STAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model
|
|
- Kathryn Gregory
- 5 years ago
- Views:
Transcription
1 STAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model In Chapter 5, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are good summaries when the histogram (or distribution) is symmetric and unimodal. When it is not symmetric, we use the median and IQR as summaries, although for most of the course, we will deal with things that are approximately symmetric and unimodal. Understanding the idea of what a Standard Deviation is, is very important as almost all statistical methods rely on this, and we will see it come up again and again throughout the course (in all of statistics, actually). Recall: The SD can be thought of as a measure of the typical deviation from the mean. We will use the standard deviation as a unit of measurement... I will explain this Example: Suppose I were at a crucial point in my life where I was trying to decide what to do with it; pursue my education or a career as a professional golfer? Suppose that in high school my graduating average was 82%, and the mean graduating average is 67% with a standard deviation of 15%. Suppose that for golfing, I have a mean score of 77, and the mean score of a competitive golfer is typically 79 with a standard deviation of 2.1. Which one am I relatively better at? (Note: These are made up numbers!) If we just look at the values it is hard to compare the two. They are measured on different scales, with different units of measurement. A grade must be between 0 100, while a golf score almost never gets below 59. So how can we compare the two? The answer is to use the Standard Deviation as a measuring stick, as it summarizes the average /typical deviation from the mean. Essentially, we will want to find out how far each one is from its respective mean, in terms of its average deviation from the mean. 1
2 The High school grade is... above the mean grade, and... in terms of standard deviation, it is... standard deviations above the mean. The evaluation score is... above the mean score, and... in terms of standard deviation, it is... standard deviations above the mean. We can compare the two in terms of each of their own respective means and average deviation from the mean (or SD)... In your every day life you are essentially using statistical tools to make decisions, without even knowing it... In my opinion, statistics is simply a discipline that tries to take the way a person thinks about things and makes logical decisions based on what they observe in every day life, and formalize these into a set of objective rules. Adding or Multiplying each value by a constant: 1. Adding a constant (shifting) If we add a constant (c) to each observation in the data then: The measures of center (mean, median, midrange) will all have the constant (c) added to them, and so will the Quartiles. The measures of spread (variance, SD, range, IQR) will all remain the same. 2. Multiplying by a constant (scaling) If we multiply each observation by a constant (c), then: The measures of center (mean, median, midrange), and the measures of spread (SD, range, IQR) and the Quartiles will all be multiplied by the constant (c). The variance will be multiplied by c 2 In short, adding changes center, but not spread. Multiplying changes the spread and center. Multiplying by a constant is how we change measurement units (eg) Kg to lbs. 2
3 Standardizing (Z-scores): Question: How can we compare observations that were measured on different scales or from two different distributions? Answer: By summarizing how far away each of the observations is from the mean, in terms of its standard deviation (or average/typical deviation from the mean)! The Z-score summarizes how far a given observation (y i ) is from its mean (ȳ), in terms of it s SD (s). Z-score (Z)= Z = y i ȳ s difference between observation and mean Standard deviation Exercise: A flight from Vancouver to Toronto usually takes 4.5 hours with a SD of 15 minutes. If my last flight took 4 hours and 10 minutes, how far is this from the mean in standard units? When we Standardize, we are adding (actually subtracting) a constant from every observation, and then multiplying (actually dividing) every observation by a constant...check rules on last page If we let M = y i ȳ, then the mean of M is ȳ ȳ = 0, and the SD of M is unchanged. If we now let Z = M, then the mean of Z is the mean of M times the constant, which SD equals 0. The SD of Z is the SD of M times the constant, which is SD = 1. SD So, Z-scores have a mean of 0 and a SD of 1. A positive Z-score means that the observation is above the mean, and a negative one means its below it. The farther an observation is from the mean, the larger the Z-score will be in absolute value. 3
