4.3 The Normal Distribution

Size: px
Start display at page:

Download "4.3 The Normal Distribution"

Transcription

1 4.3 The Normal Distribution Objectives. Definition of normal distribution. Standard normal distribution. Specialties of the graph of the standard normal distribution. Percentiles of the standard normal distribution. Z-values. Nonstandard normal distribution. Normal approximation to a discrete population. Normal approximation to a binomial distribution Definition. A continuous rv X is said to have a normal distribution with parameters and (or and ), where and 0, if the pdf of X is f x;, Abbreviation: X N, 1 x e x Meaning of the parameters and., V X E X Interactive website with normal curves: 1

2 4.3. Graphs of Normal Curves. Mean of, N determines the axis of symmetry x. Since Normal Probability Density Function is symmetric and bell-shaped, its mean is also its mode (the highest point) and its median (the point that splits the area under the curve in half). (!) Therefore, the area to each side of the is 0.5. Standard Deviation is a measure of spread (variability). When is large the normal curve is wide and, therefore, low. When is small, the normal curve is narrow and high. 1 Inflection point on the graph,.

3 The Empirical Rule for N, If the population distribution of a variable is (approximately) normal, then 1. Roughly 68% of the values are within 1 SD of the mean.. Roughly 95% of the values are within SDs of the mean. 3. Roughly 99.7% of the values are within 3 SDs of the mean. Remark When dealing with, N we need to compute x P a X b e dx b a 1 There is no antiderivative as an elementary function, so standard techniques of computing such integrals do not work. These integrals are computed by using numerical techniques and tabulations. 3

4 Standard Normal Distribution. Definitions. The normal distribution with parameters 0 and 1 is called the standard normal distribution. A random variable having a standard normal distribution is called a standard normal random variable and will be denoted by Z. The pdf of Z is The graph of ; 0,1 z 1 f z; 0,1 e z The standard normal distribution Z N0,1 z 1 f z e is called standard normal curve, or z curve. Cumulative function of a standard normal distribution. z ;0,1 z P Z z f t dt is used as a reference distribution from which information about other normal distributions can be obtained. N 0,1 Center at 0. Inflection point 1 1, 4

5 Standartization. Process of transforming, X N in Z N 0,1 Computing probabilities for Z N 0,1 Example (4.13 p. 154 textbook.) Appendix Table A3 Determine the following standard normal probabilities: a) PZ 1.5 ; b) PZ 1.5 ; c) PZ 1.5, and d) P 0.5 Z 1.5 PZ PZ

6 PZ P 0.5 Z P 0.5 Z

7 Percentiles of the Standard Normal Distribution. Z-table. For any p between 0 and 1, Appendix Table A.3 can be used to obtain the (100p) th percentile of the standard normal distribution. The 99th percentile of the standard normal distribution is the value on the horizontal axis such that the area under the z curve to the left of the value is Appendix Table A.3 gives for fixed z the area under the standard normal curve to the left of z. So, we are given the area and need to determine the value of z. This is the inverse problem. Thus, the table is used in an inverse fashion: Find in the middle of the table.9900; the row and column in which it lies identify the 99 th z percentile. Here the intersection of the row marked.3 and column marked.03, so the 99th percentile is (approximately) z

8 Example Find the 95 th percentile of N 0;1. 95 th percentile Z Values for Commonly Used Percentiles Percentile 1st Z th 5th 10th 5th 50th 75th 90th 95th 97.5th 99th The most important for theory and practice Notation for z Critical Values. will denote the value on the z-axis for which of the area under the z curve lies to the right of is equal to. Remark. The area under the curve to the left of is. This means that is the percentile of the standard normal distribution 8

9 Illustration for critical value z. Area 1 Area Example Find critical value z for Solution. 1) Critical value z ) In the table body find the value 0.85(0.8508) and consider the row and column that contain this value. 3) Read critical value z 1.04 Area Area 0.15 Critical value z

