DOCSLIB.ORG

Statistics, Measures of Central Tendency I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Statistics, Measures of Central TendencyI We are considering a random variable X with a probability distribution which has some parameters. We want to get an idea what these parameters are. We perfom an experiment n times and record the outcome. This means we have X1;:::; Xn i.i.d. random variables, with probability distribution same as X . We want to use the outcome to infer what the parameters are. Mean The outcomes are x1;:::; xn. The Sample Mean is x1+···+xn x := n . Also sometimes called the average. The expected value of X , EX , is also called the mean of X . Often denoted by µ. Sometimes called population mean. Median The number so that half the values are below, half above. If the sample is of even size, you take the average of the middle terms. Mode The number that occurs most frequently. There could be several modes, or no mode. Dan Barbasch Math 1105 Chapter 9 Week of September 25 1 / 24 Statistics, Measures of Central TendencyII Example You have a coin for which you know that P(H) = p and P(T ) = 1 − p: You would like to estimate p. You toss it n times. You count the number of heads. The sample mean should be an estimate of p: EX = p, and E(X1 + ··· + Xn) = np: So X + ··· + X E 1 n = p: n Dan Barbasch Math 1105 Chapter 9 Week of September 25 2 / 24 Descriptive StatisticsI Frequency Distribution Divide into a number of equal disjoint intervals. For each interval count the number of elements in the sample occuring. Histogram see the next slide Grouped Data Mean Essentially calculate the mean of the frequency distribution. Intervals are used, rather than single values. It is assumed that all these values are located at the midpoint of the interval. The letter xM is used to represent the midpoints and f represents the frequencies: P fi xM;i n Frequency Polygon Connect the middles of the tops of each interval. Dan Barbasch Math 1105 Chapter 9 Week of September 25 3 / 24 Histogram A histogram is a graphical representation of the distribution of numerical data. It is a kind of bar graph. To construct a histogram, the first step is to "bin" the range of values, that is, divide the entire range of values into a series of intervals, and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent, and are often (but are not required to be) of equal size. Bin Count −3:5 − 2:51 9 −2:5 − 1:51 32 −1:5 − 0:51 109 −0:5 − 0:49 180 0:5 − 1:49 132 1:5 − 2:49 34 2:5 − 3:49 4 (−3)·9+(−2)·32+(−1)·109+·(0)180+1·132+2·34+3·4 Mean: 500 Dan Barbasch Math 1105 Chapter 9 Week of September 25 4 / 24 Example The table on the next page gives the number of days in June and July of recent years in which the temperature reached 90 degrees or higher in New Yorks Central Park. Source: The New York Times and Accuweather.com. a. Prepare a frequency distribution with a column for intervals and frequencies. Use seven intervals, starting with [0 4]. b. Sketch a histogram and a frequency polygon, using the intervals in part a. c. Find the mean for the original data. d. Find the mean using the grouped data from part a. e. Explain why your answers to parts c and d are different. f. Find the median and the mode for the original data. Dan Barbasch Math 1105 Chapter 9 Week of September 25 5 / 24 9.1 Frequency Distributions; Measures of Central Tendency 417 a. Use this table to estimate the mean income for white house- Animal Number of Blood Types holds in 2008. Pig 16 b. Compare this estimate with the estimate found in Exercise 39. Discuss whether this provides evidence that white Amer- Cow 12 ican households have higher earnings than African Ameri- Chicken 11 can households. Horse 9 41. Airlines The number of consumer complaints against the top Human 8 U.S. airlines during the first six months of 2010 is given in the Sheep 7 following table. Source: U.S. Department of Transportation. Dog 7 Rhesus monkey 6 Complaints per 100,000 Mink 5 Airline Complaints Passengers Boarding Rabbit 5 Delta 1175 2.19 Mouse 4 American 660 1.56 Rat 4 United 487 1.84 Cat 2 US Airways 428 1.69 Continental 350 1.64 Southwest 149 0.29 General Interest Skywest 77 0.65 44. Temperature The following table gives the number of days in American Eagle 68 0.87 June and July of recent years in which the temperature reached Expressjet 56 0.70 90 degrees or higher in New York’s Central Park. Source: The Alaska 34 0.44 New York Times and Accuweather.com. Temperature Data a. By considering the numbers in the column labeled “Com- plaints,” calculate the mean and median number of com- plaints per airline. Year Days Year Days Year Days b. Explain why the averages found in part a are not meaningful. 