세 라틴어로 國家, 狀態, 記述의미. 국가와 같은 집단의 상태를 기술.

국가성립 = 통계학 발달.

독일: H.Conring(17c.), 國狀學 [국가의 현저한 사항(토지 인구 재정 병력) 수량적 표현]

영국: J.Graunt, W.Petty, Political Arithematics : 社會 大數法則의 경험적 발전

프랑스: 고전확률론, 게임(도박)확률계산 수학적 취급, Pascal, Bernoulli, “law of large

numbers"

Laplace, 고전 확률론 집대성, A.Quetelet (1796-1874) 통합, 근대통계학 창시자.

인간사회의 해석에 자연주의적 결정론의 생각 과용 (예:범죄발생율 규칙성발견 -> 예정된 범죄?) 平均人(average man)이론 : 평균인=이상인(신장 ,체중, 도덕성 등) : 산술평균 중심의 기계적 유물론의 통계학 -> 정확한 양적 기술 -> K.Pearson 기술통계방법 정밀화

현대통계학 : W.Gosset, 소표본 처리방법 -> 생물통계 -> R.A.Fisher, 추측통계학 발전

추측통계학(=실험자료로부터 모집단에 관하여 추측하는 것이 목적), 표본조사방법 발전

1930년대 가설검정, 구간추정법 연구 => 불확실한 상황에서 현명한 의사결정을 위한

행동의 논리를 연구하는 학문.

국장학파 : 비교 표시 수량적 표현 ------------------	명칭

정치산술학파 : 수량적 관찰, 대량관찰, 초보적 추산 --------	대상

고전확률론 : 우연의 세계 처리하는 도구로 확률론 사용 ------	방법론적 기초

통계자료(=여러 가지 복합적 요인의 결과를 나타내는 사실, 수치)를 분석 처리하여 그 속에 담긴 정보를 추출할 수 있는 방법체계를 연구:

(제한된)부분 정보 -> (가능한 한 정확한) 전체 판단. 정확성 + 경제성 -> 최선의 판단.

통계학적 추론방법 = 표본을 바탕으로 하는 歸納的(inductive) 推論; 항상 오차가 수반.

통계적 추론과정에는 여러 가지 오류의 함정 : 잘못된 모집단 설정; 부적절한 표본추출;

잘못된 계산; 부적절한 기법 선택; 상관관계와 인과관계의 혼돈.

數理統計學 : 통계학적 방법론 이론적 근거를 수학적으로 유도 -> 이론체계화: 순수과학

應用통계학 : 유도 정리된 통계학이론 기법 응용, 현실문제에 관한 최선판단: 경제통계학 등

경제이론의 경험적 검증에 있어서 통계학이 분석도구로 사용 : 경제학의 주요 개념(용어)   들은 定量的(quantitative)이므로 통계학과의 결합이 용이.

경제통계학의 내용 : 계량경제학(모수 통계적 추정; 경제변수 간의 관계 가설검정 -> 현실  자료와의 적합성여부 결정; 경제모형에서의 계수 추정)을 위한 기초 과정으로서 다음과 같이 구성:

確 率 論 : 추정 검정의 기본분석개념인 확률밀도함수, 분포함수;

標本理論 : 표본추출방법, 표본분포의 특성;

推定理論 : 모수 추정방법, 통계학적으로 바람직한 특성설정;

檢定理論 : 모수에 대한 가설을 표본에서 관찰된 추정량을 중심으로 기각 채택하는 과정;

回歸分析 : 경제변수 간의 관계를 통계적으로 추정 검정하는 과정 => 計量經濟學

CHAPTER 1 FUNDAMENTALS OF STATISTICS

1.1 The Five Basic Words of Statistics

Population 모집단 (母集團) cf. universe, cohort.  finite vs. infinite

Concept: All members of a group about which you want to draw a conclusion.

Examples: All citizens (of a country) who are currently registered to vote, etc.

Sample 표본 (標本)

Concept: The part of the population selected for analysis.

Examples: The registered voters selected to participate in a recent survey concerning their intention to vote in the next election, etc.

Parameter 모수 (母數)

Concept: A numerical measure that describes a characteristic of a population.

Examples: The percentage of all registered voters who intend to vote in the next election, etc.

Statistic 통계량 (統計量)

Concept: A numerical measure that describes a characteristic of a sample.

Examples: The percentage in a sample of registered voters who intend to vote in the next election, etc.

Interpretation: Calculating statistics for a sample is the most common activity, because collecting population data is impractical for most actual decision-making situations.

Variable 변수 (變數)

Concept: A characteristic of an item or an individual that will be analyzed using statistics.

Examples: Gender, the household income of the citizens who voted in the last (presidential) election, etc.