10 Approximating the Binomial Distribution. The probability histogram of Binomial Distribution is a bit skewed and the normal curve gives a very good approximation for it. Let X be a binomial rv based on n trials with success probability p. Then if the binomial probability histogram is not too skewed, X has approximately a normal distribution with np and npq. area under the normal curve P X x B x, n p to the left of x 0.5 x 0.5 np npq The good approximation is provided when np 10 and nq 10. Example (Example 4.0 p.161). Suppose that 5% of all students at a large public university receive financial aid. Let X be the number of students in a random sample of size 50 who receive financial aid, so that p 0.5. Then 1.5 and Since np nq , the approximation can safely be applied. 10

11 For example, the probability that at most 10 students receive aid is P X 10 B10; 50,

6-1 THE STANDARD NORMAL DISTRIBUTION

6-1 THE STANDARD NORMAL DISTRIBUTION 6-1 THE STANDARD NORMAL DISTRIBUTION The major focus of this chapter is the concept of a normal probability distribution, but we begin with a uniform distribution so that we can see the following two very

More information

Chapter 6 Normal Probability Distributions

Chapter 6 Normal Probability Distributions Chapter 6 Normal Probability Distributions 6-1 Review and Preview 6-2 The Standard Normal Distribution 6-3 Applications of Normal Distributions 6-4 Sampling Distributions and Estimators 6-5 The Central

More information

Chapter 6. The Normal Distribution. McGraw-Hill, Bluman, 7 th ed., Chapter 6 1

Chapter 6. The Normal Distribution. McGraw-Hill, Bluman, 7 th ed., Chapter 6 1 Chapter 6 The Normal Distribution McGraw-Hill, Bluman, 7 th ed., Chapter 6 1 Bluman, Chapter 6 2 Chapter 6 Overview Introduction 6-1 Normal Distributions 6-2 Applications of the Normal Distribution 6-3

More information

Chapter 2 Modeling Distributions of Data

Chapter 2 Modeling Distributions of Data Chapter 2 Modeling Distributions of Data Section 2.1 Describing Location in a Distribution Describing Location in a Distribution Learning Objectives After this section, you should be able to: FIND and

More information

10.4 Measures of Central Tendency and Variation

10.4 Measures of Central Tendency and Variation 10.4 Measures of Central Tendency and Variation Mode-->The number that occurs most frequently; there can be more than one mode ; if each number appears equally often, then there is no mode at all. (mode

More information

10.4 Measures of Central Tendency and Variation

10.4 Measures of Central Tendency and Variation 10.4 Measures of Central Tendency and Variation Mode-->The number that occurs most frequently; there can be more than one mode ; if each number appears equally often, then there is no mode at all. (mode

More information

Chapter 6: Continuous Random Variables & the Normal Distribution. 6.1 Continuous Probability Distribution

Chapter 6: Continuous Random Variables & the Normal Distribution. 6.1 Continuous Probability Distribution Chapter 6: Continuous Random Variables & the Normal Distribution 6.1 Continuous Probability Distribution and the Normal Probability Distribution 6.2 Standardizing a Normal Distribution 6.3 Applications

More information

Lecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 6.2-1

Lecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 6.2-1 Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Section 6.2-1 Chapter 6 Normal Probability Distributions 6-1 Review and Preview 6-2 The Standard

More information

CHAPTER 2 Modeling Distributions of Data

CHAPTER 2 Modeling Distributions of Data CHAPTER 2 Modeling Distributions of Data 2.2 Density Curves and Normal Distributions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Density Curves

More information

Chapter 6. THE NORMAL DISTRIBUTION

Chapter 6. THE NORMAL DISTRIBUTION Chapter 6. THE NORMAL DISTRIBUTION Introducing Normally Distributed Variables The distributions of some variables like thickness of the eggshell, serum cholesterol concentration in blood, white blood cells

More information

CHAPTER 2: Describing Location in a Distribution

CHAPTER 2: Describing Location in a Distribution CHAPTER 2: Describing Location in a Distribution 2.1 Goals: 1. Compute and use z-scores given the mean and sd 2. Compute and use the p th percentile of an observation 3. Intro to density curves 4. More

More information

Density Curve (p52) Density curve is a curve that - is always on or above the horizontal axis.