1972 11 1985 4 1998 5 c. Find the mean and median of the numbers in the column 1973 8 1986 8 1999 24 labeled “Complaints per 100,000 Passengers Boarding.” 1974 11 1987 14 2000 3 Discuss whether these averages are meaningful. 1975 3 1988 21 2001 4 Life Sciences 1976 8 1989 10 2002 13 1977 11 1990 6 2003 11 42. Pandas The size of the home ranges (in square kilometers) of several pandas were surveyed over a year’s time, with the fol- 1978 5 1991 21 2004 1 lowing results. 1979 7 1992 4 2005 12 1980 12 1993 25 2006 5 Home Range Frequency 1981 12 1994 16 2007 4 0.1–0.5 11 1982 11 1995 14 2008 10 0.6–1.0 12 1983 20 1996 0 2009 0 1.1–1.5 7 1984 7 1997 10 2010 20 1.6–2.0 6 Dan Barbasch Math 1105 Chapter 9 Week of September 25 6 / 24 2.1–2.5 2 2.6–3.0 1 a. Prepare a frequency distribution with a column for intervals 3.1–3.5 1 and frequencies. Use six intervals, starting with 0–4. b. Sketch a histogram and a frequency polygon, using the a. Sketch a histogram and frequency polygon for the data. intervals in part a. b. Find the mean for the data. c. Find the mean for the original data. 43. Blood Types The number of recognized blood types varies by d. Find the mean using the grouped data from part a. species, as indicated by the table below. Find the mean, median, and mode of this data. Source: The Handy Science e. Explain why your answers to parts c and d are different. Answer Book. f. Find the median and the mode for the original data. Measures of Variation Summary of Section 9.2 Range The difference Largest Data - Smallest Data in a Sample. Deviation from the Mean P x2−nx2 P(x −x)2 1 Variance σ2 = s2 = i = i n−1 p n−1 2 Standard Deviation σ = s = s2 These are random variables called Sample Variance and Sample Standard Deviation. For a random variable X ; µ = E(X ) is called the mean. The variance Var(X ) is σ2 = Var(X ) = E((X − µ)2). Main Property/ Explanation for dividing by n − 1: If Xi are P 2 2 (Xi −X ) i.i.d with distribution X ; then if you set S = n−1 , its expected value is E(S2) = σ2: This is not true for the standard deviation, E(S) 6= σ: s P 2 2 fi x − nx Grouped Data s = M;i : n − 1 Dan Barbasch Math 1105 Chapter 9 Week of September 25 7 / 24 ExamplesI Example (Range) Data 15; −3; 4; 7; 18. The smallest is −3, the largest 18 so Range = 18 − (−3) = 21: Always a nonnegative number. Example (Deviation from the Mean) 15−3+4+7+18 In the previous example, x = 5 = 8:2. So 15 − 8:2 = 6:8; −3 − 8:2 = −11:2; 4 − 8:2 = −3:8; 7 − 8:2 = −1:2; 18 − 8:2 = 9:8: Example (Variance and Standard Deviation) 2 2 2 2 2 2 2 2 2 2 2 s2 = 6:8 +11:2 +3:8 +1:2 +9:8 = 15 +3 +4 +7 +18 −5·8:2 p 4 4 s = s2: Dan Barbasch Math 1105 Chapter 9 Week of September 25 8 / 24 ExamplesII Example (Binomial Distribution) P(X = 1) = p; P(X = 0) = 1 − p: Then µ = E(X ) = p; and σ2 = E((X − p)2) = (1 − p)2p + (0 − p)2(1 − p) = p(1 − p): This is the same as E(X 2 − p2) = (1 − p2)p + (−p2)(1 − p) = (1 − p)p: Remark: Note that the formula for variance and standard deviation only holds for n > 2: Otherwise, for n = 1; you would be dividing by 0. For one random variable, the variance is defined as Var(X ) = E((X − E(X ))2): For X1; X2;, two independent random variables, Var(X1 + X2) = Var(X1) + Var(X2): Suppose X is a random variable. We can write a table X a1 a2 ::: an P(X ) p1 p2 ::: pn Dan Barbasch Math 1105 Chapter 9 Week of September 25 9 / 24 Examples III For the expected value µ = E(X ); you multiply the two terms in each column, and add X ai × pn = a1p1 + ··· + anpn: i In a spreadsheet program, the data would be in columns and you would add over the products from the rows.
Recommended publications
  • 5. the Student T Distribution