4 The Normal Model (Bell Curve, Normal Distribution): This is where we take a small step into the theoretical world of statistics. Many types of data one collects have a distribution that is bell shaped and roughly symmetric, and the Normal Model is appropriate for summarizing these (note that we are dealing with only quantitative variables here). (eg) weight, IQ scores,... Characteristics of Normal Model: 1. It is bell-shaped, unimodal, and perfectly symmetric about the mean (Ȳ or µ). 2. The spread of the distribution is determined by the standard deviation (s or σ). 3. This model is denoted by: N(µ, σ 2 ), where µ=mean, σ 2 =Variance, and σ is the SD. 4. The total area under the curve is 100% (just as the total area of the bars for a histogram is 100%) Theoretical Normal Models Porbability (%) N(2, 36) N( 4, 9) N(2, 4) Values Notes: For the Normal Model, we use (µ) for the mean instead of (ȳ), and (σ) for the SD instead of (s), why??? The (ȳ) and (s) are Sample Statistics; numerical summaries of the observed data. (sample) The (µ) and (σ) are Population Parameters; that specify the theoretical model. (population) 4
5 Standardized Values (for the Normal Model: ) Z = y µ σ When we standardize an observation from a Normal Model, the Z-score is N(0, 1). What we do is we use a theoretical Normal Model to describe the distribution of an observed variable. One must check the histogram to make sure that such a model is appropriate (symmetric and unimodal). We take the observed estimates of the mean and SD, and if a Normal Model seems appropriate, then we use the Normal Model (with the same mean and SD to approximate the observed data. We then standardize the value(s) of interest, so that we can use a Standard Normal variable (N(0, 1)). We can then answer questions such as: What proportion of males have weights above 190lbs? How many between 210 and 220? and so on... The Rule: Approximately 68% of the data will be within +/ 1 SD of the mean. Approximately 95% of the data will be within +/ 2 SDs of the mean. Approximately 99.7% of the data will be within +/ 3 SDs of the mean. (eg) if a class has a mean grade of 70% and a SD of 5% and the grades are normally distributed, then approximately 68% of students will receive grades between 65-75%, approx. 95% will receive grades between 60-80%, and 99.7% between 55-85%. Let s Draw a Picture: 5
6 Finding Percentages Under the Normal Model: 1. Draw a Normal Model and label where the mean is. Then shade the area of interest. 2. Standardize the y-value(s) that are at the boundaries of the area of interest. 3. Use the Normal Table in Appendix E of the Text Book to find the area of the shaded region. Example: What is the area (probability) below a Z-score of Z = 1.52? What is the area (probability) between Z-scores of and 1.23? Summary: 1. We estimate the mean and SD for our observed data. 2. Check if a Normal Model is appropriate (symmetric, unimodal) 3. If it is, then we standardize the values of interest. 4. Use the Normal Table to find the percentages we are interested in. (the Normal Model is HUGE in statistics, so make sure to practice many of these problems) 6
7 Exercises: 1. Suppose that math SAT scores follow the normal model. The past results of the math SAT exams show that males and females have mean scores of 500 and 455 and standard deviations of 100 and 120, respectively. Joe and Linda took the math SAT exam, and they both scored 620. (a) Compare their scores using the z-score. (b) What percentage of males score over 600 on the math SAT test? (c) What percentage of females score between 255 and 555 on the math SAT test? 2. Find the area under the Normal Model for the following Z-scores. (a) smaller than (b) bigger than (c) bigger than 2.15 (d) between 0 and 1.18 (e) between and 1.62 (f) smaller than (g) bigger than Find the z-scores corresponding to the following percentiles: (a) 50 th (b) 70 th (c) 15 th 4. Suppose that scores on a standard IQ test approximately follow the normal model with mean µ = 110 and standard deviation σ = 25. (a) What percentage of people have IQ scores above 100? (b) What percentage have scores between 90 and 120? (c) Find the interquartile range for the IQ scores. 7
8 5. The length of human pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 266 days and standard deviation 16 days. (a) Between what values do the lengths of the middle 95% of all pregnancies fall? Use the rule to answer this question. (b) How short are the shortest 1% of all pregnancies? 8
STAT Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model
STAT 203 - Chapter 6 The Standard Deviation (SD) as a Ruler and The Normal Model In Chapter 5, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are good
More informationBIOL The Normal Distribution and the Central Limit Theorem
BIOL 300 - The Normal Distribution and the Central Limit Theorem In the first week of the course, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are
More informationNormal Model (Part 1)
Normal Model (Part 1) Formulas New Vocabulary The Standard Deviation as a Ruler The trick in comparing very different-looking values is to use standard deviations as our rulers. The standard deviation
More informationBoth the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need.
