DOCSLIB.ORG

Measures of Central Tendency for Censored Wage Data

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Measures of Central Tendency for Censored Wage Data MEASURES OF CENTRAL TENDENCY FOR CENSORED WAGE DATA Sandra West, Diem-Tran Kratzke, and Shail Butani, Bureau of Labor Statistics Sandra West, 2 Massachusetts Ave. N.E. Washington, D.C. 20212 Key Words: Mean, Median, Pareto Distribution medians can lead to misleading results. This paper tested the linear interpolation method to estimate the median. I. INTRODUCTION First, the interval that contains the median was determined, The research for this paper began in connection with the and then linear interpolation is applied. The occupational need to measure the central tendency of hourly wage data minimum wage is used as the lower limit of the lower open from the Occupational Employment Statistics (OES) survey interval. If the median falls in the uppermost open interval, at the Bureau of Labor Statistics (BLS). The OES survey is all that can be said is that the median is equal to or above a Federal-State establishment survey of wage and salary the upper limit of hourly wage. workers designed to produce data on occupational B. MEANS employment for approximately 700 detailed occupations by A measure of central tendency with desirable properties is industry for the Nation, each State, and selected areas within the mean. In addition to the usual desirable properties of States. It provided employment data without any wage means, there is another nice feature that is proven by West information until recently. Wage data that are produced by (1985). It is that the percent difference between two means other Federal programs are limited in the level of is bounded relative to the percent difference between occupational, industrial, and geographic coverage. To subgroup means, if the proportion of units in each subgroup address this critical void in the Federal statistical efforts, the remains the same for the two groups. Since the problem OES program tested the feasibility of incorporating wage considered in this paper deals with grouped data that have questions into the survey in 1989 and 1990 pilot studies. lower and upper open intervals, it is not possible to compute The 1992 OES survey collects data in 15 States on an exact mean. The problem will be considered from the occupational hourly wage by industry in nonagricultural point of view of computing a population mean from fight establishments. The data are collected in eleven intervals, and left censored data. First the problem will be formulated rather than in exact dollar amounts, with the lowest and and two methods for computing the mean will be discussed. uppermost intervals open. One method results in the Winsorized mean and the other Research was conducted to f'md a suitable estimate of method uses a classical Pareto distribution. central tendency for the occupational wage data of the OES The population of true earnings data are denoted by survey. It was determined that both mean and median x,,x~ ..... x~. would be measured. Each has advantages and disadvantages The mean of the population is desired; that is, which will be discussed. The first part of the research focused on the problem of estimating the overall occupational wage mean for each X = industry, according to the OES survey's objective. The second part of the research explores the best method for estimating the wage mean of an upper open interval. This The data actually observed are the frequencies of each of would be useful for analyses such as regression where the intervals 11, 12 ..... It, which are mutually exclusive, and interval wages are used as dependent variables. The two measures of central tendency are described in exhaustive of the earnings scale. I~ and I r are open Section II. The empirical studies and results are given in intervals where 11 contains all xj less than some fixed Section III. The conclusions are presented in Section IV. Section V contains plans for future research. number U~ and Ir contains all xj greater than or equal to IL MEASURES OF CENTRAL TENDENCY A. MEDIANS some fixed number U(~_1). For i=2,3 ..... (r-l), I i are In elementary theory the median has considerable claims bounded intervals containing all xj between some fixed to be used as a measure of location for unimodal distributions. It is readily interpretable in terms of ordinary numbers U(i_l) and U i . ideas. What gives the arithmetic mean the greater importance in advanced theory is its superior mathematical Let f denote the observed frequency of interval I i for tractability and certain sampling properties. The median has i = 1,2 ..... r and let M~ be the mid-point of the i-th interval, a compensating advantage in that it is less sensitive to the configuration of the outlying parts of the frequency then the usual estimator of X is the grouped mean, Xg • distribution than is the mean. This is especially important with earnings data and in particular, with the censored data, _ r M/I//lV the median is a logical choice. However, the median is X = Z sensitive to the way the data are grouped and operating with g i=1 643 Since intervals 11 and I r are not bounded, M~ and M r will need to be estimated. 