Density Curve (p52) Density curve is a curve that - is always on or above the horizontal axis. 1.3 Density curves p50 Some times the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve. It is easier to work with a smooth curve, because the histogram

More information

Lecture 3 Questions that we should be able to answer by the end of this lecture:

Lecture 3 Questions that we should be able to answer by the end of this lecture: Lecture 3 Questions that we should be able to answer by the end of this lecture: Which is the better exam score? 67 on an exam with mean 50 and SD 10 or 62 on an exam with mean 40 and SD 12 Is it fair

More information

Chapter 2: The Normal Distributions

Chapter 2: The Normal Distributions Chapter 2: The Normal Distributions Measures of Relative Standing & Density Curves Z-scores (Measures of Relative Standing) Suppose there is one spot left in the University of Michigan class of 2014 and

More information

MAT 102 Introduction to Statistics Chapter 6. Chapter 6 Continuous Probability Distributions and the Normal Distribution

MAT 102 Introduction to Statistics Chapter 6. Chapter 6 Continuous Probability Distributions and the Normal Distribution MAT 102 Introduction to Statistics Chapter 6 Chapter 6 Continuous Probability Distributions and the Normal Distribution 6.2 Continuous Probability Distributions Characteristics of a Continuous Probability

More information

Probability & Statistics Chapter 6. Normal Distribution

Probability & Statistics Chapter 6. Normal Distribution I. Graphs of Normal Probability Distributions Normal Distribution Studied by French mathematician Abraham de Moivre and German mathematician Carl Friedrich Gauss. Gauss work was so important that the normal

More information

Chapter 6. THE NORMAL DISTRIBUTION

Chapter 6. THE NORMAL DISTRIBUTION Chapter 6. THE NORMAL DISTRIBUTION Introducing Normally Distributed Variables The distributions of some variables like thickness of the eggshell, serum cholesterol concentration in blood, white blood cells

More information

Lecture 6: Chapter 6 Summary

Lecture 6: Chapter 6 Summary 1 Lecture 6: Chapter 6 Summary Z-score: Is the distance of each data value from the mean in standard deviation Standardizes data values Standardization changes the mean and the standard deviation: o Z

More information

Lecture 3 Questions that we should be able to answer by the end of this lecture:

Lecture 3 Questions that we should be able to answer by the end of this lecture: Lecture 3 Questions that we should be able to answer by the end of this lecture: Which is the better exam score? 67 on an exam with mean 50 and SD 10 or 62 on an exam with mean 40 and SD 12 Is it fair

More information

Ch6: The Normal Distribution

Ch6: The Normal Distribution Ch6: The Normal Distribution Introduction Review: A continuous random variable can assume any value between two endpoints. Many continuous random variables have an approximately normal distribution, which

More information

Student Learning Objectives

Student Learning Objectives Student Learning Objectives A. Understand that the overall shape of a distribution of a large number of observations can be summarized by a smooth curve called a density curve. B. Know that an area under

More information

Stat 528 (Autumn 2008) Density Curves and the Normal Distribution. Measures of center and spread. Features of the normal distribution

Stat 528 (Autumn 2008) Density Curves and the Normal Distribution. Measures of center and spread. Features of the normal distribution Stat 528 (Autumn 2008) Density Curves and the Normal Distribution Reading: Section 1.3 Density curves An example: GRE scores Measures of center and spread The normal distribution Features of the normal

More information

Learning Objectives. Continuous Random Variables & The Normal Probability Distribution. Continuous Random Variable

Learning Objectives. Continuous Random Variables & The Normal Probability Distribution. Continuous Random Variable Learning Objectives Continuous Random Variables & The Normal Probability Distribution 1. Understand characteristics about continuous random variables and probability distributions 2. Understand the uniform