Interpretation: All the variables taken together form the data of an analysis.

You should distinguish between a variable, such as gender, and its value 값 for an individual, such as male. An observation 관찰치 is all the values for an individual item in the sample.

Variables can be divided into the following types:

	Categorical Variables	Numerical Variables
Concept	The values are selected from an established list of categories	The values involve a counted or measured value
Subtypes	None	Discrete values; 이산적 (離散的) Continuous values; 연속적 (連續的)
Examples	Gender; Academic major	The number of a citizen voted; The household income of a citizen who voted

All variables should have an operational definition 조작적 정의(操作的 定義) - that is, a universally accepted meaning that is clear to all associated with an analysis. Without an operational definitions, confusion can occur. Give some examples of such confusion.

1.2 The Branches of Statistics

Descriptive Statistics  기술(記述)통계학  cf. deductive

Concept: The branch of statistics that focuses on collecting, summarizing, and presenting a set of data.

Example: The average age of citizens who voted for the winning candidate.

Interpretation: Descriptive statistics forms the basis for analysis and discussion in such diverse fields as the social sciences, government, and professional sports.

Inferential Statistics  추측(推測)통계학  cf. statistical inference: probability

Concept: The branch of statistics that analyzes sample data to draw conclusions about a population.

Example: A survey that sampled 2,000 full-or part-time workers ages 50 to 70, conducted by the Association of Retired Persons, discovered that 70% of those polled planned to work past the traditional mid-60s retirement age.

Interpretation: When you use inferential statistics, you start with a hypothesis and look to see whether the data are consistent with that hypothesis.

1.3 Sources of Data 자료출처

Published Sources 출판된 형태 자료원천 (문서, 웹),  1차적 vs. 2차적 자료

Concept: Data available in print or in electronic form (including data found on Internet Web sites). Primary data sources are those published by the individual or group that collected the data. Secondary data sources are those compiled from primary sources.

Example: Many government agencies, including the Bureau of Census, publish primary data sources that are available at the Web site www.bos.gov. Business news sections of daily newspapers publish secondary source data.

Interpretation: You should always consider the possible bias of the publisher and whether the data contain all the necessary and relevant variables when using published data. Remember that anyone can publish data on the Internet.

Experiments 실험 (實驗), 실험집단 대 통제집단,  cf. Placebo effect

Concept: A process that studies the effect on a variable of varying the value(s) of another variable(s), while keeping all other things equal. The treatment group consists of those individuals or things that receive the treatment(s) being studies. The control group consists of those that do not receive the treatment(s).

Example: Pharmaceutical companies use experimental studies to determine whether a new drug is effective. A group of patients who have many similar characteristics is divided into two subgroups.

Interpretation: Proper experiments are either single-blind or double-blind. A study is a single-blind experiment if only the researcher conducting the study knows the identities of the members of the treatment and control groups.

Surveys  조사 (調査)

Concept: A Process that uses questionnaires or similar means to gather values for the responses from a set of participants.

Example: A Web site instant poll or "question of the day."

Interpretation: Surveys are either informal, open to anyone who wishes to participate; targeted, directed toward a specific group of individuals; or include people chosen at random. The type of survey affects how the data collected can be used and interpreted.

1.4 Sampling Concepts

Sampling  표본추출 (標本抽出)

Concept: The process by which members of a population are selected for a sample.

Example: Choosing every fifth voter who leaves a polling place to interview, drawing playing cards randomly from a deck, etc.

Interpretation: The method by which sampling occur, the identification of all items in a population, and the techniques used to select individual observations all affect sampling.

Probability Sampling  확률적(確率的) 표본추출  cf. 비확률적 표본추출을 하는 경우:

Concept: A sampling process that takes into consideration the chance of occurrence of each item being selected. Probability sampling increase your chances that the sample will be representative of the population.

Example: The registered voters selected to participate in a recent survey concerning their intention to vote in the next election, etc.

Interpretation: You should use probability sampling whenever possible, because only this type of sampling allows you to apply inferential statistical methods to the data you collect. In contrast, you should use nonprobability sampling, in which the chance of occurrence of each item being selected is not known, to obtain rough approximations of results at low cost or for small-scale, initial, or pilot studies that will later be followed up by a more rigorous analysis.

Simple Random Sampling (SRS) 단순 무작위(無作爲)추출

Concept: The Probability sampling process in which every individual or item from a population has the same chance of selection as every other individual or item.

Example: Selecting a playing card from a shuffled deck, or using a statistical device such as a table of random numbers.  난수표(亂數表)

Interpretation: SRS forms the basis for other random sampling techniques. In statistics, random implies no repeating patterns - that is, in a given sequence, a given pattern is equally likely (or unlikely) as another.