Both the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need. For exams (MD1, MD2, and Final): You may bring one 8.5 by 11 sheet of
More informationChapter 6. y y. Standardizing with z-scores. Standardizing with z-scores (cont.)
Starter Ch. 6: A z-score Analysis Starter Ch. 6 Your Statistics teacher has announced that the lower of your two tests will be dropped. You got a 90 on test 1 and an 85 on test 2. You re all set to drop
More informationMath 243 Lecture Notes
Assume the average annual rainfall for in Portland is 36 inches per year with a standard deviation of 9 inches. Also assume that the average wind speed in Chicago is 10 mph with a standard deviation of
More informationThe Standard Deviation as a Ruler and the Normal Model. Copyright 2009 Pearson Education, Inc.
The Standard Deviation as a Ruler and the Normal Mol Copyright 2009 Pearson Education, Inc. The trick in comparing very different-looking values is to use standard viations as our rulers. The standard
More informationShifting and rescaling data distributions
Shifting and rescaling data distributions It is useful to consider the effect of systematic alterations of all the values in a data set. The simplest such systematic effect is a shift by a fixed constant.
More informationDensity curves. (James Madison University) February 4, / 20
Density curves Figure 6.2 p 230. A density curve is always on or above the horizontal axis, and has area exactly 1 underneath it. A density curve describes the overall pattern of a distribution. Example
More informationSection3-2: Measures of Center
Chapter 3 Section3-: Measures of Center Notation Suppose we are making a series of observations, n of them, to be exact. Then we write x 1, x, x 3,K, x n as the values we observe. Thus n is the total number
More informationThe Normal Distribution
Stat 6 Introduction to Business Statistics I Spring 009 Professor: Dr. Petrutza Caragea Section A Tuesdays and Thursdays 9:300:50 a.m. Chapter, Section.3 The Normal Distribution Density Curves So far we
More informationChapter 3. Numerical Descriptive Measures. Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 1
Chapter 3 Numerical Descriptive Measures Copyright 2016 Pearson Education, Ltd. Chapter 3, Slide 1 Objectives In this chapter, you learn to: Describe the properties of central tendency, variation, and
More informationDescribing Data: One Quantitative Variable
STAT 250 Dr. Kari Lock Morgan The Big Picture Describing Data: One Quantitative Variable Population Sampling SECTIONS 2.2, 2.3 One quantitative variable (2.2, 2.3) Statistical Inference Sample Descriptive
More informationStatistics 511 Supplemental Materials
Gaussian (or Normal) Random Variable In this section we introduce the Gaussian Random Variable, which is more commonly referred to as the Normal Random Variable. This is a random variable that has a bellshaped
More informationApplications of Data Dispersions
1 Applications of Data Dispersions Key Definitions Standard Deviation: The standard deviation shows how far away each value is from the mean on average. Z-Scores: The distance between the mean and a given
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More informationTerms & Characteristics
NORMAL CURVE Knowledge that a variable is distributed normally can be helpful in drawing inferences as to how frequently certain observations are likely to occur. NORMAL CURVE A Normal distribution: Distribution
More informationChapter 6. The Normal Probability Distributions
Chapter 6 The Normal Probability Distributions 1 Chapter 6 Overview Introduction 6-1 Normal Probability Distributions 6-2 The Standard Normal Distribution 6-3 Applications of the Normal Distribution 6-5
More informationAP Stats ~ Lesson 6B: Transforming and Combining Random variables
AP Stats ~ Lesson 6B: Transforming and Combining Random variables OBJECTIVES: DESCRIBE the effects of transforming a random variable by adding or subtracting a constant and multiplying or dividing by a
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 6 The Normal Distribution 2 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find
More information8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions 8-1
8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions For Example: On August 8, 2011, the Dow dropped 634.8 points, sending shock waves through the financial community.
