MEASURES OF DISPERSION (VARIABILITY)
|
|
- Elizabeth Richard
- 5 years ago
- Views:
Transcription
1 POLI 300 Hadout #7 N. R. Miller MEASURES OF DISPERSION (VARIABILITY) While measures of cetral tedecy idicate what value of a variable is (i oe sese or other, e.g., mode, media, mea), average or cetral or typical i a set of data, measures of dispersio (or variability or spread) idicate (i oe sese or other) the extet to which the observed values are spread out aroud that ceter how far apart observed values typically are from each other or from some average value (i particular, the mea). Thus: (a) (b) (c) if all cases have idetical observed values (ad thereby also all have the average value), dispersio is zero; if most cases have observed values that are quite close together (thereby also quite close to the average value), dispersio is low (but greater tha zero); but if may cases have observed values that are quite far apart from may others (or from the average value), dispersio is high. A measure of dispersio provides a summary statistic that idicates the magitude of such dispersio ad, like a measure of cetral tedecy, is a uivariate statistic. Because dispersio is cocered with how close together or far apart observed values are (i.e., with the magitude of the itervals betwee them), it should be apparet that the otio of dispersio make sese ad measures of dispersio are defied oly for iterval (or ratio) variables. (There is oe exceptio: a very crude measure of dispersio called the variatio ratio, which is defied for ordial ad eve omial variables. It will be discussed briefly i the Aswers & Discussio to PS #7.) There are two pricipal types of measures of dispersio: rage measures ad deviatio measures. Rage Measures Rage measures are based o the distace betwee (relatively) extreme values observed i the data ad are coceptually coected with the media as a measure of cetral tedecy (See the data illustratig Percetiles, the Media, ad Rages o the back page of the Hadout #6 o Measures of Cetral Tedecy.) The ( total or simple ) rage is the maximum (highest) value observed i the data (the value of the case at the 100th percetile) mius the miimum (lowest) value observed i the data (the value of the case at the 0th percetile) that is, the distace or iterval betwee the values of these two extreme cases. (Note that this may be less tha the rage of the possible values of the variable, sice logically possible extreme values may ot be observed i actual data; for example, the variable LEVEL OF TURNOUT has logically possible values ragig from 0% to 100%, but i U.S. Presidetial electios, the rage of observed values [as covetioally measured, i.e., as Total Vote for Presidet divided by Votig Age Populatio] over the past 60 years or so rages from a miimum observed of about 48% (i 1996) to about 64% (i 1960). The problem with the (total or simple) rage as a measure of dispersio is that it depeds o the values of just two cases cases that by defiitio have atypical (ad perhaps extraordiarily atypical) values. I particular, the rage
2 #7 Dispersio page 2 makes o distictio betwee a polarized distributio i which almost all observed values are close to either the miimum or maximum values ad a distributio i which almost all observed values are buched together but there are a few extreme outliers. Also the rage is udefied for theoretical distributios that are ope-eded (the techical term is asymptotic), like the ormal distributio (that we will take up i the ext topic) or the upper ed of a icome distributio type of curve (see PS #5C). Therefore other variats of the rage measure that do ot reach etirely out to the extremes of the frequecy distributio are ofte used i place of the total rage. The iterdecile rage is the value of the case that stads at the 90th percetile of the distributio mius the value of the case that stads at the 10th percetile that is, the distace or iterval betwee the values of these two less extreme cases. I like maer, the iterquartile rage is the value of the case that stads at the 75th percetile of the distributio mius the value of the case that stads at the 25th percetile. (The first quartile is the media observatio amog all cases that lie below the overall media ad the third quartile is the media observatio amog all cases that lie above the overall media. I these terms, the iterquartile rage is third quartile mius the first quartile.) We have previously used a rage measure i a special cotext. The hadout o Radom Samplig said the followig: Suppose the