Frame  (표본추출) 틀

Concept: The list of all items in the population from which samples will be selected.

Example: Voter registration list, customer or human resource database, etc.

Interpretation: Frames influence the results of an analysis, and using two different frames can lead to different conclusions.  You should always be careful to make sure your frame completely represents a population.  모집단의 대표성.

1.5 Sample Selection Methods  표본추출 방법

Sampling With Replacement  복원(復元)표본추출

Concept: A sampling method in which each selected item is returned to the frame from which it was selected so that it has the same probability of being selected again.

Example: Selecting entries from a fishbowl and returning each entry to the fishbowl after it is drawn.

Sampling Without Replacement  비(非)복원 표본추출

Concept: A Sampling method in which each selected item is not returned to the frame from which it was selected. Using this technique, an item can be selected no more than one item.

Example: Selecting numbers in lottery games, etc. 로또 복권

Interpretation: Sampling without replacement means that an item can be selected no more than one time.

Other, more complex, sampling methods are also used in survey sampling. In a stratified sample 층화(層化), the items in the frame are first subdivided into separate subpopulations, or strata, and a simple random sample is conducted within each of the strata . In a cluster군집(群集) sample, the items in the frame are divided into several clusters so that each cluster is representative of the entire population. A random sampling of cluster is then taken, and all the items in each selected cluster or a sample from each cluster are then studied.

* snowball sampling

CHAPTER 2 PRESENTING DATA IN CHARTS AND TABLES

2.1 Presenting Categorical Data 범주별 자료의 제시

You present categorical data by sorting responses by categories. the count, amount, or percentage of responses by category is then placed into a summary table or into one of several forms of charts.

Summary Table  요약표.  제1열에는 범주별 자료 이름, 2열에는 값

Concept: A two-column table in which the names of the categories are listed in the first column and the count, amount, or percentage of reponses are listed in a second column.

Example: Blood Donation Behavior

Behavior	Percentage (%)
Donate regularly Have donated and may again Have not but may in future Never have, never will Want to, but think they cannot	11 31 17 13 28

Interpretation: Summary tables enable you to see the big picture about a set of data. In the example, you can conclude that there seems a good opportunity to increase the percentage of people giving blood donations because only 11% donate regularly, and an equally small group (13%) say they will never donate.

The Bar Chart

Concept: A Chart containing rectangles ("bars") in which the length of each bar represents the count, amount, or percentage of responses of one category.

Example:

Blood Donation Behavior

Want to, think they cannot

Never have, never will

Have not but may in future

Have donated and may again

Donate regularly

0     5     10     15     20     25     30     35

Percent

Interpretation: A bar chart better makes the point that the category “have donated and may donate again” is the single largest category for this example. For most people, scanning a bar chart is easier than scanning a column of numbers.

The Pie Chart  원(파이) 그림

Concept: A circle chart in which wedge-shaped area - pie slices - represent the count, amount, or percentage of each category and the entire circle (“pie”) represents the total.

Example:

Blood Donation Behavior

Donate regularly

11%

Want to, think they cannot

28%

Have donated and may again

31%

Never have, never will

13%

Have not but may in future

17%

Interpretation: By ordering the categories carefully, you can make the point that a majority of those surveyed either want to donate but think they cannot or are people who have donated in the past and may do so again.  Although you will probably produce most of your pie charts using computers or calculators, you can also produce it using a protractor to divide up a drawn circle. To produce it in this way, first calculate percentages for each category. Then multiply each percentage by 360, the number of degrees in a circle, to get the number of degrees for the arc (part of circle) portion of each category's pie slice. For example, for the "have donated blood regularly" category, multiply 11% by 360 degrees to get 39.6 degrees.  계산기나 컴퓨터로 도출하지 않고, 각도기 사용:

The Pareto Diagram  파레토 그림

Concept: A special type of bar chart in which the counts, amounts, or percentages of each category are presented in descending order left to right, along with a superimposed plotted line that represents a running cumulative percentage.

Example: Computer Keyboards Defects for a Three-Month Period

Defect	Frequency	Percentage
Black spot Damage Jetting Pin mark Scratches Shot mold Silver streak Sink mark Spray mark Warpage Total	413 1,039 258 834 442 275 413 371 292 1,987 6,342	6.53 16.43 4.08 13.19 6.99 4.35 6.53 5.87 4.62 31.42 100.01

Interpretation: When there are many categories, Pareto diagrams enable you to focus on the most important categories by visually separating the "vital few" from the "trivial many" categories.