More information

appstats6.notebook September 27, 2016

appstats6.notebook September 27, 2016 Chapter 6 The Standard Deviation as a Ruler and the Normal Model Objectives: 1.Students will calculate and interpret z scores. 2.Students will compare/contrast values from different distributions using

More information

Name: Date: Period: Chapter 2. Section 1: Describing Location in a Distribution

Name: Date: Period: Chapter 2. Section 1: Describing Location in a Distribution Name: Date: Period: Chapter 2 Section 1: Describing Location in a Distribution Suppose you earned an 86 on a statistics quiz. The question is: should you be satisfied with this score? What if it is the

More information

Chapter 5: The standard deviation as a ruler and the normal model p131

Chapter 5: The standard deviation as a ruler and the normal model p131 Chapter 5: The standard deviation as a ruler and the normal model p131 Which is the better exam score? 67 on an exam with mean 50 and SD 10 62 on an exam with mean 40 and SD 12? Is it fair to say: 67 is

More information

AP Statistics. Study Guide

AP Statistics. Study Guide Measuring Relative Standing Standardized Values and z-scores AP Statistics Percentiles Rank the data lowest to highest. Counting up from the lowest value to the select data point we discover the percentile

More information

STA Module 2B Organizing Data and Comparing Distributions (Part II)

STA Module 2B Organizing Data and Comparing Distributions (Part II) STA 2023 Module 2B Organizing Data and Comparing Distributions (Part II) Learning Objectives Upon completing this module, you should be able to 1 Explain the purpose of a measure of center 2 Obtain and

More information

STA Learning Objectives. Learning Objectives (cont.) Module 2B Organizing Data and Comparing Distributions (Part II)

STA Learning Objectives. Learning Objectives (cont.) Module 2B Organizing Data and Comparing Distributions (Part II) STA 2023 Module 2B Organizing Data and Comparing Distributions (Part II) Learning Objectives Upon completing this module, you should be able to 1 Explain the purpose of a measure of center 2 Obtain and

More information

STP 226 ELEMENTARY STATISTICS NOTES PART 2 - DESCRIPTIVE STATISTICS CHAPTER 3 DESCRIPTIVE MEASURES

STP 226 ELEMENTARY STATISTICS NOTES PART 2 - DESCRIPTIVE STATISTICS CHAPTER 3 DESCRIPTIVE MEASURES STP 6 ELEMENTARY STATISTICS NOTES PART - DESCRIPTIVE STATISTICS CHAPTER 3 DESCRIPTIVE MEASURES Chapter covered organizing data into tables, and summarizing data with graphical displays. We will now use

More information

Distributions of random variables

Distributions of random variables Chapter 3 Distributions of random variables 31 Normal distribution Among all the distributions we see in practice, one is overwhelmingly the most common The symmetric, unimodal, bell curve is ubiquitous

More information

Name Date Types of Graphs and Creating Graphs Notes

Name Date Types of Graphs and Creating Graphs Notes Name Date Types of Graphs and Creating Graphs Notes Graphs are helpful visual representations of data. Different graphs display data in different ways. Some graphs show individual data, but many do not.

More information

height VUD x = x 1 + x x N N 2 + (x 2 x) 2 + (x N x) 2. N

height VUD x = x 1 + x x N N 2 + (x 2 x) 2 + (x N x) 2. N Math 3: CSM Tutorial: Probability, Statistics, and Navels Fall 2 In this worksheet, we look at navel ratios, means, standard deviations, relative frequency density histograms, and probability density functions.