Gallup Poll takes a radom sample of respodets ad reports that the Presidet's curret approval ratig is 62% ad that this sample statistic has a margi of error of ± 3 %. Here is what this meas: if (hypothetically) Gallup were to take a great may radom samples of the same size from the same populatio (e.g., the America VAP o a give day), the differet samples would give differet statistics (approval ratigs), but 95% of these samples would give approval ratigs withi 3 percetage poits of the true populatio parameter. Thus, if our data is the list of sample statistics produced by the (hypothetical) great may radom samples, the margi or error specifies the rage betwee the value of the sample statistic that stads at the 97.5th percetile mius the sample statistic that stads at the 2.5th percetile (so that 95% of the sample statistics lie withi the rage). Specifically (ad lettig P be the value of the populatio parameter) this rage is (P + 3%)!(P! 3%) = 6%, i.e., twice the margi error. Deviatio Measures Deviatio measures are based o average deviatios from some average value. (Recall the discussio of Deviatios from the Average i Hadout #6 o Measures of Cetral Tedecy.) Sice we are dealig with iterval variables, we ca calculate meas, ad deviatio measures are typically based o the mea deviatio from the mea value. Thus the usual deviatio measures are coceptually coected with the mea as a measure of cetral tedecy. Suppose we have a variable X ad a set of cases umbered 1,2,...,. Let the observed value of the variable i each case be desigated x 1, x 2, etc. Thus: x 1 + x x 3 x mea of X = xg = =.
3 #7 Dispersio page 3 The deviatio from the mea for a represetative case i is (x i! xg ). If almost all of these deviatios are small (if almost all cases are close to the mea value), dispersio is small; but if may of these deviatios are large (if may cases are much above or below the mea), dispersio is large. This suggests we could costruct a measure D of dispersio that would simply be the average (mea) of all the deviatios: (x 1! xg ) + (x 2! xg ) (x! xg ) 3 (x i! xg ) D = =. But this will ot work, because some of the deviatio are positive ad others are egative ad, as we saw earlier (Hadout #6, poit (d) uder Deviatios from the Average), these positive ad egative deviatios ecessarily balace out ad add up to zero, i.e., for ay distributio of observed values 3(x i! xg ) = 0. A practical way aroud this problem is simply to igore the fact that some deviatios are egative while others are positive by averagig the absolute values of the deviatios (i effect, by igorig the egative sig before each egative deviatio): 3 *x i! xg* MD =. This measure (called the mea deviatio) tells us the average (mea) amout that the values for all cases deviate (regardless of whether they are higher or lower) from the average (mea) value. Ideed, this is a ituitive, uderstadable, ad perfectly reasoable measure of dispersio, ad it is occasioally used i research. However, statisticias are mathematicias, ad they dislike this measure because the formula is mathematically messy by virtue of beig o-algebraic (i that it igores egative sigs). Therefore statisticias, ad most researchers, use aother slightly differet deviatio measure of dispersio that is algebraic, ad that makes use of the fact that the square of ay (positive or egative) umber (i.e., the umber multiplied by itself) other tha zero is itself always positive. This formula is based o fidig the average of the squared deviatios; sice these are all o-egative, they do ot balace out. This measure of dispersio is called the variace of the variable. 3 (x i! xg ) 2 Variace of X = Var(X) = s 2 =. That is, the variace is the average squared deviatio from the mea. Remember from Hadout #6 (poit (e) uder Deviatios from the Average) that the average squared deviatio from the mea value of X is smaller tha the average squared deviatio from ay other value of X. The variace is the usual measure of dispersio i statistical theory, but it has a drawback whe researchers wat to describe the dispersio i data i a practical way. Whatever uits the origial data (ad its average values ad its mea dispersio) are expressed i, the variace is expressed i the square of those uits, ad thus it does't make much ituitive or practical sese. This ca be remedied by fidig the (positive) square root of the variace (which takes us back to the origial uits). This measure of dispersio is called stadard deviatio of the variable:
4 #7 Dispersio page 4 / 3 (x i! xg ) Stadard Deviatio of X = SD(X) = s = / 2. r I order to iterpret a stadard deviatio, or to make a plausible estimate of the SD of some data, it is useful to thik of the mea deviatio because (i) it is easier to estimate the magitude of the mea deviatio ad (ii) the stadard deviatio has approximately the same umerical magitude as the mea deviatio. More precisely, give ay distributio of data, the stadard deviatio is ever less tha the mea deviatio; it is equal to the mea deviatio if the data is distributed i a maximally polarized fashio; otherwise the SD is somewhat larger typically about 20-50% larger. Sample Estimates of Populatio Dispersio Radom sample statistics that are percetages or averages provide ubiased estimates of the correspodig populatio parameters. However, sample statistics that are dispersio measures provide estimates of populatio dispersio that are biased (at least slightly) dowward. This is most obvious i the case of the rage; it should be evidet that a sample rage is almost always smaller, ad ca ever be larger, tha the correspodig populatio rage. The sample stadard deviatio (or variace) is also biased slightly dowward. (While the SD of a particular sample ca be larger tha the populatio SD, sample SDs are o average slightly smaller tha the correspodig populatio SDs). However, the sample SD ca be adjusted to provide a ubiased estimate of the populatio SD; this adjustmet cosists of dividig the sum of the squared deviatios by!1, rather tha by. (Clearly this adjustmet makes o practical differece uless the sample is quite small. Notice that if you apply the SD formula i the evet that you have just a sigle observatio i your sample, i.e., = 1, it must give SD = 0 regardless of what the observed value is. More ituitively, you ca get o sese of how much dispersio there is i a populatio with respect to some variable util you observe at least two cases ad ca see how far apart they are.) This is why you will ofte see the formula for the variace ad SD with a!1 divisor (ad scietific calculators ofte build i this formula). However, for POLI 300 problem sets ad tests, you should use the formula give i the previous sectio of this hadout. Dispersio i Ratio Variables Give a ratio variable (e.g. icome), the iterestig dispersio questio may pertai ot to the iterval betwee two observed values or betwee a observed value ad the mea value but to the ratio betwee the two values. (For example, oe household poverty level is defied as oe half the media household icome, ad households with more tha twice the media icome are sometimes characterized as well off. The average compesatio of CEOs today is about 250 times that of the average worker, whereas 50 years it was oly about 40 times that of the average worker.) The degree of dispersio i ratio variables ca aturally be referred to as the degree iequality. Oe ratio measure of dispersio/iequality is the coefficiet of variatio, which is simply the stadard deviatio divided by the mea. Aother is the Gii Idex of Iequality, which is based o a compariso betwee the actual cumulative distributio whe cases are raked ordered from lowest
5 #7 Dispersio page 5 to highest value (e.g., from poorest to richest) ad the cumulative distributio that would exist if all cases had the same value. How to Compute a Stadard Deviatio The formula for the stadard deviatio is: SD(X) = s = 3 (x i! xg ) / 2. r Here is how to use the formula. 1. Set up a worksheet like the oe show below. 2. I the first colum, list the values of the variable X for each of the cases. (This is the raw data.) 3. Fid the mea value of the variable i the data, by addig up the values i each case ad dividig by the umber of cases. 4. I the secod colum, subtract the mea from each value to get, for each case, the deviatio from the mea. Some deviatios are positive, others egative, ad (apart from roudig error) they must add up to zero; add them up as a arithmetic check. 5. I the third colum, square each deviatio from the mea, i.e., multiply the deviatio by itself. Sice the product of two egative umbers is positive, every squared deviatio is oegative, i.e., either positive or (i the evet a case has a value that coicides with the mea value). 6. Add up the squared deviatios over all cases. 7. Divide the sum of the squared deviatios by the umber of cases; this gives the average squared deviatio from the mea, commoly called the variace. 8. The stadard deviatio is the (positive) square root of the variace. (The square root of x is that umber which whe multiplied by itself gives x.)