More informationWhat was in the last lecture?
What was in the last lecture? Normal distribution A continuous rv with bell-shaped density curve The pdf is given by f(x) = 1 2πσ e (x µ)2 2σ 2, < x < If X N(µ, σ 2 ), E(X) = µ and V (X) = σ 2 Standard
More informationThe Normal Distribution
5.1 Introduction to Normal Distributions and the Standard Normal Distribution Section Learning objectives: 1. How to interpret graphs of normal probability distributions 2. How to find areas under the
More information3.1 Measures of Central Tendency
3.1 Measures of Central Tendency n Summation Notation x i or x Sum observation on the variable that appears to the right of the summation symbol. Example 1 Suppose the variable x i is used to represent
More informationMeasures of Center. Mean. 1. Mean 2. Median 3. Mode 4. Midrange (rarely used) Measure of Center. Notation. Mean
Measure of Center Measures of Center The value at the center or middle of a data set 1. Mean 2. Median 3. Mode 4. Midrange (rarely used) 1 2 Mean Notation The measure of center obtained by adding the values
More informationMidterm Exam III Review
Midterm Exam III Review Dr. Joseph Brennan Math 148, BU Dr. Joseph Brennan (Math 148, BU) Midterm Exam III Review 1 / 25 Permutations and Combinations ORDER In order to count the number of possible ways
More informationSTAT:2010 Statistical Methods and Computing. Using density curves to describe the distribution of values of a quantitative
STAT:10 Statistical Methods and Computing Normal Distributions Lecture 4 Feb. 6, 17 Kate Cowles 374 SH, 335-0727 kate-cowles@uiowa.edu 1 2 Using density curves to describe the distribution of values of
More informationChapter 3. Descriptive Measures. Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 1
Chapter 3 Descriptive Measures Copyright 2016, 2012, 2008 Pearson Education, Inc. Chapter 3, Slide 1 Chapter 3 Descriptive Measures Mean, Median and Mode Copyright 2016, 2012, 2008 Pearson Education, Inc.
More informationIOP 201-Q (Industrial Psychological Research) Tutorial 5
IOP 201-Q (Industrial Psychological Research) Tutorial 5 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. To establish a cause-and-effect relation between two variables,
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationMath 2311 Bekki George Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment
Math 2311 Bekki George bekki@math.uh.edu Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment Class webpage: http://www.math.uh.edu/~bekki/math2311.html Math 2311 Class
More informationNORMAL RANDOM VARIABLES (Normal or gaussian distribution)
NORMAL RANDOM VARIABLES (Normal or gaussian distribution) Many variables, as pregnancy lengths, foot sizes etc.. exhibit a normal distribution. The shape of the distribution is a symmetric bell shape.
More informationThe Range, the Inter Quartile Range (or IQR), and the Standard Deviation (which we usually denote by a lower case s).