    5. the Student T Distribution

    Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way.
  • Chapter 8 Fundamental Sampling Distributions And

    Chapter 8 Fundamental Sampling Distributions And

    CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Random Sampling pling procedure, it is desirable to choose a random sample in the sense that the observations are made The basic idea of the statistical inference is that we independently and at random. are allowed to draw inferences or conclusions about a Random Sample population based on the statistics computed from the sample data so that we could infer something about Let X1;X2;:::;Xn be n independent random variables, the parameters and obtain more information about the each having the same probability distribution f (x). population. Thus we must make sure that the samples Define X1;X2;:::;Xn to be a random sample of size must be good representatives of the population and n from the population f (x) and write its joint proba- pay attention on the sampling bias and variability to bility distribution as ensure the validity of statistical inference. f (x1;x2;:::;xn) = f (x1) f (x2) f (xn): ··· 8.2 Some Important Statistics It is important to measure the center and the variabil- ity of the population. For the purpose of the inference, we study the following measures regarding to the cen- ter and the variability. 8.2.1 Location Measures of a Sample The most commonly used statistics for measuring the center of a set of data, arranged in order of mag- nitude, are the sample mean, sample median, and sample mode. Let X1;X2;:::;Xn represent n random variables. Sample Mean To calculate the average, or mean, add all values, then Bias divide by the number of individuals.
  • Chapter 4: Central Tendency and Dispersion

    Chapter 4: Central Tendency and Dispersion

    Central Tendency and Dispersion 4 In this chapter, you can learn • how the values of the cases on a single variable can be summarized using measures of central tendency and measures of dispersion; • how the central tendency can be described using statistics such as the mode, median, and mean; • how the dispersion of scores on a variable can be described using statistics such as a percent distribution, minimum, maximum, range, and standard deviation along with a few others; and • how a variable’s level of measurement determines what measures of central tendency and dispersion to use. Schooling, Politics, and Life After Death Once again, we will use some questions about 1980 GSS young adults as opportunities to explain and demonstrate the statistics introduced in the chapter: • Among the 1980 GSS young adults, are there both believers and nonbelievers in a life after death? Which is the more common view? • On a seven-attribute political party allegiance variable anchored at one end by “strong Democrat” and at the other by “strong Republican,” what was the most Democratic attribute used by any of the 1980 GSS young adults? The most Republican attribute? If we put all 1980 95 96 PART II :: DESCRIPTIVE STATISTICS GSS young adults in order from the strongest Democrat to the strongest Republican, what is the political party affiliation of the person in the middle? • What was the average number of years of schooling completed at the time of the survey by 1980 GSS young adults? Were most of these twentysomethings pretty close to the average on
  • Notes Mean, Median, Mode & Range

    Notes Mean, Median, Mode & Range

    Notes Mean, Median, Mode & Range How Do You Use Mode, Median, Mean, and Range to Describe Data? There are many ways to describe the characteristics of a set of data. The mode, median, and mean are all called measures of central tendency. These measures of central tendency and range are described in the table below. The mode of a set of data Use the mode to show which describes which value occurs value in a set of data occurs most frequently. If two or more most often. For the set numbers occur the same number {1, 1, 2, 3, 5, 6, 10}, of times and occur more often the mode is 1 because it occurs Mode than all the other numbers in the most frequently. set, those numbers are all modes for the data set. If each number in the set occurs the same number of times, the set of data has no mode. The median of a set of data Use the median to show which describes what value is in the number in a set of data is in the middle if the set is ordered from middle when the numbers are greatest to least or from least to listed in order. greatest. If there are an even For the set {1, 1, 2, 3, 5, 6, 10}, number of values, the median is the median is 3 because it is in the average of the two middle the middle when the numbers are Median values. Half of the values are listed in order. greater than the median, and half of the values are less than the median.
  • Weighted Means and Grouped Data on the Graphing Calculator