Two-Way Cross-Classification Tables  2원(元)교차(交叉)분류(分類)표

Concept: A multicolumn table that presents the count or percentage of responses to two categorical variables. In two-way tables, the categories of one of the variables form the rows of the table, while the categories of the second variable form the columns. Cross-classification tables are also known as cross-tabulation tables.

Example: Counts of Particles Found Cross-Classified by Wafer Condition

예: 실리콘 웨이퍼에서 발견된 티끌이 웨이퍼의 성능에 미치는 영향조사

	Wafer Condition Good Bad	Total
Particles Found Yes No	14 36 320 80	50 400
Total	334 116	450

Row percentages Table 행 백분율 표. 티끌이 발견된 50개 중에서 성능이 좋은 것이 14개로 28%

	Wafer Condition Good Bad	Total
Particles Found Yes No	28.0 72.0 80.0 20.0	100.0 100.0
Total	74.2 25.8	100.0

Column Percentage Table 열 백분율 표.

웨이퍼성능이 좋은 것 334개 중에 티끌이 발견된 것이 14개로 4.2%

	Wafer Condition Good Bad	Total
Particles Found Yes No	4.2 31.0 95.8 69.0	11.1 88.9
Total	100.0 100.0	100.0

Overall Total Percentages Table 전반적 총 백분율 표

기간 중 생산된 450개의 웨이퍼 중 티끌이 발견되었지만 성능이 좋은 것이 14개로 3.1%

	Wafer Condition Good Bad	Total
Particles Found Yes No	3.1 8.0 71.1 17.8	11.1 88.9
Total	74.2 25.8	100.0

Interpretation: The simplest two-way table has but two rows and two columns in its inner part.

Each cell in the inner part of the table represents the count or percentage of a pairing, or cross-classifying, of categories from each variable. Two-way tables reveal what combination of values is most prevalent in data. In the example, the tables reveal that bad wafers are much more likely to have particles than the good wafers. Because the number of good and bad wafers was unequal in this example, you can best see this pattern in the Row Percentage table. That table shows that nearly three-quaters of the wafers that had particles were bad, but only 20% of wafers that did not have particles were bad.

2.2 Presenting Numerical Data  수치적 자료의 제시

This section reviews the commonly used frequency and percentage distribution tables and histogram and dot scale diagram charts. You present numerical data by either organizing the responses into groups that contain specific ranges of values or producing charts that represent each response as a point or a symbol.

The Frequency and Percentage Distribution  도수(度數) 및 백분율 분포표, 히스토그램, 점도표 등

Concept: A three-column table of grouped numerical data that contains the names of each group in the first column, the counts (frequencies) of each group in the second, and the percentages of each group in the third.

Example: Viscosity measurements taken from 120 manufacturing batches, ordered from lowest to highest viscosity:  예: 점성(黏性)측정 자료

----------------------------------------------------- 12.6  12.8  13.0  13.1  13.3  13.3  13.4  13.5  13.6  13.7  13.7  13.7  13.8  13.8  16.5

13.9  13.9  14.0  14.0  14.0  14.1  14.1  14.1  14.2  14.2  14.2  14.3  14.3  14.3  16.6

14.3  14.3  14.3  14.4  14.4  14.4  14.4  14.4  14.4  14.4  14.4  14.5  14.5  14.5  16.8

14.5  14.5  14.5  14.6  14.6  14.6  14.7  14.7  14.8  14.8  14.8  14.8  14.9  14.9  16.9

14.9  14.9  14.9  14.9  14.9  15.0  15.0  15.0  15.0  15.1  15.1  15.1  15.1  15.2  16.9

15.2  15.2  15.2  15.2  15.2  15.2  15.2  15.3  15.3  15.3  15.3  15.3  15.4  15.4  17.0

15.4  15.4  15.5  15.5  15.6  15.6  15.6  15.6  15.6  15.7  15.7  15.7  15.8  15.8  17.6

15.9  15.9  16.0  16.0  16.0  16.0  16.1  16.1  16.1  16.2  16.3  16.4  16.4  16.5  18.6

-----------------------------------------------------

Frequency and Percentage Distribution for the Viscosity of a Chemical

위에서 제시한 자료를 이용하여 표를 작성함.

Viscosity	Frequency	Percentage
12.0 to under 13.0 13.0 to under 14.0 14.0 to under 15.0 15.0 to under 16.0 16.0 to under 17.0 17.0 to under 18.0 18.0 to under 19.0

Interpretation: Frequency and percentage distributions enable you to quickly determine differences among the many values. In this example, you can quickly see that most of the viscosities are between 14.0 and 16.0, and that there are very few viscosities that are either below 13.0 or above 18.0.  대부분 14.0~16.0사이.  주의점 : 계급설정 주의.