1 ! has a natural lower bound, M r = ~x f(xl Uir_,))dx = U(,._I)o~l(a- 1). either 0 or the minimum wage; for this study the Federal U{,-i) minimum occupational wage , We1) , was used. Thus the Note that for the mean to be positive, a > 1. Many methods exist for estimating the parameter tZ. The estimate for M I is: one most used and recommended in the literature is the quantile method, which is described next. /~1 ----(W(1) +Vl )/2. Let Mp and Mq denote the p-th and q- th quantile An obvious estimate for M r is: respectively; that is, ~'Ir.w = U(r_l)' F(Mv) = P(M <_ My)= 1-(KiMv) °< = p. which would lead to the Winsorized group mean: Similarly, for Mq. r--] /N Letting My and it~/q be estimators of Mp and M q Xg,w -- {/~lJ~ + Z Mifi + U(r-1)fr } " respectively, leads to the following estimator of a: i=2 Clearly,/¢/,.w will underestimate M, . With the Winsorized q ll,.I,aql ,l. mean a straight line is used for the missing values; a natural Most researchers seem to use this method with either the extension, now to be considered, is to fit a curve for the mid-points of the last two bounded intervals or the last missing values. bounded interval and the upper open interval. Specifically, if the mid-points of the last two bounded intervals are used, For the estimator Of M, , consider fitting a theoretical the estimator of t~ becomes: distribution to the (r-1) mid-points, and take/17/r as the mean d< = ln[(f,)/(fr + ir_l)]/ln[Mr_~lMr_,]. of the conditional distribution, P( X < xl X > U(r_,) ). ^ This will lead to Mr,q, 2 and Xg,q,2 as estimators for M r That is, and Xg, respectively. That is, M, = x dP(X < xlX >_ U(r_l)) = f(xlU(,_,)) dx f X A r-! /N = + Z M,,, + 7=2 where f(xlU(r_~) ) is the conditional density of X given that where X is greater than or equal to the fixed number U(r_~). = t4r-,) 1). Another possibility for /~/, is the median of the This method is referred to as the quantile II method, and conditional density. Parker and Fenwiek (1983) found that ^ Xg,q,2 is referred to as the Qnt II estimator in this paper. If this estimator performed better than the mean, but this was not the ease with the method and data considered in West the lower bounds of the last bounded interval and the open (1985). interval are used, then the estimator of ~ becomes: Many distributions have been proposed for earnings data, but the literature indicates that the researchers are satisfied 07o = ~l(/,_, + i,)l(i,)Vl,l u<,_,)lU(,_~> i. with the Pareto distribution as a fit to the upper portion of the earnings curve. Consider the Pareto distribution: In the literature the estimator ,d o seems to be the one most recommended, for example, see Shryock (1975), F(x) = P(X < x) = 1 -(K[x) ~ for x > K > O, a > 0 Parker and Fenwick (1983). This will lead to 34r,q.~ and =0 for x<K. ^ Noting that, Xg.q,l as estimators for M r and Xg, respectively. The P(X >_ xlX >_ V(,_1)) = P(X >_ x)/P(X >_ U(r_1)) = (V(r_l))tt x -t* method is referred to as the quantile I method and Xg,q.1 is then referred to as the Qnt I estimator in this paper. f(xlU(,._,)) = -dP(X > xlX > U(r_,))ldx An alternative estimator for t;t is a modified maximum = a(u(,_,)yx-:-', for x )_ U(r_l ) . likelihood estimator developed in West (1985). A brief THUS, description of the estimator will be given here. Since the Pareto distribution is considered a good fit for the 644 distribution of higher earnings, the parameter will be We define the error of estimation to be the difference estimated from the left truncated distribution. Let between the estimated value and the true value, which is M., M.+, ..... M._, assumed to be known. The relative error is defined as the ratio of the error of estimation to the true value. We be the left tnmcated mid-points of the bounded intervals, compare different estimators by looking at the absolute then the modified maximum likelihood estimator is: values of the relative errors of their estimates.