    Weighted Means and Grouped Data on the Graphing Calculator

    Weighted Means and Grouped Data on the Graphing Calculator Weighted Means. Suppose that, in a recent month, a bank customer had $2500 in his account for 5 days, $1900 for 2 days, $1650 for 3 days, $1375 for 9 days, $1200 for 1 day, $900 for 6 days, and $675 for 4 days. To calculate the average balance for that month, you would use a weighted mean: ∑ ̅ ∑ To do this automatically on a calculator, enter the account balances in L1 and the number of days (weight) in L2: As before, go to STAT CALC and 1:1-Var Stats. This time, type L1, L2 after 1-Var Stats and ENTER: The weighted (sample) mean is ̅ (so that the average balance for the month is $1430.83.) We can also see that the weighted sample standard deviation is . Estimating the Mean and Standard Deviation from a Frequency Distribution. If your data is organized into a frequency distribution, you can still estimate the mean and standard deviation. For example, suppose that we are given only a frequency distribution of the heights of the 30 males instead of the list of individual heights: Height (in.) Frequency (f) 62 - 64 3 65- 67 7 68 - 70 9 71 - 73 8 74 - 76 3 ∑ n = 30 We can just calculate the midpoint of each height class and use that midpoint to represent the class. We then find the (weighted) mean and standard deviation for the distribution of midpoints with the given frequencies as in the example above: Height (in.) Height Class Frequency (f) Midpoint 62 - 64 (62 + 64)/2 = 63 3 65- 67 (65 + 67)/2 = 66 7 68 - 70 69 9 71 - 73 72 8 74 - 76 75 3 ∑ n = 30 The approximate sample mean of the distribution is ̅ , and the approximate sample standard deviation of the distribution is .
  • Lecture 3: Measure of Central Tendency

    Lecture 3: Measure of Central Tendency

    Lecture 3: Measure of Central Tendency Donglei Du ([email protected]) Faculty of Business Administration, University of New Brunswick, NB Canada Fredericton E3B 9Y2 Donglei Du (UNB) ADM 2623: Business Statistics 1 / 53 Table of contents 1 Measure of central tendency: location parameter Introduction Arithmetic Mean Weighted Mean (WM) Median Mode Geometric Mean Mean for grouped data The Median for Grouped Data The Mode for Grouped Data 2 Dicussion: How to lie with averges? Or how to defend yourselves from those lying with averages? Donglei Du (UNB) ADM 2623: Business Statistics 2 / 53 Section 1 Measure of central tendency: location parameter Donglei Du (UNB) ADM 2623: Business Statistics 3 / 53 Subsection 1 Introduction Donglei Du (UNB) ADM 2623: Business Statistics 4 / 53 Introduction Characterize the average or typical behavior of the data. There are many types of central tendency measures: Arithmetic mean Weighted arithmetic mean Geometric mean Median Mode Donglei Du (UNB) ADM 2623: Business Statistics 5 / 53 Subsection 2 Arithmetic Mean Donglei Du (UNB) ADM 2623: Business Statistics 6 / 53 Arithmetic Mean The Arithmetic Mean of a set of n numbers x + ::: + x AM = 1 n n Arithmetic Mean for population and sample N P xi µ = i=1 N n P xi x¯ = i=1 n Donglei Du (UNB) ADM 2623: Business Statistics 7 / 53 Example Example: A sample of five executives received the following bonuses last year ($000): 14.0 15.0 17.0 16.0 15.0 Problem: Determine the average bonus given last year. Solution: 14 + 15 + 17 + 16 + 15 77 x¯ = = = 15:4: 5 5 Donglei Du (UNB) ADM 2623: Business Statistics 8 / 53 Example Example: the weight example (weight.csv) The R code: weight <- read.csv("weight.csv") sec_01A<-weight$Weight.01A.2013Fall # Mean mean(sec_01A) ## [1] 155.8548 Donglei Du (UNB) ADM 2623: Business Statistics 9 / 53 Will Rogers phenomenon Consider two sets of IQ scores of famous people.
  • Volatility Modeling Using the Student's T Distribution