Care should be taken in forming groups for distributions, because the ranges of the group will affect how you perceive the data. For example, had the viscosity data been grouped into only two groups, below 15 and 15-and-above, you would see no pattern to the data.

Histogram  히스토그램  막대간격이 없도록 그림

Concept: A special bar chart for grouped numerical data in which the frequencies or percentages of each group of numerical data are represented as individual bars on the vertical Y-axis and the variable is plotted on the horizontal X-axis. In a histogram, there are no gaps between adjacent bars as there would be in a bar chart of categorical data.

Example:

The values below the bars (12.5, 13.5, 14.5, 15.5, 16.5, 17.5, and 18.5) are mid- points, the approximate middle value for each group of data.

Interpretation: Histogram reveal the overall shape of the frequencies for the groups.

The Dot Scale Diagram  점 도표.  두 개 이상의 경우 점들을 수직으로 표시

Concept: A chart in which each response is represented as a dot above a horizontal line that extends through the range of all values. Should two or more response values be identical, the dots for these responses are stacked (placed vertically) above each other.

Example:

Interpretation: By avoiding the grouping of data, the dot scale diagram avoids the misleading patterns that histograms sometime produce. This dot scale diagram shows that the viscosity data is not as nicely symmetrical as the histogram would suggest. Note, however, that both charts clearly show a concentration of values in the center of the distribution between 14 and 16.

The Time-Series Plot  시계열(時系列)도(圖)

Concept: A chart in which each point represents the value of a numerical variable at a specific time. By convention, the X-axis (the horizontal axis) always represents units of time, and the Y-axis (the vertical axis) always represents units of the variable.

Example:  1988-2002년, 뉴욕 지역 주택 mortgage 지출액

Interpretation: Time-series plots can reveal patterns over time, patterns that can be quite hard to see when looking at a long list of numerical values. In this example, the plot reveals that monthly mortgage payments (when considered in 2002 constant dollars) dropped steadily in the late 1980s and early 1990s, only to level off and start rising again since 2000.

The Scatter Plot  산포도(散布圖).  두 변수 사이의 관계를 알고자 함

Concept: A chart that plots the values of two numerical variables for each response. In a scatter plot, the X-axis always represents units of one variable, and the Y-axis always represents units of the second variable.

Example:

Interpretation: Scatter plots help reveal patterns in the relationship between two numerical variables. The scatter plot for this example reveals a strong positive linear (straight line) relationship between the number of cubic feet moved and the number of labor hours required. Based on these data, a manager at the urban moving company could conclude that number of cubic feet moved in a specific job will be a useful predictor of the number of labor hours that will be needed.

Labor Hours and Cubic Feet Moved for 36 Moving Job

Hours	Cu. Feet	Hours	Cu. Feet
24	545	25	557
13.5	400	45	1028
26.25	562	29	793
25	540	21	523
9	220	22	564
20	344	16.5	312
22	569	37	757
11.25	340	32	600
50	900	34	796
12	285	25	577
38.75	865	31	500
40	831	24	695
19.5	344	40	1054
18	360	27	486
28	750	18	442
27	650	62.5	1249
21	415	53.75	995
15	275	79.5	1397

2.3  Misusing Graphs

Good graphical displays, such as those presented in this chapter, reveal what the data are conveying. Unfortunately, many graphs that you will see in the news media or in formal reports either are incorrect, misleading, or are so unnecessarily complicated that they never should be used.

When producing your own graphs, consider the following guidelines:

- Avoid unnecessary decorations or illustration around the borders or in the background.

- Avoid the use of fancy pictorial symbols to represent data values.

- In two-dimensional graphs, always include a scale for each axis.

- When charting non-negative values, the scale on the vertical axis should begin at zero.

- Always label every axis.

- Always supply a title.

- Always choose the simplest graph that can present your data.

CHAPTER 3 DESCRIPTIVE STATISTICS FOR NUMERICAL VARIABLES 기술(記述)통계

3.1 Measures of Central Tendency  중심경향치 (中心傾向値) (산술)평균, 중앙값, 최빈값

Because the data values of most numerical variables show a tendency to group around a specific value, statisticians use a set of methods, collectively known as measures of central tendency, to help identify the general properties of that data. Three commonly used measures are the arithmetic mean, also known simply as the mean or average, the median, and the mode. You can calculate these measures as either sample statistics or population parameters.

The Mean 평균

Concept: A number equal to the sum of the data value for a variable, divided by the number of data values that were summed.

Example: Many sports statistic (including baseball batting average, etc), average SAT score for incoming freshmen at a college, average age of the workers in a company, average waiting times at a bank.