More information

Chapter 2: The Normal Distribution

Chapter 2: The Normal Distribution Chapter 2: The Normal Distribution 2.1 Density Curves and the Normal Distributions 2.2 Standard Normal Calculations 1 2 Histogram for Strength of Yarn Bobbins 15.60 16.10 16.60 17.10 17.60 18.10 18.60

More information

STA Rev. F Learning Objectives. Learning Objectives (Cont.) Module 3 Descriptive Measures

STA Rev. F Learning Objectives. Learning Objectives (Cont.) Module 3 Descriptive Measures STA 2023 Module 3 Descriptive Measures Learning Objectives Upon completing this module, you should be able to: 1. Explain the purpose of a measure of center. 2. Obtain and interpret the mean, median, and

More information

Frequency Distributions

Frequency Distributions Displaying Data Frequency Distributions After collecting data, the first task for a researcher is to organize and summarize the data so that it is possible to get a general overview of the results. Remember,

More information

Chapter 2 Describing, Exploring, and Comparing Data

Chapter 2 Describing, Exploring, and Comparing Data Slide 1 Chapter 2 Describing, Exploring, and Comparing Data Slide 2 2-1 Overview 2-2 Frequency Distributions 2-3 Visualizing Data 2-4 Measures of Center 2-5 Measures of Variation 2-6 Measures of Relative

More information

BIOL Gradation of a histogram (a) into the normal curve (b)

BIOL Gradation of a histogram (a) into the normal curve (b) (التوزيع الطبيعي ( Distribution Normal (Gaussian) One of the most important distributions in statistics is a continuous distribution called the normal distribution or Gaussian distribution. Consider the

More information

Unit 7 Statistics. AFM Mrs. Valentine. 7.1 Samples and Surveys

Unit 7 Statistics. AFM Mrs. Valentine. 7.1 Samples and Surveys Unit 7 Statistics AFM Mrs. Valentine 7.1 Samples and Surveys v Obj.: I will understand the different methods of sampling and studying data. I will be able to determine the type used in an example, and

More information

Normal Distribution. 6.4 Applications of Normal Distribution

Normal Distribution. 6.4 Applications of Normal Distribution Normal Distribution 6.4 Applications of Normal Distribution 1 /20 Homework Read Sec 6-4. Discussion question p316 Do p316 probs 1-10, 16-22, 31, 32, 34-37, 39 2 /20 3 /20 Objective Find the probabilities

More information

What s Normal Anyway?

What s Normal Anyway? Name Class Problem 1 A Binomial Experiment 1. When rolling a die, what is the theoretical probability of rolling a 3? 2. When a die is rolled 100 times, how many times do you expect that a 3 will be rolled?

More information

Section 10.4 Normal Distributions

Section 10.4 Normal Distributions Section 10.4 Normal Distributions Random Variables Suppose a bank is interested in improving its services to customers. The manager decides to begin by finding the amount of time tellers spend on each

More information

Chapter 1. Looking at Data-Distribution

Chapter 1. Looking at Data-Distribution Chapter 1. Looking at Data-Distribution Statistics is the scientific discipline that provides methods to draw right conclusions: 1)Collecting the data 2)Describing the data 3)Drawing the conclusions Raw

More information

CHAPTER 2 DESCRIPTIVE STATISTICS

CHAPTER 2 DESCRIPTIVE STATISTICS CHAPTER 2 DESCRIPTIVE STATISTICS 1. Stem-and-Leaf Graphs, Line Graphs, and Bar Graphs The distribution of data is how the data is spread or distributed over the range of the data values. This is one of

More information

Measures of Central Tendency. A measure of central tendency is a value used to represent the typical or average value in a data set.

Measures of Central Tendency. A measure of central tendency is a value used to represent the typical or average value in a data set. Measures of Central Tendency A measure of central tendency is a value used to represent the typical or average value in a data set. The Mean the sum of all data values divided by the number of values in

More information

Measures of Central Tendency

Measures of Central Tendency Page of 6 Measures of Central Tendency A measure of central tendency is a value used to represent the typical or average value in a data set. The Mean The sum of all data values divided by the number of

More information

L E A R N I N G O B JE C T I V E S

L E A R N I N G O B JE C T I V E S 2.2 Measures of Central Location L E A R N I N G O B JE C T I V E S 1. To learn the concept of the center of a data set. 2. To learn the meaning of each of three measures of the center of a data set the