Load more

Recommended publications
  • 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
    [Show full text]
  • Central Tendency, Dispersion, Correlation and Regression Analysis Case Study for MBA Program
    Business Statistic Central Tendency, Dispersion, Correlation and Regression Analysis Case study for MBA program CHUOP Theot Therith 12/29/2010 Business Statistic SOLUTION 1. CENTRAL TENDENCY - The different measures of Central Tendency are: (1). Arithmetic Mean (AM) (2). Median (3). Mode (4). Geometric Mean (GM) (5). Harmonic Mean (HM) - The uses of different measures of Central Tendency are as following: Depends upon three considerations: 1. The concept of a typical value required by the problem. 2. The type of data available. 3. The special characteristics of the averages under consideration. • If it is required to get an average based on all values, the arithmetic mean or geometric mean or harmonic mean should be preferable over median or mode. • In case middle value is wanted, the median is the only choice. • To determine the most common value, mode is the appropriate one. • If the data contain extreme values, the use of arithmetic mean should be avoided. • In case of averaging ratios and percentages, the geometric mean and in case of averaging the rates, the harmonic mean should be preferred. • The frequency distributions in open with open-end classes prohibit the use of arithmetic mean or geometric mean or harmonic mean. Prepared by: CHUOP Theot Therith 1 Business Statistic • If the distribution is bell-shaped and symmetrical or flat-topped one with few extreme values, the arithmetic mean is the best choice, because it is less affected by sampling fluctuations. • If the distribution is sharply peaked, i.e., the data cluster markedly at the middle or if there are abnormally small or large values, the median has smaller sampling fluctuations than the arithmetic mean.
    [Show full text]
  • Redalyc.Robust Analysis of the Central Tendency, Simple and Multiple
    International Journal of Psychological Research ISSN: 2011-2084 [email protected] Universidad de San Buenaventura Colombia Courvoisier, Delphine S.; Renaud, Olivier Robust analysis of the central tendency, simple and multiple regression and ANOVA: A step by step tutorial. International Journal of Psychological Research, vol. 3, núm. 1, 2010, pp. 78-87 Universidad de San Buenaventura Medellín, Colombia Available in: http://www.redalyc.org/articulo.oa?id=299023509005 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative International Journal of Psychological Research, 2010. Vol. 3. No. 1. Courvoisier, D.S., Renaud, O., (2010). Robust analysis of the central tendency, ISSN impresa (printed) 2011-2084 simple and multiple regression and ANOVA: A step by step tutorial. International ISSN electrónica (electronic) 2011-2079 Journal of Psychological Research, 3 (1), 78-87. Robust analysis of the central tendency, simple and multiple regression and ANOVA: A step by step tutorial. Análisis robusto de la tendencia central, regresión simple y múltiple, y ANOVA: un tutorial paso a paso. Delphine S. Courvoisier and Olivier Renaud University of Geneva ABSTRACT After much exertion and care to run an experiment in social science, the analysis of data should not be ruined by an improper analysis. Often, classical methods, like the mean, the usual simple and multiple linear regressions, and the ANOVA require normality and absence of outliers, which rarely occurs in data coming from experiments. To palliate to this problem, researchers often use some ad-hoc methods like the detection and deletion of outliers.
    [Show full text]
  • Robust Statistical Procedures for Testing the Equality of Central Tendency Parameters Under Skewed Distributions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository@USM ROBUST STATISTICAL PROCEDURES FOR TESTING THE EQUALITY OF CENTRAL TENDENCY PARAMETERS UNDER SKEWED DISTRIBUTIONS SHARIPAH SOAAD SYED YAHAYA UNIVERSITI SAINS MALAYSIA 2005 ii ROBUST STATISTICAL PROCEDURES FOR TESTING THE EQUALITY OF CENTRAL TENDENCY PARAMETERS UNDER SKEWED DISTRIBUTIONS by SHARIPAH SOAAD SYED YAHAYA Thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy May 2005 ACKNOWLEDGEMENTS Firstly, my sincere appreciations to my supervisor Associate Professor Dr. Abdul Rahman Othman without whose guidance, support, patience, and encouragement, this study could not have materialized. I am indeed deeply indebted to him. My sincere thanks also to my co-supervisor, Associate Professor Dr. Sulaiman for his encouragement and support throughout this study. I would also like to thank Universiti Utara Malaysia (UUM) for sponsoring my study at Universiti Sains Malaysia. Thanks are due to Professor H.J. Keselman for according me access to his UNIX server to run the relevant programs; Professor R.R. Wilcox and Professor R.G. Staudte for promptly responding to my queries via emails; Pn. Azizah Ahmad and Pn. Ramlah Din for patiently helping me with some of the computer programs and En. Sofwan for proof reading the chapters of this thesis. I am deeply grateful to my parents and my wonderful daughter, Nur Azzalia Kamaruzaman for their inspiration, patience, enthusiasm and effort. I would also like to thank my brothers for their constant support. Finally, my grateful recognition are due to (in alphabetical order) Azilah and Nung, Bidin and Kak Syera, Izan, Linda, Min, Shikin, Yen, Zurni and to those who had directly or indirectly lend me their friendship, moral support and endless encouragement during my study.