6 #7 Dispersio page 6
Median and IQR The median is the value which divides the ordered data values in half.
STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationChapter 2 Descriptive Statistics
Chapter 2 Descriptive Statistics Statistics Most commoly, statistics refers to umerical data. Statistics may also refer to the process of collectig, orgaizig, presetig, aalyzig ad iterpretig umerical data
More informationData Description. Measure of Central Tendency. Data Description. Chapter x i
Data Descriptio Describe Distributio with Numbers Example: Birth weights (i lb) of 5 babies bor from two groups of wome uder differet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Chapter 3
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationChapter If n is odd, the median is the exact middle number If n is even, the median is the average of the two middle numbers
Chapter 4 4-1 orth Seattle Commuity College BUS10 Busiess Statistics Chapter 4 Descriptive Statistics Summary Defiitios Cetral tedecy: The extet to which the data values group aroud a cetral value. Variatio:
More informationAnna Janicka Mathematical Statistics 2018/2019 Lecture 1, Parts 1 & 2
Aa Jaicka Mathematical Statistics 18/19 Lecture 1, Parts 1 & 1. Descriptive Statistics By the term descriptive statistics we will mea the tools used for quatitative descriptio of the properties of a sample
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the
More informationParameter, Statistic and Random Samples
Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,
More informationEconomics 250 Assignment 1 Suggested Answers. 1. We have the following data set on the lengths (in minutes) of a sample of long-distance phone calls
Ecoomics 250 Assigmet 1 Suggested Aswers 1. We have the followig data set o the legths (i miutes) of a sample of log-distace phoe calls 1 20 10 20 13 23 3 7 18 7 4 5 15 7 29 10 18 10 10 23 4 12 8 6 (1)
More informationCHAPTER 2. Mean This is the usual arithmetic mean or average and is equal to the sum of the measurements divided by number of measurements.
CHAPTER 2 umerical Measures Graphical method may ot always be sufficiet for describig data. You ca use the data to calculate a set of umbers that will covey a good metal picture of the frequecy distributio.
More information7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals
7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1 7- Sectio 1. Samplig Distributio 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses
More informationENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Solutions Descriptive Statistics. None at all!
ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Solutios Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced
More informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More information(# x) 2 n. (" x) 2 = 30 2 = 900. = sum. " x 2 = =174. " x. Chapter 12. Quick math overview. #(x " x ) 2 = # x 2 "
Chapter 12 Describig Distributios with Numbers Chapter 12 1 Quick math overview = sum These expressios are algebraically equivalet #(x " x ) 2 = # x 2 " (# x) 2 Examples x :{ 2,3,5,6,6,8 } " x = 2 + 3+
More information1 Lesson 6: Measure of Variation
1 Lesso 6: Measure of Variatio 1.1 The rage As we have see, there are several viable coteders for the best measure of the cetral tedecy of data. The mea, the mode ad the media each have certai advatages
More informationBig Picture. 5. Data, Estimates, and Models: quantifying the accuracy of estimates.
5. Data, Estimates, ad Models: quatifyig the accuracy of estimates. 5. Estimatig a Normal Mea 5.2 The Distributio of the Normal Sample Mea 5.3 Normal data, cofidece iterval for, kow 5.4 Normal data, cofidece
More informationContinuous Data that can take on any real number (time/length) based on sample data. Categorical data can only be named or categorised
Questio 1. (Topics 1-3) A populatio cosists of all the members of a group about which you wat to draw a coclusio (Greek letters (μ, σ, Ν) are used) A sample is the portio of the populatio selected for
More informationCURRICULUM INSPIRATIONS: INNOVATIVE CURRICULUM ONLINE EXPERIENCES: TANTON TIDBITS:
CURRICULUM INSPIRATIONS: wwwmaaorg/ci MATH FOR AMERICA_DC: wwwmathforamericaorg/dc INNOVATIVE CURRICULUM ONLINE EXPERIENCES: wwwgdaymathcom TANTON TIDBITS: wwwjamestatocom TANTON S TAKE ON MEAN ad VARIATION
More informationExample: Find the SD of the set {x j } = {2, 4, 5, 8, 5, 11, 7}.