More information

Downloaded from

Downloaded from UNIT 2 WHAT IS STATISTICS? Researchers deal with a large amount of data and have to draw dependable conclusions on the basis of data collected for the purpose. Statistics help the researchers in making

More information

Central Limit Theorem Sample Means

Central Limit Theorem Sample Means Date Central Limit Theorem Sample Means Group Member Names: Part One Review of Types of Distributions Consider the three graphs below. Match the histograms with the distribution description. Write the

More information

Chapter 2. Descriptive Statistics: Organizing, Displaying and Summarizing Data

Chapter 2. Descriptive Statistics: Organizing, Displaying and Summarizing Data Chapter 2 Descriptive Statistics: Organizing, Displaying and Summarizing Data Objectives Student should be able to Organize data Tabulate data into frequency/relative frequency tables Display data graphically

More information

Learner Expectations UNIT 1: GRAPICAL AND NUMERIC REPRESENTATIONS OF DATA. Sept. Fathom Lab: Distributions and Best Methods of Display

Learner Expectations UNIT 1: GRAPICAL AND NUMERIC REPRESENTATIONS OF DATA. Sept. Fathom Lab: Distributions and Best Methods of Display CURRICULUM MAP TEMPLATE Priority Standards = Approximately 70% Supporting Standards = Approximately 20% Additional Standards = Approximately 10% HONORS PROBABILITY AND STATISTICS Essential Questions &

More information

Basic Statistical Terms and Definitions

Basic Statistical Terms and Definitions I. Basics Basic Statistical Terms and Definitions Statistics is a collection of methods for planning experiments, and obtaining data. The data is then organized and summarized so that professionals can

More information

Normal Data ID1050 Quantitative & Qualitative Reasoning

Normal Data ID1050 Quantitative & Qualitative Reasoning Normal Data ID1050 Quantitative & Qualitative Reasoning Histogram for Different Sample Sizes For a small sample, the choice of class (group) size dramatically affects how the histogram appears. Say we

More information

1.3 Graphical Summaries of Data

1.3 Graphical Summaries of Data Arkansas Tech University MATH 3513: Applied Statistics I Dr. Marcel B. Finan 1.3 Graphical Summaries of Data In the previous section we discussed numerical summaries of either a sample or a data. In this

More information

Lecture Series on Statistics -HSTC. Frequency Graphs " Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.)

Lecture Series on Statistics -HSTC. Frequency Graphs  Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.) Lecture Series on Statistics -HSTC Frequency Graphs " By Dr. Bijaya Bhusan Nanda, Ph. D. (Stat.) CONTENT Histogram Frequency polygon Smoothed frequency curve Cumulative frequency curve or ogives Learning

More information

Section 9: One Variable Statistics

Section 9: One Variable Statistics The following Mathematics Florida Standards will be covered in this section: MAFS.912.S-ID.1.1 MAFS.912.S-ID.1.2 MAFS.912.S-ID.1.3 Represent data with plots on the real number line (dot plots, histograms,

More information

Distributions of Continuous Data

Distributions of Continuous Data C H A P T ER Distributions of Continuous Data New cars and trucks sold in the United States average about 28 highway miles per gallon (mpg) in 2010, up from about 24 mpg in 2004. Some of the improvement

More information

Chapter 2: Modeling Distributions of Data

Chapter 2: Modeling Distributions of Data Chapter 2: Modeling Distributions of Data Section 2.2 The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE Chapter 2 Modeling Distributions of Data 2.1 Describing Location in a Distribution

More information

Data Analysis & Probability

Data Analysis & Probability Unit 5 Probability Distributions Name: Date: Hour: Section 7.2: The Standard Normal Distribution (Area under the curve) Notes By the end of this lesson, you will be able to Find the area under the standard

More information

STA Module 4 The Normal Distribution

STA Module 4 The Normal Distribution STA 2023 Module 4 The Normal Distribution Learning Objectives Upon completing this module, you should be able to 1. Explain what it means for a variable to be normally distributed or approximately normally

More information

STA /25/12. Module 4 The Normal Distribution. Learning Objectives. Let s Look at Some Examples of Normal Curves