We will look the three common and useful measures of spread. The Range, the Inter Quartile Range (or IQR), and the Standard Deviation (which we usually denote by a lower case s). 1 Ameasure of the center
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.2 Transforming and Combining Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers 6.2 Reading Quiz (T or F)
More informationChapter 3. Lecture 3 Sections
Chapter 3 Lecture 3 Sections 3.4 3.5 Measure of Position We would like to compare values from different data sets. We will introduce a z score or standard score. This measures how many standard deviation
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More information1 Describing Distributions with numbers
1 Describing Distributions with numbers Only for quantitative variables!! 1.1 Describing the center of a data set The mean of a set of numerical observation is the familiar arithmetic average. To write
More informationSTAB22 section 1.3 and Chapter 1 exercises
STAB22 section 1.3 and Chapter 1 exercises 1.101 Go up and down two times the standard deviation from the mean. So 95% of scores will be between 572 (2)(51) = 470 and 572 + (2)(51) = 674. 1.102 Same idea
More informationAs you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
More informationappstats5.notebook September 07, 2016 Chapter 5
Chapter 5 Describing Distributions Numerically Chapter 5 Objective: Students will be able to use statistics appropriate to the shape of the data distribution to compare of two or more different data sets.
More information5.1 Mean, Median, & Mode
5.1 Mean, Median, & Mode definitions Mean: Median: Mode: Example 1 The Blue Jays score these amounts of runs in their last 9 games: 4, 7, 2, 4, 10, 5, 6, 7, 7 Find the mean, median, and mode: Example 2
More informationThe Normal Model The famous bell curve
Math 243 Sections 6.1-6.2 The Normal Model Here are some roughly symmetric, unimodal histograms The Normal Model The famous bell curve Example 1. Let s say the mean annual rainfall in Portland is 40 inches
More informationThe graph of a normal curve is symmetric with respect to the line x = µ, and has points of
Stat 400, section 4.3 Normal Random Variables notes prepared by Tim Pilachowski Another often-useful probability density function is the normal density function, which graphs as the familiar bell-shaped
More informationSection 6.2 Transforming and Combining Random Variables. Linear Transformations
Section 6.2 Transforming and Combining Random Variables Linear Transformations In Section 6.1, we learned that the mean and standard deviation give us important information about a random variable. In
More informationMLLunsford 1. Activity: Central Limit Theorem Theory and Computations
MLLunsford 1 Activity: Central Limit Theorem Theory and Computations Concepts: The Central Limit Theorem; computations using the Central Limit Theorem. Prerequisites: The student should be familiar with
More informationFigure 1: 2πσ is said to have a normal distribution with mean µ and standard deviation σ. This is also denoted
Figure 1: Math 223 Lecture Notes 4/1/04 Section 4.10 The normal distribution Recall that a continuous random variable X with probability distribution function f(x) = 1 µ)2 (x e 2σ 2πσ is said to have a
More information22.2 Shape, Center, and Spread
Name Class Date 22.2 Shape, Center, and Spread Essential Question: Which measures of center and spread are appropriate for a normal distribution, and which are appropriate for a skewed distribution? Eplore
More informationStatistics vs. statistics
Statistics vs. statistics Question: What is Statistics (with a capital S)? Definition: Statistics is the science of collecting, organizing, summarizing and interpreting data. Note: There are 2 main ways
More informationChapter 6: The Normal Distribution
Chapter 6: The Normal Distribution Diana Pell Section 6.1: Normal Distributions Note: Recall that a continuous variable can assume all values between any two given values of the variables. Many continuous
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationChapter 6: The Normal Distribution
Chapter 6: The Normal Distribution Diana Pell Section 6.1: Normal Distributions Note: Recall that a continuous variable can assume all values between any two given values of the variables. Many continuous
More informationStat 101 Exam 1 - Embers Important Formulas and Concepts 1
1 Chapter 1 1.1 Definitions Stat 101 Exam 1 - Embers Important Formulas and Concepts 1 1. Data Any collection of numbers, characters, images, or other items that provide information about something. 2.