1 (*) If a lot of the data is far from the mea, the may of the (x j x) 2 terms will be quite large, so the mea of these terms will be large ad the SD of the data will be large. (*) I particular, outliers
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationTopic 10: Introduction to Estimation
Topic 0: Itroductio to Estimatio Jue, 0 Itroductio I the simplest possible terms, the goal of estimatio theory is to aswer the questio: What is that umber? What is the legth, the reactio rate, the fractio
More informationKLMED8004 Medical statistics. Part I, autumn Estimation. We have previously learned: Population and sample. New questions
We have previously leared: KLMED8004 Medical statistics Part I, autum 00 How kow probability distributios (e.g. biomial distributio, ormal distributio) with kow populatio parameters (mea, variace) ca give
More informationMeasures of Spread: Standard Deviation
Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chi-square Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chi-square Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationStatistical Inference (Chapter 10) Statistical inference = learn about a population based on the information provided by a sample.
Statistical Iferece (Chapter 10) Statistical iferece = lear about a populatio based o the iformatio provided by a sample. Populatio: The set of all values of a radom variable X of iterest. Characterized
More informationUnderstanding Dissimilarity Among Samples
Aoucemets: Midterm is Wed. Review sheet is o class webpage (i the list of lectures) ad will be covered i discussio o Moday. Two sheets of otes are allowed, same rules as for the oe sheet last time. Office
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationCONFIDENCE INTERVALS STUDY GUIDE
CONFIDENCE INTERVALS STUDY UIDE Last uit, we discussed how sample statistics vary. Uder the right coditios, sample statistics like meas ad proportios follow a Normal distributio, which allows us to calculate
More informationEconomics Spring 2015
1 Ecoomics 400 -- Sprig 015 /17/015 pp. 30-38; Ch. 7.1.4-7. New Stata Assigmet ad ew MyStatlab assigmet, both due Feb 4th Midterm Exam Thursday Feb 6th, Chapters 1-7 of Groeber text ad all relevat lectures
More informationUnderstanding Samples
1 Will Moroe CS 109 Samplig ad Bootstrappig Lecture Notes #17 August 2, 2017 Based o a hadout by Chris Piech I this chapter we are goig to talk about statistics calculated o samples from a populatio. We
More informationLecture 1. Statistics: A science of information. Population: The population is the collection of all subjects we re interested in studying.
Lecture Mai Topics: Defiitios: Statistics, Populatio, Sample, Radom Sample, Statistical Iferece Type of Data Scales of Measuremet Describig Data with Numbers Describig Data Graphically. Defiitios. Example
More informationCensus. Mean. µ = x 1 + x x n n
MATH 183 Basic Statistics Dr. Neal, WKU Let! be a populatio uder cosideratio ad let X be a specific measuremet that we are aalyzig. For example,! = All U.S. households ad X = Number of childre (uder age
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationBUSINESS STATISTICS (PART-9) AVERAGE OR MEASURES OF CENTRAL TENDENCY: THE GEOMETRIC AND HARMONIC MEANS
BUSINESS STATISTICS (PART-9) AVERAGE OR MEASURES OF CENTRAL TENDENCY: THE GEOMETRIC AND HARMONIC MEANS. INTRODUCTION We have so far discussed three measures of cetral tedecy, viz. The Arithmetic Mea, Media
More informationSample Size Determination (Two or More Samples)
Sample Sie Determiatio (Two or More Samples) STATGRAPHICS Rev. 963 Summary... Data Iput... Aalysis Summary... 5 Power Curve... 5 Calculatios... 6 Summary This procedure determies a suitable sample sie
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationStat 139 Homework 7 Solutions, Fall 2015