STA /25/12. Module 4 The Normal Distribution. Learning Objectives. Let s Look at Some Examples of Normal Curves STA 2023 Module 4 The Normal Distribution Learning Objectives Upon completing this module, you should be able to 1. Explain what it means for a variable to be normally distributed or approximately normally

More information

The Normal Distribution

The Normal Distribution Chapter 6 The Normal Distribution Continuous random variables are used to approximate probabilities where there are many possibilities or an infinite number of possibilities on a given trial. One of the

More information

Math 120 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency

Math 120 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency Math 1 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency lowest value + highest value midrange The word average: is very ambiguous and can actually refer to the mean,

More information

UNIT 1A EXPLORING UNIVARIATE DATA

UNIT 1A EXPLORING UNIVARIATE DATA A.P. STATISTICS E. Villarreal Lincoln HS Math Department UNIT 1A EXPLORING UNIVARIATE DATA LESSON 1: TYPES OF DATA Here is a list of important terms that we must understand as we begin our study of statistics

More information

IT 403 Practice Problems (1-2) Answers

IT 403 Practice Problems (1-2) Answers IT 403 Practice Problems (1-2) Answers #1. Using Tukey's Hinges method ('Inclusionary'), what is Q3 for this dataset? 2 3 5 7 11 13 17 a. 7 b. 11 c. 12 d. 15 c (12) #2. How do quartiles and percentiles

More information

Section 1.2. Displaying Quantitative Data with Graphs. Mrs. Daniel AP Stats 8/22/2013. Dotplots. How to Make a Dotplot. Mrs. Daniel AP Statistics

Section 1.2. Displaying Quantitative Data with Graphs. Mrs. Daniel AP Stats 8/22/2013. Dotplots. How to Make a Dotplot. Mrs. Daniel AP Statistics Section. Displaying Quantitative Data with Graphs Mrs. Daniel AP Statistics Section. Displaying Quantitative Data with Graphs After this section, you should be able to CONSTRUCT and INTERPRET dotplots,

More information

1. To condense data in a single value. 2. To facilitate comparisons between data.

1. To condense data in a single value. 2. To facilitate comparisons between data. The main objectives 1. To condense data in a single value. 2. To facilitate comparisons between data. Measures :- Locational (positional ) average Partition values Median Quartiles Deciles Percentiles

More information

Prepare a stem-and-leaf graph for the following data. In your final display, you should arrange the leaves for each stem in increasing order.

Prepare a stem-and-leaf graph for the following data. In your final display, you should arrange the leaves for each stem in increasing order. Chapter 2 2.1 Descriptive Statistics A stem-and-leaf graph, also called a stemplot, allows for a nice overview of quantitative data without losing information on individual observations. It can be a good

More information

Measures of Dispersion

Measures of Dispersion Measures of Dispersion 6-3 I Will... Find measures of dispersion of sets of data. Find standard deviation and analyze normal distribution. Day 1: Dispersion Vocabulary Measures of Variation (Dispersion

More information

Normal Curves and Sampling Distributions

Normal Curves and Sampling Distributions Normal Curves and Sampling Distributions 6 Copyright Cengage Learning. All rights reserved. Section 6.2 Standard Units and Areas Under the Standard Normal Distribution Copyright Cengage Learning. All rights

More information

Measures of Central Tendency

Measures of Central Tendency Measures of Central Tendency MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Introduction Measures of central tendency are designed to provide one number which

More information

Applied Statistics for the Behavioral Sciences

Applied Statistics for the Behavioral Sciences Applied Statistics for the Behavioral Sciences Chapter 2 Frequency Distributions and Graphs Chapter 2 Outline Organization of Data Simple Frequency Distributions Grouped Frequency Distributions Graphs

More information

Math 167 Pre-Statistics. Chapter 4 Summarizing Data Numerically Section 3 Boxplots