Interpretation: The mean represents a "balance point" 무게 중심점 in a set of data values, similar to a fulcrum on a seesaw. The mean has one great weakness: individual extreme values can distort the balance point. 단점: 이상치(outlier)의 영향을 크게 받음.

Problem 1. Suppose you want to determine the typical time (in minutes) that elapses between your alarm clock's programmed wake-up time and the time you leave your home for work. You decide to measure actual times for ten consecutive working days and record the following times:

Day    1     2     3     4     5      6     7     8     9     10

Time  39    29    43    52    39    44    40    31    44    35

The result, 39.6 minutes, is the mean time to get ready.

Problem 2. Consider Problem 1 but imagine the following recorded times: 52 -> 102를 가정.

Day    1     2     3     4      5     6     7     8     9     10

Time  39    29    43   102    39    44    40    31    44    35

That would make the sum 446 minutes, and the mean 44.6 minutes. Now how one extreme value has dramatically changed the mean. Instead of being a number "somewhere" in the middle of the 10 get-ready times, the new mean of 44.6 minutes is greater than 9 of the 10 get-ready times.  In this case, the mean fails as a measure of "central tendency."

단지 자료 1개만 변하였지만, 평균은 중심 경향치의 역할을 못한다.

[eqn: bar X `=` {X sub 1 + X sub 2 + ...+ X sub n } over {n} `=` { sum _{ i=1} ^{n }X sub i } over {n}`=` {sum X} over {n}]

[eqn: bar X`]: represents the number that is the mean of a sample,  [eqn: n`]: sample size.

Median 중앙값

Concept: The middle value in a set of data values for a variable when the data values have been ordered from lowest to highest value (ascending order). When the number of data values to be summarized is even ,자료 개수가 짝수이면, you perform a special calculation to determine the median because data sets with an even number of values have no natural middle value.

Example: Median household income for a region; median age for buying a consumer product, etc.

Interpretation: The median splits the set of ranked data values into equal-in- numbers parts. Extreme values do not affect the median, making the median a good alternative to the mean when such values occur.  중앙값은 크기순으로 나열된 자료를 정확히 반으로 나누는 역할.

When the number of data values to be summarized is even, the median is calculated by taking the mean of the two values closest to the middle, when all values are ranked from lowest to highest.

To quickly determine the middle position, add 1 to the number of data values and divide that sum by 2.

Using the [eqn: n`] symbol previously defined on page 12, you can define the

median as: Median = [eqn: {n+1} over {2} th``]ranked value.

p.13의 Problem 1에서 중앙값을 구하면 39.5.

Mode 최빈치(最頻値), -값

Concept: The value (or values) in a set of data values for a variable that appears most frequently.

Example: The most common score on an exam, the most likely income, etc.

Interpretation: Similar to the median, extreme values do not affect the mode; unlike the median, however, the mode can vary much more from sample to sample than the median (or mean).

Some sets of data values have no mode - all the unique values appear the same number of times. Other sets of data values can have more than one mode.

최빈값이 없거나 두 개 이상도 가능.

cf. uniform distribution; bimodal distribution.

Quartiles 4분위수(四分位數)

Concept: The three values that split a set of ranked data values for a variable  into four equal parts - quarters, or quartiles.  The first quartile, 제1사분위수 [eqn: Q sub 1`], is the value such that 25.0% of the ranked data value are smaller and 75.0% are larger. The second quartile, [eqn: Q sub 2 `], is another name for the median 제2사분위수=중앙값, which splits the ranked values into two equal parts. The third quartile, 제3사분위수 [eqn: Q sub 3`], is the value such that 75.0% of the ranked values are smaller and 25.0% are larger.

Example: Many math and reading standardized tests for children report results  in terms of quartiles.

Interpretation: Quartiles help summarize large sets of data values by allowing you to identify the 25th, 50th, and 75th percentiles. If you scored in the third quartile on a standardized test, your score was in the top 25% of all scores.  If you did exceptionally well, and learned that your score was reported as the 99th percentile, you would know that your score was in the top 1% of all scores (and therefore greater than 99% of all scores). 백분위수

To quickly determine the 1st quartie, add 1 to the number of data values and divide that sum by 4. For example, for 11 values, add 1 to 11 to get 12 and divide 12 by 4 to get 3 and determine that the 1st quartile is the third ranked value. To quickly determine the 3rd quartile, add 1 to the number of data values and divide that sum by 4 and then multiply the quotient by 3.

Problem 1. You are asked to determine the first quartile for the ranked get- ready times:

제1사분위수 계산 예

Time	29    31    35    39    39    40    43    44    44    52

Ranked Value   1st    2nd   3rd    4th   5th   6th   7th    8th   9th   10th

- First add 1 to 10, and divide by 4 to get 2.75 to identify the second and third ranked        values, 31 and 35.