Stat 139 Homework 7 Solutios, Fall 2015 Problem 1. I class we leared that the classical simple liear regressio model assumes the followig distributio of resposes: Y i = β 0 + β 1 X i + ɛ i, i = 1,...,,
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationFrequentist Inference
Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for
More informationMATH/STAT 352: Lecture 15
MATH/STAT 352: Lecture 15 Sectios 5.2 ad 5.3. Large sample CI for a proportio ad small sample CI for a mea. 1 5.2: Cofidece Iterval for a Proportio Estimatig proportio of successes i a biomial experimet
More informationMeasures of Spread: Variance and Standard Deviation
Lesso 1-6 Measures of Spread: Variace ad Stadard Deviatio BIG IDEA Variace ad stadard deviatio deped o the mea of a set of umbers. Calculatig these measures of spread depeds o whether the set is a sample
More informationEcon 325 Notes on Point Estimator and Confidence Interval 1 By Hiro Kasahara
Poit Estimator Eco 325 Notes o Poit Estimator ad Cofidece Iterval 1 By Hiro Kasahara Parameter, Estimator, ad Estimate The ormal probability desity fuctio is fully characterized by two costats: populatio
More information(6) Fundamental Sampling Distribution and Data Discription
34 Stat Lecture Notes (6) Fudametal Samplig Distributio ad Data Discriptio ( Book*: Chapter 8,pg5) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye 8.1 Radom Samplig: Populatio:
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationCHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics
CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based
More informationVariance of Discrete Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Variace of Discrete Radom Variables Class 5, 18.05 Jeremy Orloff ad Joatha Bloom 1 Learig Goals 1. Be able to compute the variace ad stadard deviatio of a radom variable.. Uderstad that stadard deviatio
More informationLecture Note 8 Point Estimators and Point Estimation Methods. MIT Spring 2006 Herman Bennett
Lecture Note 8 Poit Estimators ad Poit Estimatio Methods MIT 14.30 Sprig 2006 Herma Beett Give a parameter with ukow value, the goal of poit estimatio is to use a sample to compute a umber that represets
More informationStatisticians use the word population to refer the total number of (potential) observations under consideration
6 Samplig Distributios Statisticias use the word populatio to refer the total umber of (potetial) observatios uder cosideratio The populatio is just the set of all possible outcomes i our sample space
More informationRecall the study where we estimated the difference between mean systolic blood pressure levels of users of oral contraceptives and non-users, x - y.
Testig Statistical Hypotheses Recall the study where we estimated the differece betwee mea systolic blood pressure levels of users of oral cotraceptives ad o-users, x - y. Such studies are sometimes viewed
More informationChapter 23: Inferences About Means
Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationSummarizing Data. Major Properties of Numerical Data
Summarizig Data Daiel A. Meascé, Ph.D. Dept of Computer Sciece George Maso Uiversity Major Properties of Numerical Data Cetral Tedecy: arithmetic mea, geometric mea, media, mode. Variability: rage, iterquartile
More informationMBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos
More informationLet us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.
Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationInferential Statistics. Inference Process. Inferential Statistics and Probability a Holistic Approach. Inference Process.
Iferetial Statistics ad Probability a Holistic Approach Iferece Process Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos Attributio-ShareAlike
More informationBecause it tests for differences between multiple pairs of means in one test, it is called an omnibus test.
Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal
More informationA quick activity - Central Limit Theorem and Proportions. Lecture 21: Testing Proportions. Results from the GSS. Statistics and the General Population
A quick activity - Cetral Limit Theorem ad Proportios Lecture 21: Testig Proportios Statistics 10 Coli Rudel Flip a coi 30 times this is goig to get loud! Record the umber of heads you obtaied ad calculate
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationCEE 522 Autumn Uncertainty Concepts for Geotechnical Engineering
CEE 5 Autum 005 Ucertaity Cocepts for Geotechical Egieerig Basic Termiology Set A set is a collectio of (mutually exclusive) objects or evets. The sample space is the (collectively exhaustive) collectio
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Aswers & Notes TI-Nspire Ivestigatio Studet 60 mi 7 8 9 0 Itroductio A 00 survey of attitudes to climate chage, coducted i Australia by the CSIRO,
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More information4 Multidimensional quantitative data
Chapter 4 Multidimesioal quatitative data 4 Multidimesioal statistics Basic statistics are ow part of the curriculum of most ecologists However, statistical techiques based o such simple distributios as
More informationFinal Examination Solutions 17/6/2010
The Islamic Uiversity of Gaza Faculty of Commerce epartmet of Ecoomics ad Political Scieces A Itroductio to Statistics Course (ECOE 30) Sprig Semester 009-00 Fial Eamiatio Solutios 7/6/00 Name: I: Istructor:
More informationEco411 Lab: Central Limit Theorem, Normal Distribution, and Journey to Girl State
Eco411 Lab: Cetral Limit Theorem, Normal Distributio, ad Jourey to Girl State 1. Some studets may woder why the magic umber 1.96 or 2 (called critical values) is so importat i statistics. Where do they
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationInstructor: Judith Canner Spring 2010 CONFIDENCE INTERVALS How do we make inferences about the population parameters?
CONFIDENCE INTERVALS How do we make ifereces about the populatio parameters? The samplig distributio allows us to quatify the variability i sample statistics icludig how they differ from the parameter
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationAP Statistics Review Ch. 8
AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More informationExam II Covers. STA 291 Lecture 19. Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Location CB 234
STA 291 Lecture 19 Exam II Next Tuesday 5-7pm Memorial Hall (Same place as exam I) Makeup Exam 7:15pm 9:15pm Locatio CB 234 STA 291 - Lecture 19 1 Exam II Covers Chapter 9 10.1; 10.2; 10.3; 10.4; 10.6
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationBasics of Probability Theory (for Theory of Computation courses)
Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.
More informationAgreement of CI and HT. Lecture 13 - Tests of Proportions. Example - Waiting Times
Sigificace level vs. cofidece level Agreemet of CI ad HT Lecture 13 - Tests of Proportios Sta102 / BME102 Coli Rudel October 15, 2014 Cofidece itervals ad hypothesis tests (almost) always agree, as log
More informationn outcome is (+1,+1, 1,..., 1). Let the r.v. X denote our position (relative to our starting point 0) after n moves. Thus X = X 1 + X 2 + +X n,
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 9 Variace Questio: At each time step, I flip a fair coi. If it comes up Heads, I walk oe step to the right; if it comes up Tails, I walk oe
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationSTAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)
STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated
More informationIntroducing Sample Proportions
Itroducig Sample Proportios Probability ad statistics Studet Activity TI-Nspire Ivestigatio Studet 60 mi 7 8 9 10 11 12 Itroductio A 2010 survey of attitudes to climate chage, coducted i Australia by the
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationTABLES AND FORMULAS FOR MOORE Basic Practice of Statistics
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x 2 + +
More informationFormulas and Tables for Gerstman
Formulas ad Tables for Gerstma Measuremet ad Study Desig Biostatistics is more tha a compilatio of computatioal techiques! Measuremet scales: quatitative, ordial, categorical Iformatio quality is primary
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationConfidence Intervals for the Population Proportion p
Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical:
More informationChapter 6 Sampling Distributions
Chapter 6 Samplig Distributios 1 I most experimets, we have more tha oe measuremet for ay give variable, each measuremet beig associated with oe radomly selected a member of a populatio. Hece we eed to
More informationProbability and statistics: basic terms
Probability ad statistics: basic terms M. Veeraraghava August 203 A radom variable is a rule that assigs a umerical value to each possible outcome of a experimet. Outcomes of a experimet form the sample
More information