    [Show full text]
  • Fitting Models (Central Tendency)
    FITTING MODELS (CENTRAL TENDENCY) This handout is one of a series that accompanies An Adventure in Statistics: The Reality Enigma by me, Andy Field. These handouts are offered for free (although I hope you will buy the book).1 Overview In this handout we will look at how to do the procedures explained in Chapter 4 using R an open-source free statistics software. If you are not familiar with R there are many good websites and books that will get you started; for example, if you like An Adventure In Statistics you might consider looking at my book Discovering Statistics Using R. Some basic things to remember • RStudio: I assume that you're working with RStudio because most sane people use this software instead of the native R interface. You should download and install both R and RStudio. A few minutes on google will find you introductions to RStudio in the event that I don't write one, but these handouts don't particularly rely on RStudio except in settingthe working directory (see below). • Dataframe: A dataframe is a collection of columns and rows of data, a bit like a spreadsheet (but less pretty) • Variables: variables in dataframes are referenced using the $ symbol, so catData$fishesEaten would refer to the variable called fishesEaten in the dataframe called catData • Case sensitivity: R is case sensitive so it will think that the variable fishesEaten is completely different to the variable fisheseaten. If you get errors make sure you check for capitals or lower case letters where they shouldn't be. • Working directory: you should place the data files that you need to use for this handout in a folder, then set it as the working directory by navigating to that folder when you execute the command accessed through the Session>Set Working Directory>Choose Directory ..
    [Show full text]
  • Variance Difference Between Maximum Likelihood Estimation Method and Expected a Posteriori Estimation Method Viewed from Number of Test Items
    Vol. 11(16), pp. 1579-1589, 23 August, 2016 DOI: 10.5897/ERR2016.2807 Article Number: B2D124860158 ISSN 1990-3839 Educational Research and Reviews Copyright © 2016 Author(s) retain the copyright of this article http://www.academicjournals.org/ERR Full Length Research Paper Variance difference between maximum likelihood estimation method and expected A posteriori estimation method viewed from number of test items Jumailiyah Mahmud*, Muzayanah Sutikno and Dali S. Naga 1Institute of Teaching and Educational Sciences of Mataram, Indonesia. 2State University of Jakarta, Indonesia. 3 University of Tarumanegara, Indonesia. Received 8 April, 2016; Accepted 12 August, 2016 The aim of this study is to determine variance difference between maximum likelihood and expected A posteriori estimation methods viewed from number of test items of aptitude test. The variance presents an accuracy generated by both maximum likelihood and Bayes estimation methods. The test consists of three subtests, each with 40 multiple-choice items of 5 alternatives. The total items are 102 and 3159 respondents which were drawn using random matrix sampling technique, thus 73 items were generated which were qualified based on classical theory and IRT. The study examines 5 hypotheses. The results are variance of the estimation method using MLE is higher than the estimation method using EAP on the test consisting of 25 items with F= 1.602, variance of the estimation method using MLE is higher than the estimation method using EAP on the test consisting of 50 items with F= 1.332, variance of estimation with the test of 50 items is higher than the test of 25 items, and variance of estimation with the test of 50 items is higher than the test of 25 items on EAP method with F=1.329.
