1、1Guojun Wang(), Ph.D.School of Economics and ManagementOffice: Room 1616, Tongji Building Suite AEmail：Wechat: GuojunWangEconBasics of Calculus, Statistics & Analysis 2About MeJoined SEMTongji University in September, 2014Education BackgroundUC Davis Ph.D. in EconomicsUC Berkeley Exchange student in

2、 the finance Ph.D. programUniversity of Central FloridaM.S. in Financial MathematicsZhejiang University B.S. & M.S. in MathematicsWorking ExperienceQuantitative Strategies Group-CalPERS (largest US pension fund)Infrastructure Investment Group-CalSTRS (2nd largest US Pension fund)Hedge Fund Group-Chi

3、na Investment Corporation (sovereign wealth fund of China)Fixed Income Group-Matthews Asia (top 5 Asia-focused mutual fund)Wealth Management-Morgan Stanley Smith Barney3Part 2 Basic Statistical Techniques in Business and ManagementBasic Statistical Techniques in Business and Management 4ContentsWhat

4、 is Statistics?Descriptive StatisticsInferential StatisticsDescriptive Statistics MethodsFrequency Distribution and Statistical ChartsMeasure of Central TendencyMeasures of dispersion and skewness5Statistics is only one kind of, but most often used quantitative method in B&M. The definition and uses

5、 of the statisticsl Statistics in our life: economic growth, consumer price index, stock market price index, the structure of products, age structure of company employee, 6What is Statistics? As a kind of science, statistics is the science of collecting, organizing, presenting, analyzing and interpr

6、eting numerical data for the purpose of assisting in business analysis and decision making. Statistics tell you how to collect data you need, how to organize and interpret the data, how to analyze the data and get more and useful information, how to understand comprehensively the issues you interest

7、 in.7l Every one uses the statistics more or less. Most often users: Financial analysts, Economists, Market administration, Financial institutions, Marketing managers, accountants, quality managers, business executives, educators, gamers, governments, hospital administration, Teachers, Doctors, farm

8、ers, individual investors, bankers, social workers, also use the statistics in their work and even in the life. 8Two types statisticsl Descriptive Statistics: Statistical methods of describing data that have been collected. Statistical Techniques for organizing, summarizing, and presenting data in a

9、n informative way. It can help us know the key information of the population.9l Inferential Statistics (Statistical Inference or Inductive Statistics;推斷統(tǒng)計(jì)學(xué)) Inferential Statistics is about methods used to find out some meaningful information about a population based on a sample.What is Population? A

10、 collection of all-possible individuals、events、objects、or measurements of interesting. A particular object to be studied. Such as the annual income of PT11 participants, the profitability of FORTUNE500、the work load of employee in your companies, and so on.10What is Sample? A portion, or part of the

11、 population we are interest. Such as the annual income of participants in G5, the work load who work in marketing department, the participants who are working for Fortune 500 in PT13. Case Study: Case 2 The descriptive analysis of profitability for HSI constituents 11Chapter 3 Descriptive Statistica

12、l Methods: means of communication3.1 Frequency Distribution and Statistical Charts We will learn how to summarize data. There are two types of method summarizing raw data: frequency distribution and charts.Showing , profitability of HIS constituents in 2011What information you can have from these ta

13、bles?123.1.1 Frequency distribution In practice, we often face a big pile of data,we cant get any information from these data. So we need some kind methods to summarize these data and get useful information about overall of the object. The frequency distribution is one of the methods.13A. Constructi

14、on of a Frequency DistributionYou can construct a frequency distribution by following steps when you have many data already:Step1: Setting up groupings (called classes). Grouping criterion (age of students) Determine a group interval (5 years)Step2: Tallying observations into their appropriate class

15、;(computer can do it for you)Step3: Counting the number of tallies in every class. Then we get the number of observation in each class, this called class frequency Example: “P/Eof HS300.EXL14B. Class Interval and MidpointClass interval (25-20=30-25=40-35=5) is the size between the lower limit and up

16、per limit of the class; this size must be same in each classMidpoint is in the middle of the class (23 or 28 or 33 or 38 etc.), also called class mark. C. Suggestions on Constructing a Frequency Distribution . Use equal-size class interval. Find the suggested class interval (highest value-lowest val

17、ue)/number of classes). Choose a appropriate number of classes (in general, 5-15 classes are suggested). 15 3.1.2 Graphing the frequency distribution We have three commonly used graphic forms to graphing frequency distribution: A. Histogram See EXEL sheet B. Frequency polygon It consists of line seg

18、ments connecting the points formed by intersection of the class midpoint and class frequency. Frequency polygon preferred when comparing distribution.163.1.3 Relative frequency distribution and cumulative polygonA. Relative frequency distribution Sometimes, describing the frequency distribution with

19、 relative scale may be very effective, it shows the percent of the total number of observations in each class. To convert the class frequency to relative frequency, each class of the frequency is divided by the total number of the frequency.17 Cumulative frequency polygonIn many cases, we are intere

20、sted in such topics as “how many students are younger than 30?” or “how many students are older than 35?” Developing a cumulative frequency distribution and portraying it in a cumulative frequency polygon can answer those questions. Less-than cumulative polygona. Draw a cumulative polygon for “stude

21、nts age”b. See the example in page 4118frequency polygon19Similarly you can do a More-than cumulative polygonExample: the cumulative distribution of work age of PT6.(PT3participants data.exl/Distr.workage) 3.1.4 Line, Bar, Pie ChartA. Line chart (see EXEL sheets)B. Bar chart (see EXEL sheets)C. Pie

