Weighted Means and Grouped Data on the Graphing Calculator

Total Page：16

File Type：pdf, Size：1020Kb

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 . (These are only estimates of the true mean and standard deviation of the sample because we are basing the calculations on the frequency distribution, which is just a summary of the data, instead of the actual data.) .

Copy Link

Recommended publications

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.
[Show full text]
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.
[Show full text]
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.
[Show full text]
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.
[Show full text]
Statistics, Measures of Central Tendency I
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.
[Show full text]
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.
[Show full text]
Propensity Score Matching Regression Discontinuity Limited Dependent Variables
Propensity Score Matching Regression Discontinuity Limited Dependent Variables Christopher F Baum EC 823: Applied Econometrics Boston College, Spring 2013 Christopher F Baum (BC / DIW) PSM, RD, LDV Boston College, Spring 2013 1 / 99 Propensity score matching Propensity score matching Policy evaluation seeks to determine the effectiveness of a particular intervention. In economic policy analysis, we rarely can work with experimental data generated by purely random assignment of subjects to the treatment and control groups. Random assignment, analogous to the ’randomized clinical trial’ in medicine, seeks to ensure that participation in the intervention, or treatment, is the only differentiating factor between treatment and control units. In non-experimental economic data, we observe whether subjects were treated or not, but in the absence of random assignment, must be concerned with differences between the treated and non-treated. For instance, do those individuals with higher aptitude self-select into a job training program? If so, they are not similar to corresponding individuals along that dimension, even though they may be similar in other aspects. Christopher F Baum (BC / DIW) PSM, RD, LDV Boston College, Spring 2013 2 / 99 Propensity score matching The key concern is that of similarity. How can we find individuals who are similar on all observable characteristics in order to match treated and non-treated individuals (or plants, or firms...) With a single measure, we can readily compute a measure of distance between a treated unit and each candidate match. With multiple measures defining similarity, how are we to balance similarity along each of those dimensions? The method of propensity score matching (PSM) allows this matching problem to be reduced to a single dimension: that of the propensity score.
[Show full text]
Causal Inference Using Educational Observational Data: Statistical Bias Reduction Methods and Multilevel Data Extensions
Causal Inference Using Educational Observational Data: Statistical Bias Reduction Methods and Multilevel Data Extensions Jose M. Hernandez A Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2015 Reading Committee: Robert Abbott, Chair Elizabeth Sanders Joe Lott Program Authorized to Offer Degree: College of Education © Copyright 2015 Jose M. Hernandez University of Washington Abstract Causal Inference Using Educational Observational Data: Statistical Bias Reduction Methods and Multilevel Data Extensions Jose M. Hernandez Chair of the Supervisory Committee: Professor Robert Abbott, Ph.D. College of Education This study utilizes a data driven simulation design, which deviates from the traditional model-based approaches most commonly adopted in quasi-experimental Monte Carlo (MC) simulation studies, to answer two main questions. First, this study explores the finite sample properties of the most utilized quasi-experimental methods that control for observable selection bias in the field of education and compares them to traditional regression methods. Second, this study lends an insight into the effects of ignoring the multilevel structure of data commonly found in the field when using quasi-experimental methods. Specifically, treatment effects were estimated using (1) Ordinary Least Squares (OLS) multiple linear regression (treatment effects, adjusted for mean differences on confounders), (2) Propensity Score Matching (PSM) using nearest neighbor 1:n with replacement,
[Show full text]
Chapter 4 Classical Methods of Analysis of Grouped Data
4. CLASSICAL METHODS OF ANALYSIS OF GROUPED DATA 4.1 The Ille-et-Vilaine study of oesophageal cancer 4.2 Exact statistical inference for a single 2 x 2 table 4.3 Approximate statistical inference for a 2 x 2 table 4.4 Combinatioq of results from a series of 2 x 2 tables; control of confounding 4.5 Exposure at several levels: the 2 x K table 4.6 Joint effects of several risk factors ' CHAPTER IV CLASSICAL METHODS OF ANALYSIS OF GROUPED DATA This chapter presents the traditional methods of analysis of case-control studies based on a grouping or cross-classification of the data. The main outlines of this approach, which has proved extremely useful for practising epidemiologists, are contained in the now classic paper by Mantel and Haenszel (1959). Most of the required calculations are elementary and easily performed on a pocket calculator, especially one that is equipped with log, exponential and square-root keys. Rothman and Boice (1979) have recently published a set of programmes for such a calculator which facilitates many of the analyses presented below. The statistical procedures introduced in this chapter are appropriate for study designs in which stratification or "group-matching" is used to balance the case and control samples vis-ci-vis the confounding variables. The following chapter develops these same methods of analysis for use with designs in which there is individual matching of cases to controls. While our treatment of this material attempts to give the reader some appreciation of its logical foundations, the emphasis is on methodological aspects rather than statistical theory.