    [Show full text]
  • Robust Analysis of the Central Tendency, ISSN Impresa (Printed) 2011-2084 Simple and Multiple Regression and ANOVA: a Step by Step Tutorial
    International Journal of Psychological Research, 2010. Vol. 3. No. 1. Courvoisier, D.S., Renaud, O., (2010). Robust analysis of the central tendency, ISSN impresa (printed) 2011-2084 simple and multiple regression and ANOVA: A step by step tutorial. International ISSN electrónica (electronic) 2011-2079 Journal of Psychological Research, 3 (1), 78-87. Robust analysis of the central tendency, simple and multiple regression and ANOVA: A step by step tutorial. Análisis robusto de la tendencia central, regresión simple y múltiple, y ANOVA: un tutorial paso a paso. Delphine S. Courvoisier and Olivier Renaud University of Geneva ABSTRACT After much exertion and care to run an experiment in social science, the analysis of data should not be ruined by an improper analysis. Often, classical methods, like the mean, the usual simple and multiple linear regressions, and the ANOVA require normality and absence of outliers, which rarely occurs in data coming from experiments. To palliate to this problem, researchers often use some ad-hoc methods like the detection and deletion of outliers. In this tutorial, we will show the shortcomings of such an approach. In particular, we will show that outliers can sometimes be very difficult to detect and that the full inferential procedure is somewhat distorted by such a procedure. A more appropriate and modern approach is to use a robust procedure that provides estimation, inference and testing that are not influenced by outlying observations but describes correctly the structure for the bulk of the data. It can also give diagnostic of the distance of any point or subject relative to the central tendency.
    [Show full text]
  • Generalized Linear Models Link Function the Logistic Equation Is
    Newsom Psy 525/625 Categorical Data Analysis, Spring 2021 1 Generalized Linear Models Link Function The logistic equation is stated in terms of the probability that Y = 1, which is π, and the probability that Y = 0, which is 1 - π. π ln =αβ + X 1−π The left-hand side of the equation represents the logit transformation, which takes the natural log of the ratio of the probability that Y is equal to 1 compared to the probability that it is not equal to one. As we know, the probability, π, is just the mean of the Y values, assuming 0,1 coding, which is often expressed as µ. The logit transformation could then be written in terms of the mean rather than the probability, µ ln =αβ + X 1− µ The transformation of the mean represents a link to the central tendency of the distribution, sometimes called the location, one of the important defining aspects of any given probability distribution. The log transformation represents a kind of link function (often canonical link function)1 that is sometimes given more generally as g(.), with the letter g used as an arbitrary name for a mathematical function and the use of the “.” within the parentheses to suggest that any variable, value, or function (the argument) could be placed within. For logistic regression, this is known as the logit link function. The right hand side of the equation, α + βX, is the familiar equation for the regression line and represents a linear combination of the parameters for the regression. The concept of this logistic link function can generalized to any other distribution, with the simplest, most familiar case being the ordinary least squares or linear regression model.
    [Show full text]
  • Statistics: a Brief Overview
    Statistics: A Brief Overview Katherine H. Shaver, M.S. Biostatistician Carilion Clinic Statistics: A Brief Overview Course Objectives • Upon completion of the course, you will be able to: – Distinguish among several statistical applications – Select a statistical application suitable for a research question/hypothesis/estimation – Identify basic database structure / organization requirements necessary for statistical testing and interpretation What Can Statistics Do For You? • Make your research results credible • Help you get your work published • Make you an informed consumer of others’ research Categories of Statistics • Descriptive Statistics • Inferential Statistics Descriptive Statistics Used to Summarize a Set of Data – N (size of sample) – Mean, Median, Mode (central tendency) – Standard deviation, 25th and 75th percentiles (variability) – Minimum, Maximum (range of data) – Frequencies (counts, percentages) Example of a Summary Table of Descriptive Statistics Summary Statistics for LDL Cholesterol Std. Lab Parameter N Mean Dev. Min Q1 Median Q3 Max LDL Cholesterol 150 140.8 19.5 98 132 143 162 195 (mg/dl) Scatterplot Box-and-Whisker Plot Histogram Length of Hospital Stay – A Skewed Distribution 60 50 P 40 e r c 30 e n t 20 10 0 2 4 6 8 10 12 14 Length of Hospital Stay (Days) Inferential Statistics • Data from a random sample used to draw conclusions about a larger, unmeasured population • Two Types – Estimation – Hypothesis Testing Types of Inferential Analyses • Estimation – calculate an approximation of a result and the precision of the approximation [associated with confidence interval] • Hypothesis Testing – determine if two or more groups are statistically significantly different from one another [associated with p-value] Types of Data The type of data you have will influence your choice of statistical methods… Types of Data Categorical: • Data that are labels rather than numbers –Nominal – order of the categories is arbitrary (e.g., gender) –Ordinal – natural ordering of the data.