22、chart (see EXEL sheets)203.2 Measures of Central TendencyNow we want to know some numerical features of the distribution. The first feature of distribution is its central tendency. It shows where is the central of the data or the population. It is a typical value that represents all the values. 21Th

23、e MEASURES OF CENTRAL TENDENCY are a single value that represents a set of data. In general, there are three measures of central tendency: Mean, Median and Mode. 3.2.1 Mean Mean is also called average, arithmetic mean, geometric mean and weighted average.22A. Arithmetic meanFor different objects, th

24、ere tow kinds of mean: population mean and sample mean.Population mean pinpoint the center of the population. To find the mean, use the following formula:Population mean=(Sum of all the values in the population)/(Number of values in the population)NX23Sample mean is the center of the sample, its cal

25、culation formula is same with population mean, but the shorthand notations are different.nXXSample mean =(Sum of all the values in the sample)/(Number of values in the sample)Note: There are differences in notations: N and n, and X. These notations will often be used in the later. 24 The properties

26、of the arithmetic mean A . All the values are included in computing the mean B. Each set of data has unique mean C. The deviation of each value from the mean will always be zero, that is: D. Mean is a balance point of the data E. Affected by unusually large or small values0XX25B. Geometric meanGeome

27、tric mean has no property c of the arithmetic mean. If all X are positive, thennnXXXXMeanGeom.321This is especially useful in measuring the average annual percent increase in business or economic data, from one time period to another. X is usually the gross change of a value, i.e. 1+net change.It is

28、 always less than or equal to the arithmetic mean. e.g. An 5% increase followed by an 10% increase implies an AM=7.5% and a GM=7.47%26 The properties of the weighted mean are: a. More subjective in determining the weights; b. Can be applied to some complicate issues, (Index of stock market)iiiwXwXC.

29、 Weighted mean is fund by multiplying each observation by its corresponding weights, summing these values, then dividing by the sum of the weights:27B. Geometric mean 283.2.2 The MedianThe median is the middle value in a set of values ordered from smallest to largest. (Example in EXEL sheet)3.2.3 Th

30、e ModeThe mode is the value that occurs most in a set of d a t a . F o r t h e d a t a s e t f o l l o w i n g : 2,5,1,2,7,5,2,5,4,6,5,8,2,2,2. The mode is 2, it occurs 6 times.29 The properties of mode: a. It can be determined for all levels of data; b. A set of data can have more than one mode. Bi

31、modal issue: If there are tow values occur a large number of times in a set of data, the distribution is called bimodal. If two values occur same times, then the data set has tow modes.If the data set has more than tow modes (or peaks), the distribution is referred to multimode. In such cases we wou

32、ld probably not consider any of the mode being representative of the central value of the data.303.2.4 The relationship between the mean, median and mode These three measures are all central tendency measures; they are same or equal when the frequency distribution is symmetric, i.e. Mean=Median=Mode

33、 . (see chart3-2, page 67) Symmetric distribution is a very frequently used concept in quantitative analysis; it has same shape on either side of the center axis. But in many cases, the distributions are asymmetrical, or skewed. The relationship between three averages changes on types of asymmetry;

34、31 For negatively skewed: mode=median=mean; For positively skewed: mean= median=modepage 81,e49; P.92, e58, ?323.3 Measures of dispersion and skewnessA simple question:Would you walk across a lake that has a average depth of 4 feet? “Small dispersion” means that values in data are quite closing each

35、 other, and the central tendency is meaningful or reliable. “Big dispersion” means that the mean or median is not very reliable.333.3.1 Measures of dispersion for raw data There are four main measures of dispersion: Range, Mean deviation, Variance and Standard deviation.A. Range: Highest value Lowes

36、t valuel It is the simplest measure of dispersionl Example: What is the highest price and lowest price in the Shanghai stock market? The range is ?34B. Mean deviation: Arithmetic mean of the deviations from the mean: MD=Understandable, MD is the average distance from values to mean, very meaningful.

37、 But the absolute values are difficult to work with, so it is not as frequently used as other measures, such as standard deviation.nXX35C. Variance and Standard Deviation Both of these tow measures are based on squared deviations: Variance: The arithmetic mean of the squared deviations from the mean

38、. Squaring here is to eliminate the chance of having negative numbers of deviation.l Population Variance: l Sample Variance: NXX22122nXXs236Why is n-1 in the denominator? It is because of making s as a unbiased estimator of the population variance. The is usually unknown, in practice, is estimated b

39、y s, so it must be unbiased.37Standard deviation: the square root of the variance.Variance is very difficult to interpret because it is squared, has the different unit with original data.(years, or dollars). To eliminate the dilemma, standard deviation takes the square root of the population varianc

40、e. It is called population standard deviation 222NXNXNXX38For samples, similarly, the sample standard deviation:The sample standard deviation is also the unbiased estimator of population standard deviation. This is the main role of it. 11222nnXXnxxs39Interpretation and uses of the standard deviation

41、 what are they used for? In generally, standard deviation measure the spread of values in the population or sample, small S.D. means the values are quite close each other, and vice versa. 403.3.1 Other measures of dispersion Variance and standard deviation are most important and often used measures

42、of dispersion. But, except them, there are also other useful measures of dispersion, such as interquartile range, quartile deviation, percentile range and so on.A. Interquartile rangeQuartile: divide the whole observations into four parts with equal size, i.e., 25% percent of observations in each pa

43、rt.41The interquartile range=Third quartile-First quartile=Q3-Q1A. Percentile rangePercentile is similarly with quartile, it divide a distribution into 100 parts and more exactly describe the data set. P10 represents the 10th percentile and so on. 42The percentile range P90 - P10 include 80% of the observations, if the