[Show full text]
Uptick in Income Segregation: Real Trend Or Random Sampling Variance
NBER WORKING PAPER SERIES THE UPTICK IN INCOME SEGREGATION: REAL TREND OR RANDOM SAMPLING VARIANCE John R. Logan Andrew Foster Jun Ke Fan Li Working Paper 23656 http://www.nber.org/papers/w23656 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts Avenue Cambridge, MA 02138 August 2017, Revised January 2018 This research was supported by a research grant from National Institutes of Health (1R21HD078762-01A1), the Sociology Program of the National Science Foundation, and by the staff of the research initiative on Spatial Structures in the Social Sciences at Brown University. Cici Bauer provided helpful suggestions for calculation of segregation measures. The Population Studies and Training Center at Brown University (R24 HD041020) provided general support. The authors have full responsibility for the findings and interpretations reported here. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research. NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications. © 2017 by John R. Logan, Andrew Foster, Jun Ke, and Fan Li. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source. The Uptick in Income Segregation: Real Trend or Random Sampling Variance John R. Logan, Andrew Foster, Jun Ke, and Fan Li NBER Working Paper No. 23656 August 2017, Revised January 2018 JEL No. R23 ABSTRACT Recent studies have reported a reversal of an earlier trend in income segregation in metropolitan regions, from a decline in the 1990s to an increase in the 2000-2010 decade.
[Show full text]
Treatment Effects
treatment effects The term ‘treatment effect’ refers to the causal effect of a binary (0–1) variable on an outcome variable of scientific or policy interest. Economics examples include the effects of government programmes and policies, such as those that subsidize training for disadvantaged workers, and the effects of individual choices like college attendance. The principal econometric problem in the estimation of treatment effects is selection bias, which arises from the fact that treated individuals differ from the non-treated for reasons other than treatment status per se. Treatment effects can be estimated using social experiments, regression models, matching estimators, and instrumental variables. A ‘treatment effect’ is the average causal effect of a binary (0–1) variable on an outcome variable of scientific or policy interest. The term ‘treatment effect’ originates in a medical literature concerned with the causal effects of binary, yes-or-no ‘treatments’, such as an experimental drug or a new surgical procedure. But the term is now used much more generally. The causal effect of a subsidized training programme is probably the mostly widely analysed treatment effect in economics (see, for example, Ashenfelter, 1978, for one of the first examples, or Heckman and Robb, 1985 for an early survey). Given a data-set describing the labour market circumstances of trainees and a non-trainee comparison group, we can compare the earnings of those who did participate in the programme and those who did not. Any empirical study of treatment effects would typically start with such simple comparisons. We might also use regression methods or matching to control for demographic or background characteristics.
[Show full text]
Robustifying Independent Component Analysis by Adjusting for Group-Wise Stationary Noise
Journal of Machine Learning Research 20 (2019) 1{50 Submitted 6/18; Revised 4/19; Published 10/19 Robustifying Independent Component Analysis by Adjusting for Group-Wise Stationary Noise Niklas Pfister∗ [email protected] Seminar for Statistics, ETH Zürich Rämisstrasse 101, 8092 Zürich,Switzerland Sebastian Weichwald∗ [email protected] Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5, 2100 Copenhagen, Denmark Peter Bühlmann [email protected] Seminar for Statistics, ETH Zürich Rämisstrasse 101, 8092 Zürich,Switzerland Bernhard Schölkopf [email protected] Empirical Inference Department, Max Planck Institute for Intelligent Systems Max-Planck-Ring 4, 72076 Tübingen,Germany Editor: Kenji Fukumizu Abstract We introduce coroICA, confounding-robust independent component analysis, a novel ICA algorithm which decomposes linearly mixed multivariate observations into independent components that are corrupted (and rendered dependent) by hidden group-wise stationary confounding. It extends the ordinary ICA model in a theoretically sound and explicit way to incorporate group-wise (or environment-wise) confounding. We show that our proposed general noise model allows to perform ICA in settings where other noisy ICA procedures fail. Additionally, it can be used for applications with grouped data by adjusting for different stationary noise within each group. Our proposed noise model has a natural relation to causality and we explain how it can be applied in the context of causal inference. In addition to our theoretical framework, we provide an efficient estimation procedure and prove identifiability of the unmixing matrix under mild assumptions. Finally, we illustrate the performance and robustness of our method on simulated data, provide audible and visual examples, and demonstrate the applicability to real-world scenarios by experiments on publicly available Antarctic ice core data as well as two EEG data sets.
[Show full text]