    Volatility Modeling Using the Student's T Distribution

    Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics Aris Spanos, Chair Richard Ashley Raman Kumar Anya McGuirk Dennis Yang August 29, 2003 Blacksburg, Virginia Keywords: Student’s t Distribution, Multivariate GARCH, VAR, Exchange Rates Copyright 2003, Maria S. Heracleous Volatility Modeling Using the Student’s t Distribution Maria S. Heracleous (ABSTRACT) Over the last twenty years or so the Dynamic Volatility literature has produced a wealth of uni- variateandmultivariateGARCHtypemodels.Whiletheunivariatemodelshavebeenrelatively successful in empirical studies, they suffer from a number of weaknesses, such as unverifiable param- eter restrictions, existence of moment conditions and the retention of Normality. These problems are naturally more acute in the multivariate GARCH type models, which in addition have the problem of overparameterization. This dissertation uses the Student’s t distribution and follows the Probabilistic Reduction (PR) methodology to modify and extend the univariate and multivariate volatility models viewed as alternative to the GARCH models. Its most important advantage is that it gives rise to internally consistent statistical models that do not require ad hoc parameter restrictions unlike the GARCH formulations. Chapters 1 and 2 provide an overview of my dissertation and recent developments in the volatil- ity literature. In Chapter 3 we provide an empirical illustration of the PR approach for modeling univariate volatility. Estimation results suggest that the Student’s t AR model is a parsimonious and statistically adequate representation of exchange rate returns and Dow Jones returns data.
  • The Value at the Mode in Multivariate T Distributions: a Curiosity Or Not?

    The Value at the Mode in Multivariate T Distributions: a Curiosity Or Not?

    The value at the mode in multivariate t distributions: a curiosity or not? Christophe Ley and Anouk Neven Universit´eLibre de Bruxelles, ECARES and D´epartement de Math´ematique,Campus Plaine, boulevard du Triomphe, CP210, B-1050, Brussels, Belgium Universit´edu Luxembourg, UR en math´ematiques,Campus Kirchberg, 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Grand-Duchy of Luxembourg e-mail: [email protected]; [email protected] Abstract: It is a well-known fact that multivariate Student t distributions converge to multivariate Gaussian distributions as the number of degrees of freedom ν tends to infinity, irrespective of the dimension k ≥ 1. In particular, the Student's value at the mode (that is, the normalizing ν+k Γ( 2 ) constant obtained by evaluating the density at the center) cν;k = k=2 ν converges towards (πν) Γ( 2 ) the Gaussian value at the mode c = 1 . In this note, we prove a curious fact: c tends k (2π)k=2 ν;k monotonically to ck for each k, but the monotonicity changes from increasing in dimension k = 1 to decreasing in dimensions k ≥ 3 whilst being constant in dimension k = 2. A brief discussion raises the question whether this a priori curious finding is a curiosity, in fine. AMS 2000 subject classifications: Primary 62E10; secondary 60E05. Keywords and phrases: Gamma function, Gaussian distribution, value at the mode, Student t distribution, tail weight. 1. Foreword. One of the first things we learn about Student t distributions is the fact that, when the degrees of freedom ν tend to infinity, we retrieve the Gaussian distribution, which has the lightest tails in the Student family of distributions.
  • Random Variables and Applications

    Random Variables and Applications

    Random Variables and Applications OPRE 6301 Random Variables. As noted earlier, variability is omnipresent in the busi- ness world. To model variability probabilistically, we need the concept of a random variable. A random variable is a numerically valued variable which takes on different values with given probabilities. Examples: The return on an investment in a one-year period The price of an equity The number of customers entering a store The sales volume of a store on a particular day The turnover rate at your organization next year 1 Types of Random Variables. Discrete Random Variable: — one that takes on a countable number of possible values, e.g., total of roll of two dice: 2, 3, ..., 12 • number of desktops sold: 0, 1, ... • customer count: 0, 1, ... • Continuous Random Variable: — one that takes on an uncountable number of possible values, e.g., interest rate: 3.25%, 6.125%, ... • task completion time: a nonnegative value • price of a stock: a nonnegative value • Basic Concept: Integer or rational numbers are discrete, while real numbers are continuous. 2 Probability Distributions. “Randomness” of a random variable is described by a probability distribution. Informally, the probability distribution specifies the probability or likelihood for a random variable to assume a particular value. Formally, let X be a random variable and let x be a possible value of X. Then, we have two cases. Discrete: the probability mass function of X specifies P (x) P (X = x) for all possible values of x. ≡ Continuous: the probability density function of X is a function f(x) that is such that f(x) h P (x < · ≈ X x + h) for small positive h.
  • Mistaking the Forest for the Trees: the Mistreatment of Group-Level Treatments in the Study of American Politics

    Mistaking the Forest for the Trees: the Mistreatment of Group-Level Treatments in the Study of American Politics