More informationUnit 2 Statistics of One Variable
Unit 2 Statistics of One Variable Day 6 Summarizing Quantitative Data Summarizing Quantitative Data We have discussed how to display quantitative data in a histogram It is useful to be able to describe
More informationLecture 6: Normal distribution
Lecture 6: Normal distribution Statistics 101 Mine Çetinkaya-Rundel February 2, 2012 Announcements Announcements HW 1 due now. Due: OQ 2 by Monday morning 8am. Statistics 101 (Mine Çetinkaya-Rundel) L6:
More informationLECTURE 6 DISTRIBUTIONS
LECTURE 6 DISTRIBUTIONS OVERVIEW Uniform Distribution Normal Distribution Random Variables Continuous Distributions MOST OF THE SLIDES ADOPTED FROM OPENINTRO STATS BOOK. NORMAL DISTRIBUTION Unimodal and
More informationLecture 2 Describing Data
Lecture 2 Describing Data Thais Paiva STA 111 - Summer 2013 Term II July 2, 2013 Lecture Plan 1 Types of data 2 Describing the data with plots 3 Summary statistics for central tendency and spread 4 Histograms
More informationTHE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management
THE UNIVERSITY OF TEXAS AT AUSTIN Department of Information, Risk, and Operations Management BA 386T Tom Shively PROBABILITY CONCEPTS AND NORMAL DISTRIBUTIONS The fundamental idea underlying any statistical
More informationSection Introduction to Normal Distributions
Section 6.1-6.2 Introduction to Normal Distributions 2012 Pearson Education, Inc. All rights reserved. 1 of 105 Section 6.1-6.2 Objectives Interpret graphs of normal probability distributions Find areas
More informationNOTES TO CONSIDER BEFORE ATTEMPTING EX 2C BOX PLOTS
NOTES TO CONSIDER BEFORE ATTEMPTING EX 2C BOX PLOTS A box plot is a pictorial representation of the data and can be used to get a good idea and a clear picture about the distribution of the data. It shows
More informationFall 2011 Exam Score: /75. Exam 3
Math 12 Fall 2011 Name Exam Score: /75 Total Class Percent to Date Exam 3 For problems 1-10, circle the letter next to the response that best answers the question or completes the sentence. You do not
More informationNormal Distribution. Notes. Normal Distribution. Standard Normal. Sums of Normal Random Variables. Normal. approximation of Binomial.
Lecture 21,22, 23 Text: A Course in Probability by Weiss 8.5 STAT 225 Introduction to Probability Models March 31, 2014 Standard Sums of Whitney Huang Purdue University 21,22, 23.1 Agenda 1 2 Standard
More informationIn a binomial experiment of n trials, where p = probability of success and q = probability of failure. mean variance standard deviation
Name In a binomial experiment of n trials, where p = probability of success and q = probability of failure mean variance standard deviation µ = n p σ = n p q σ = n p q Notation X ~ B(n, p) The probability
More informationSince his score is positive, he s above average. Since his score is not close to zero, his score is unusual.
Chapter 06: The Standard Deviation as a Ruler and the Normal Model This is the worst chapter title ever! This chapter is about the most important random variable distribution of them all the normal distribution.
More informationAnnouncements. Unit 2: Probability and distributions Lecture 3: Normal distribution. Normal distribution. Heights of males
Announcements Announcements Unit 2: Probability and distributions Lecture 3: Statistics 101 Mine Çetinkaya-Rundel First peer eval due Tues. PS3 posted - will be adding one more question that you need to
More informationMath 140 Introductory Statistics. First midterm September
Math 140 Introductory Statistics First midterm September 23 2010 Box Plots Graphical display of 5 number summary Q1, Q2 (median), Q3, max, min Outliers If a value is more than 1.5 times the IQR from the
More informationChapter 2: Descriptive Statistics. Mean (Arithmetic Mean): Found by adding the data values and dividing the total by the number of data.
-3: Measure of Central Tendency Chapter : Descriptive Statistics The value at the center or middle of a data set. It is a tool for analyzing data. Part 1: Basic concepts of Measures of Center Ex. Data
More informationUniversity of California, Los Angeles Department of Statistics. The central limit theorem The distribution of the sample mean
University of California, Los Angeles Department of Statistics Statistics 12 Instructor: Nicolas Christou First: Population mean, µ: The central limit theorem The distribution of the sample mean Sample
More informationReview of commonly missed questions on the online quiz. Lecture 7: Random variables] Expected value and standard deviation. Let s bet...