    [Show full text]
  • Measures of Central Tendency & Dispersion
    Measures of Central Tendency & Dispersion Measures that indicate the approximate center of a distribution are called measures of central tendency. Measures that describe the spread of the data are measures of dispersion. These measures include the mean, median, mode, range, upper and lower quartiles, variance, and standard deviation. A. Finding the Mean The mean of a set of data is the sum of all values in a data set divided by the number of values in the set. It is also often referred to as an arithmetic average. The Greek letter (“mu”) is used as the symbol for population mean and the symbol ̅ is used to represent the mean of a sample. To determine the mean of a data set: 1. Add together all of the data values. 2. Divide the sum from Step 1 by the number of data values in the set. Formula: ∑ Example: Consider the data set: 17, 10, 9, 14, 13, 17, 12, 20, 14 ∑ The mean of this data set is 14. B. Finding the Median The median of a set of data is the “middle element” when the data is arranged in ascending order. To determine the median: 1. Put the data in order from smallest to largest. 2. Determine the number in the exact center. i. If there are an odd number of data points, the median will be the number in the absolute middle. ii. If there is an even number of data points, the median is the mean of the two center data points, meaning the two center values should be added together and divided by 2.
    [Show full text]
  • Chapter 3 : Central Tendency
    Chapter 3 : Central Tendency OiOverview • Definition: Central tendency is a statistical measure to determine a single score tha t dfidefines the center of a distribution. – The goal of central tendency is to find the single score that is most typical or most representative of the entire group. – Measures of central tendency are also useful for making comparisons between groups of individuals or between sets of figures. • For example, weather data indicate that for Seattle, Washington, the average yearly temperature is 53° and the average annual precipitation is 34 inches. OiOverview cont. – By comparison, the average temperature in Phoenix, Arizona, is 71 ° and the average precipitation is 7.4 inches. • Clearly, there are problems defining the "center" of a distribution. • Occasionally, you will find a nice, neat distribution like the one shown in Figure 3.2(a), for which everyone will agree on the center. (See next slide.) • But you should realize that other distr ibut ions are possible and that there may be different opinions concerning the definition of the center. (See in two slides) OiOverview cont. • To deal with these problems, statisticians have developed three different methods for measuring central tendency: – Mean – MdiMedian – Mode OiOverview cont. Negatively Skewed Distribution Bimodal Distribution The Mean • The mean for a population will be identified by the Greek letter mu, μ (pronounced "mew"), and the mean for a sample is identified by M or (read "x- bar"). • Definition: The mean for a distribution is the sum of the scores divided by the number of scores. • The formula for the population mean is • The formula for the sample mean uses symbols that signify sample values: The Mean cont.
    [Show full text]
  • Measures of Central Tendency
    Measures of central tendency Questions such as: \how many calories do I eat per day?" or \how much time do I spend talking per day?" can be hard to answer because the answer will vary from day to day. It's sometimes more sensible to ask \how many calories do I consume on a typical day?" or \on average, how much time do I spend talking per day?". In this section we will study three ways of measuring central tendency in data, the mean, the median and the mode. Each measure give us a single value? that might be considered typical. Each measure has its own strengths and weaknesses ?: some exclusions apply Measures of central tendency A population of books, cars, people, polar bears, all games played by Babe Ruth throughout his career etc.... is the entire collection of those objects. For any given variable under consideration, each member of the population has a particular value of the variable associated to them, for example the number of home runs scored by Babe Ruth for each game played by him during his career. These values are called data and we can apply our measures of central tendency to the entire population, to get a single value (maybe more than one for the mode) measuring central tendency for the entire population; or we can apply our measures to a subset or sample of the population, to get an estimate of the central tendency for the population. Measures of central tendency A sample is a subset of the population, for example, we might collect data on the number of home runs hit by Miguel Cabrera in a random sample of 20 games.
    [Show full text]