- Multiply 35, the larger value, by the decimal fraction 0.75 to get 26.25.

- Multiply 31, the smaller value, by the fraction 0.25 to get 7.75.

- Then add 26.25 and 7.75 to produce 34, the first quartile value, indicating that 25% of the    get-ready times are 34 minutes or below and that the other 75% are 34 minutes or above.

제3사분위를 계산하고자 한다.

Using the equation for the median, Median = [eqn: {n+1} over {2} th``]ranked value

you can express the first quartile [eqn: Q sub 1`] as [eqn: Q sub 1 = {n+1} over {4} th``]ranked value

and the third quartile[eqn: Q sub 3], as [eqn: Q sub 3 = {3(n+1)} over {4} th] ranked value.

Problem 2. You want to undertake a study that compares the cost for a restaurant meal in a major city to the cost of a similar meal in the suburbs ourside the city. You collect data about the cost of a meal per person from a sample of 50 city restaurants and 50 suburban restaurants and arrange the 100 numbers in the two ranked sets as follows:

도심과 교외지역 식비를 비교:

City

14  22  23  25  26  27  30  31  31  32  33  34  34  35  35  35  36  36  37  37  38  38  38  39  39  39  39  40  41  42  43  44  44  44  44  45  45  48  48  49  50  50  50  50  51  51  53  53  56  63

Suburban

23  23  24  24  25  25  26  26  26  26  27  27  28  28  29  29  29  30  30  30  30  31  31  32  32  32  33  33  34  34  36  37  37  37  38  38  38  38  38  38  39  39  41  43  44  44  48  51  51  55

Due to the many data values involved, you decide to use Microsoft Excel to calculate the mean, median, and the 1st and 3rd quartiles of the two groups of cost data.

Excel을 이용하여 평균, 중앙값, 제1, 제3사분위를 계산하기.

From the results, you note the following:

- The mean cost of city meals, 39.74, is higher than the mean cost suburban meals,          33.74.

- The median cost of a city meal, 39, is also higher than the median suburban cost, 32.

- The 1st and 3rd quartiles for city meals (34 and 48) are also higher than their suburban

counterparts (28 and 38).

3.2 Measures of Variation  변동(變動)의 척도

A second important property that describes a set of numerical data is variation. Variation is the amount of dispersion, or "spread," in the data. Four frequently used measures of variation are the range, the variance, the standard deviation, and the Z score, all of which can be calculated as either sample statistics or population parameters.

산포(散布), 범위(範圍), 분산(分散), 표준편차(標準偏差),  Z-값

The Range  범위=최대값-최소값

Concept: The difference between the largest and smallest data values in a set  of values for a variable.

Example: Daily hight and low temperatures, stock market high and low closing  prices, etc.

Interpretation: The range is the number that represents the largest possible difference between any two values in a set of data values for a variable.

The Variance and the Standard Deviation  분산과 표준편차

Concept: Two related numbers that each individually measures how a set of data values for a variable fluctuate around the mean of that variable.

Example: The variance among SAT scores for incoming freshmen at a college, the standard deviation in the age of the workers in a company, etc.

Interpretation: The variance and standard deviation help you to know how a set of data values distributes around its mean. For almost all sets of data values, the majority of the values lie within an interval of plus and minus one standard deviation above and below the mean. 평균을 중심으로 분포된 정도. 거의 모든 자료의 대부분이 평균 [eqn: ±] 1표준 편차 내에 놓여있다.

The simplest measure of variation might take the difference between each value and the mean and sum these differences. However, by the properties of arithmetic and the definition of mean, the result of such calculations would be zero for every set of data values - not very helpful in comparing one set to another. 편차의 합은 항상 “0"임.

Instead, statisticians developed a method in which the difference between each data value and the mean is squared and the squared numbers are summed. The sum of squares is then divided by either one less than the number of data values, for sample data, or the number of data value, for population data, to produce the variance. The variance itself can never be negative. The standard deviation is then the positive square root of the variance.

편차의 제곱 합을 이용. 표본과 모집단의 경우 각각 [eqn: n-1]과 [eqn: N]을 이용.