Recap Review of commonly missed questions on the online quiz Lecture 7: ] Statistics 101 Mine Çetinkaya-Rundel OpenIntro quiz 2: questions 4 and 5 September 20, 2011 Statistics 101 (Mine Çetinkaya-Rundel)
More informationExamples of continuous probability distributions: The normal and standard normal
Examples of continuous probability distributions: The normal and standard normal The Normal Distribution f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.
More informationCHAPTER 6 Random Variables
CHAPTER 6 Random Variables 6.2 Transforming and Combining Random Variables The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Transforming and Combining
More informationUnit2: Probabilityanddistributions. 3. Normal distribution
Announcements Unit: Probabilityanddistributions 3 Normal distribution Sta 101 - Spring 015 Duke University, Department of Statistical Science February, 015 Peer evaluation 1 by Friday 11:59pm Office hours:
More informationSome estimates of the height of the podium
Some estimates of the height of the podium 24 36 40 40 40 41 42 44 46 48 50 53 65 98 1 5 number summary Inter quartile range (IQR) range = max min 2 1.5 IQR outlier rule 3 make a boxplot 24 36 40 40 40
More informationNormal distribution. We say that a random variable X follows the normal distribution if the probability density function of X is given by
Normal distribution The normal distribution is the most important distribution. It describes well the distribution of random variables that arise in practice, such as the heights or weights of people,
More informationSection 7.4 Transforming and Combining Random Variables (DAY 1)
Section 7.4 Learning Objectives (DAY 1) After this section, you should be able to DESCRIBE the effect of performing a linear transformation on a random variable (DAY 1) COMBINE random variables and CALCULATE
More informationSTOR 155 Practice Midterm 1 Fall 2009
STOR 155 Practice Midterm 1 Fall 2009 INSTRUCTIONS: BOTH THE EXAM AND THE BUBBLE SHEET WILL BE COLLECTED. YOU MUST PRINT YOUR NAME AND SIGN THE HONOR PLEDGE ON THE BUBBLE SHEET. YOU MUST BUBBLE-IN YOUR
More informationPercentiles, STATA, Box Plots, Standardizing, and Other Transformations
Percentiles, STATA, Box Plots, Standardizing, and Other Transformations Lecture 3 Reading: Sections 5.7 54 Remember, when you finish a chapter make sure not to miss the last couple of boxes: What Can Go
More information3) Marital status of each member of a randomly selected group of adults is an example of what type of variable?
MATH112 STATISTICS; REVIEW1 CH1,2,&3 Name CH1 Vocabulary 1) A statistics student wants to find some information about all college students who ride a bike. She collected data from other students in her
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name The bar graph shows the number of tickets sold each week by the garden club for their annual flower show. ) During which week was the most number of tickets sold? ) A) Week B) Week C) Week 5
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More informationChapter 5 The Standard Deviation as a Ruler and the Normal Model
Chapter 5 The Standard Deviation as a Ruler and the Normal Model 55 Chapter 5 The Standard Deviation as a Ruler and the Normal Model 1. Stats test. Nicole scored 65 points on the test. That is one standard
More informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 6.3 Binomial and
More informationA LEVEL MATHEMATICS ANSWERS AND MARKSCHEMES SUMMARY STATISTICS AND DIAGRAMS. 1. a) 45 B1 [1] b) 7 th value 37 M1 A1 [2]
1. a) 45 [1] b) 7 th value 37 [] n c) LQ : 4 = 3.5 4 th value so LQ = 5 3 n UQ : 4 = 9.75 10 th value so UQ = 45 IQR = 0 f.t. d) Median is closer to upper quartile Hence negative skew [] Page 1 . a) Orders
More informationUniversity of California, Los Angeles Department of Statistics
University of California, Los Angeles Department of Statistics Statistics 13 Instructor: Nicolas Christou The central limit theorem The distribution of the sample proportion The distribution of the sample
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter7 Probability Distributions and Statistics Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number of boys in
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Chapter 6 Exam A Name The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. 1) The probability of
More informationDot Plot: A graph for displaying a set of data. Each numerical value is represented by a dot placed above a horizontal number line.