    Mistaking the Forest for the Trees: The Mistreatment of Group-Level Treatments in the Study of American Politics Kelly T. Rader Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2012 c 2012 Kelly T. Rader All Rights Reserved ABSTRACT Mistaking the Forest for the Trees: The Mistreatment of Group-Level Treatments in the Study of American Politics Kelly T. Rader Over the past few decades, the field of political science has become increasingly sophisticated in its use of empirical tests for theoretical claims. One particularly productive strain of this devel- opment has been the identification of the limitations of and challenges in using observational data. Making causal inferences with observational data is difficult for numerous reasons. One reason is that one can never be sure that the estimate of interest is un-confounded by omitted variable bias (or, in causal terms, that a given treatment is ignorable or conditionally random). However, when the ideal hypothetical experiment is impractical, illegal, or impossible, researchers can often use quasi-experimental approaches to identify causal effects more plausibly than with simple regres- sion techniques. Another reason is that, even if all of the confounding factors are observed and properly controlled for in the model specification, one can never be sure that the unobserved (or error) component of the data generating process conforms to the assumptions one must make to use the model. If it does not, then this manifests itself in terms of bias in standard errors and incor- rect inference on statistical significance of quantities of interest.
  • Principal Component Analysis: Application to Statistical Process Control

    Principal Component Analysis: Application to Statistical Process Control

    Chapter 1 Principal Component Analysis: Application to Statistical Process Control 1.1. Introduction Principal component analysis (PCA) is an exploratory statistical method for graphical description of the information present in large datasets. In most applications, PCA consists of studying p variables measured on n individuals. When n and p are large, the aim is to synthesize the huge quantity of information into an easy and understandable form. Unidimensional or bidimensional studies can be performed on variables using graphical tools (histograms, box plots) or numerical summaries (mean, variance, correlation). However, these simple preliminary studies in a multidimensional context are insufficient since they do not take into account the eventual relationships between variables, which is often the most important point. Principal component analysis is often considered as the basic method of factor analysis, which aims to find linear combinations of the p variables called components used to visualize the observations in a simple way. Because it transforms a large number of correlated variables into a few uncorrelated principal components, PCA is a dimension reduction method. However, PCA can also be used as a multivariate outlier detection method, especially by studying the last principal components. This property is useful in multidimensional quality control. Chapter written by Gilbert SAPORTA and Ndèye NIANG. 1 2 Data Analysis 1.2. Data table and related subspaces 1.2.1. Data and their characteristics Data are generally represented in a rectangulartable with n rows for the individuals and p columns corresponding to the variables. Choosing individuals and variables to analyze is a crucial phase which has an important influence on PCA results.
  • Lesson 16.5 Measures of Central Tendency and Grouped Data

    Lesson 16.5 Measures of Central Tendency and Grouped Data

    NAME: INTEGRATED ALGEBRA 1 DATE: MR. THOMPSON LESSON 16.5 MEASURES OF CENTRAL TENDENCY AND GROUPED DATA HOMEWORK ASSIGNMENT # 126: PAGES 695-697: # 2 - 12 EVENS EXAMPLE I OiliflitEMIZZEITIOMMESEMZESEM72ra, In the table, the data indicate the heights, in inches, of 17 basketball players. For these data find: a. the mode b. the median c. the mean 77 2 Solution a. The greatest frequency, 5, occurs for the height 76 0 of 75 inches. The mode, or height appearing most often, is 75. 75 5 74 3 b.For 17 players, the median is the 9th number, so there are 8 heights greater than or equal to the 73 4 median and 8 heights less than or equal to the 72 2 median. Counting the frequencies going down, 1 we have 2 + 0 + 5 = 7. Since the frequency of 71 the next interval is 3, the 8th, 9th, and 10th heights are in this interval, 74. Counting the frequencies going up, we have 1 + 2 + 4 = 7. Again, the fre- quency of the next interval is 3, and the 8th, 9th, and 10th heights are in this interval. The 9th height, the median, is 74. c.(1) Multiply each height by its corresponding frequency: 77 x 2 = 154 76 x 0 = 0 75 x 5 375 74 x 3 = 222 73 X 4 = 292 72 X 2 = 144 71 x 1 = 71 (2) Fmd the total of these products: 154 + 0 + 375 + 222 + 292 + 144 + 71 = 1,258 (3) Divide this total, 1,258, by the total frequency, 17 to obtain the mean: 1258 +47 = 74 14.