Problem 1. First Step Calculating the Variance and Standard Deviation for the Get-Ready Times Data  분산과 표준편차 계산:

Day	Time	Difference: Time - Mean(39.6)	Square of Difference
1 2 3 4 5 6 7 8 9 10	39 29 43 52 39 44 40 31 44 35	-0.6 -4.6	0.36 21.16
sum of squares:			412.40

분산:[eqn: {412.40} over {10-1} =45.82]

표준편차:[eqn: sqrt {45.82} CONG 6.77]

평균[eqn: ±]1표준편차 범위: [eqn: (39.6+-6.77)]

Sample Variance=[eqn: S sup 2 = {sum (X sub i - bar X ) sup 2 } over {n-1}]

Population Variance=[eqn: sigma sup 2 = {sum (X sub i - mu ) sup 2} over {N}]

Sample standard deviation=[eqn: S= sqrt{ {sum (X sub i - bar X ) sup 2 } over {n-1}}]

Population Standard deviation=[eqn: sigma = sqrt {{sum (X sub i - mu) sup 2 } over {N}}]

Standard (Z) Score  표준점수 (Z-값)

Concept: The number that is the difference between a data value for a variable and the mean of the variable, divided by the standard deviation, is its Z score.

Example: The Z score for a particular incoming freshman's SAT score, etc.

Interpretation: Z score help you determine whether a data value is an extreme value, or outlier. As a general rule, a Z score that is less than -3 or greater than +3 indicates that the data value its represents is an extreme value.

절대값 3보다 큰 Z 점수는 곧 그 값이 이상치라는 것을 말해준다

Problem 1. Table of Z Score Calculation for the Get-Ready Time Sample

Using symbols, you can express the Z score as follows: [eqn: Z={X- bar X} over S]

Day	Time	Difference: Time - Mean(39.6)	Z Score
1 2 3 4 5 6 7 8 9 10	39 29 43 52 39 44 40 31 44 35	-0.6 -4.6	-0.09 -0.68

3.3 Shape of Distribution  분포의 형태  중심경향, 산포도, 분포형태

A third important property that describes a set of numerical data is shape. Shape describes the pattern of the distribution of data values through the range of the data values.  The shape may be either symmetrical, left-skewed, or right-skewed.

Symmetrical Shape  대칭분포 형태  평균=중앙값(=최빈값)

Concept: A set of data values in which the mean equals the median value.

Example: Scores on a standardized exam, actual amount of soft drink in a one-liter bottle.

Left-Skewed Shape  왼 꼬리분포  평균 < 중앙값

Concept: A set of data values in which the mean is less than the median value. Also known as negative skew.

Example: Scores on an exam in which most students score between 80 and 100, whereas a few student score between 10 and 79.

Right-Skewed Shape  오른 꼬리분포  평균 > 중앙값

Concept: A set of data value in which the mean is greater than the median value. Also know as positive skew.

Example: Prices of new homes, annual family income, etc.

Interpretation: Right (or positive) skewness occurs when the data set contains some extremely high data values (that increase the mean).

To see the shape more clearly, statisticians create area-under-the curve or distribution graphs, in which a plotted, curved line represents the tops of all the bars of the histogram.

분포곡선 하의 면적:

The Box-and-Whisker Plot  Box-Whisker 그림 최소값,[eqn: Q _{1}], 중앙값, [eqn: Q _{3}] 최대값 이용 대칭정도 판별.

Concept: For a set of data values for a variable, the five numbers that correspond to the smallest value, the first quartile [eqn: Q sub 1], the median, the third quartile [eqn: Q sub 3], and the largest value.

Interpretation: The five-number summary concisely summarizes the shape of a set of data values for a variable. This method determines the degree of symmetry (or skewness) based on the distances that separate the five numbers. To compare these distances effectively, you can create a box-and-whisker plot as shown below in which the five numbers are plotted as vertical lines, interconnected so as, with some imagination, to form a "box" from which a pair of cat whiskers sprout.

Smallest             First                   Third               Largest

Value              Quartile     Median     Quartile              Value

X

A box-and-whisker plot shows a symmetric shape for a set of data values if both of the following relationships are present in the plot: 대칭분포를 위한 조건:

- The distance from the line that represents the smallest value to the line that represents     the median equals the distance from the line that represents the median to the line that     represents the largest value.

- The distance from the line than represents the smallest value to the line that represents     the first quartile equals the distance from the line that represents the third quartile to      the line that represents the largest value.

A box-and-whisker plot shows a right-skewed shape for a set of data values if the following relationships are both present in the plot: 오른 꼬리 분포의 조건:

- The distance from the line that represents the median to the line that represents the        largest value is greater than the distance from the line that represents the smallest         value to the line that represents the median.

- The distance from the line that represents the third quartile to the line that represents      the greatest value is greater than the distance from the line that represents the smallest     value to the line that represents the first quartile.

A box-and-whisker plot shows a left-skewed shape ..if the following ..: 왼 꼬리분포 조건?

Problem 1. Produce the box-and-whisker plots for the restaurant meal cost of both the city and suburban groups. What do you find in examining the box-and-whisker plots? Give your conclusion.