Introduction We continue our study of descriptive statistics with measures of dispersion, such as dot plots, stem and leaf displays, quartiles, percentiles, and box plots. Dot plots, a stem-and-leaf display,
More informationChapter 6: Random Variables
Chapter 6: Random Variables Section 6.2 The Practice of Statistics, 4 th edition For AP* STARNES, YATES, MOORE Chapter 6 Random Variables 6.1 Discrete and Continuous Random Variables 6.2 6.3 Binomial and
More informationData that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc.
Chapter 8 Measures of Center Data that can be any numerical value are called continuous. These are usually things that are measured, such as height, length, time, speed, etc. Data that can only be integer
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationLecture 6: Chapter 6
Lecture 6: Chapter 6 C C Moxley UAB Mathematics 3 October 16 6.1 Continuous Probability Distributions Last week, we discussed the binomial probability distribution, which was discrete. 6.1 Continuous Probability
More information6.2.1 Linear Transformations
6.2.1 Linear Transformations In Chapter 2, we studied the effects of transformations on the shape, center, and spread of a distribution of data. Recall what we discovered: 1. Adding (or subtracting) a
More informationChapter 15: Graphs, Charts, and Numbers Math 107
Chapter 15: Graphs, Charts, and Numbers Math 107 Data Set & Data Point: Discrete v. Continuous: Frequency Table: Ex 1) Exam Scores Pictogram: Misleading Graphs: In reality, the data looks like this 45%
More informationDATA SUMMARIZATION AND VISUALIZATION
APPENDIX DATA SUMMARIZATION AND VISUALIZATION PART 1 SUMMARIZATION 1: BUILDING BLOCKS OF DATA ANALYSIS 294 PART 2 PART 3 PART 4 VISUALIZATION: GRAPHS AND TABLES FOR SUMMARIZING AND ORGANIZING DATA 296
More informationUnit2: Probabilityanddistributions. 3. Normal and binomial distributions
Announcements Unit2: Probabilityanddistributions 3. Normal and binomial distributions Sta 101 - Summer 2017 Duke University, Department of Statistical Science PS: Explain your reasoning + show your work
More informationI. Standard Error II. Standard Error III. Standard Error 2.54
1) Original Population: Match the standard error (I, II, or III) with the correct sampling distribution (A, B, or C) and the correct sample size (1, 5, or 10) I. Standard Error 1.03 II. Standard Error
More informationExample - Let X be the number of boys in a 4 child family. Find the probability distribution table:
Chapter8 Probability Distributions and Statistics Section 8.1 Distributions of Random Variables tthe value of the result of the probability experiment is a RANDOM VARIABLE. Example - Let X be the number
More informationMEASURES OF DISPERSION, RELATIVE STANDING AND SHAPE. Dr. Bijaya Bhusan Nanda,
MEASURES OF DISPERSION, RELATIVE STANDING AND SHAPE Dr. Bijaya Bhusan Nanda, CONTENTS What is measures of dispersion? Why measures of dispersion? How measures of dispersions are calculated? Range Quartile
More informationDATA HANDLING Five-Number Summary
DATA HANDLING Five-Number Summary The five-number summary consists of the minimum and maximum values, the median, and the upper and lower quartiles. The minimum and the maximum are the smallest and greatest
More informationBiostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras
Biostatistics and Design of Experiments Prof. Mukesh Doble Department of Biotechnology Indian Institute of Technology, Madras Lecture - 05 Normal Distribution So far we have looked at discrete distributions
More information