Math 167 Pre-Statistics. Chapter 4 Summarizing Data Numerically Section 3 Boxplots Math 167 Pre-Statistics Chapter 4 Summarizing Data Numerically Section 3 Boxplots Objectives 1. Find quartiles of some data. 2. Find the interquartile range of some data. 3. Construct a boxplot to describe

More information

CHAPTER 2: DESCRIPTIVE STATISTICS Lecture Notes for Introductory Statistics 1. Daphne Skipper, Augusta University (2016)

CHAPTER 2: DESCRIPTIVE STATISTICS Lecture Notes for Introductory Statistics 1. Daphne Skipper, Augusta University (2016) CHAPTER 2: DESCRIPTIVE STATISTICS Lecture Notes for Introductory Statistics 1 Daphne Skipper, Augusta University (2016) 1. Stem-and-Leaf Graphs, Line Graphs, and Bar Graphs The distribution of data is

More information

MTH 3210: PROBABILITY AND STATISTICS DESCRIPTIVE STATISTICS WORKSHEET

MTH 3210: PROBABILITY AND STATISTICS DESCRIPTIVE STATISTICS WORKSHEET MTH 3210: PROBABILITY AND STATISTICS DESCRIPTIVE STATISTICS WORKSHEET Before you work on the practice problems (Section 3) please make sure that you read the supplementary notes (Section 1) and work through

More information

Goals. The Normal Probability Distribution. A distribution. A Discrete Probability Distribution. Results of Tossing Two Dice. Probabilities involve

Goals. The Normal Probability Distribution. A distribution. A Discrete Probability Distribution. Results of Tossing Two Dice. Probabilities involve Goals The Normal Probability Distribution Chapter 7 Dr. Richard Jerz Understand the difference between discrete and continuous distributions. Compute the mean, standard deviation, and probabilities for

More information

Section 2.2 Normal Distributions

Section 2.2 Normal Distributions Section 2.2 Mrs. Daniel AP Statistics We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ). Any particular Normal distribution is completely specified by two numbers: its

More information

MAT 110 WORKSHOP. Updated Fall 2018

MAT 110 WORKSHOP. Updated Fall 2018 MAT 110 WORKSHOP Updated Fall 2018 UNIT 3: STATISTICS Introduction Choosing a Sample Simple Random Sample: a set of individuals from the population chosen in a way that every individual has an equal chance

More information

The Normal Probability Distribution. Goals. A distribution 2/27/16. Chapter 7 Dr. Richard Jerz

The Normal Probability Distribution. Goals. A distribution 2/27/16. Chapter 7 Dr. Richard Jerz The Normal Probability Distribution Chapter 7 Dr. Richard Jerz 1 2016 rjerz.com Goals Understand the difference between discrete and continuous distributions. Compute the mean, standard deviation, and

More information

Univariate Statistics Summary

Univariate Statistics Summary Further Maths Univariate Statistics Summary Types of Data Data can be classified as categorical or numerical. Categorical data are observations or records that are arranged according to category. For example:

More information

MATH 1070 Introductory Statistics Lecture notes Descriptive Statistics and Graphical Representation

MATH 1070 Introductory Statistics Lecture notes Descriptive Statistics and Graphical Representation MATH 1070 Introductory Statistics Lecture notes Descriptive Statistics and Graphical Representation Objectives: 1. Learn the meaning of descriptive versus inferential statistics 2. Identify bar graphs,

More information

Chapter 5: The normal model

Chapter 5: The normal model Chapter 5: The normal model Objective (1) Learn how rescaling a distribution affects its summary statistics. (2) Understand the concept of normal model. (3) Learn how to analyze distributions using the

More information

Math 14 Lecture Notes Ch. 6.1

Math 14 Lecture Notes Ch. 6.1 6.1 Normal Distribution What is normal? a 10-year old boy that is 4' tall? 5' tall? 6' tall? a 25-year old woman with a shoe size of 5? 7? 9? an adult alligator that weighs 200 pounds? 500 pounds? 800

More information

So..to be able to make comparisons possible, we need to compare them with their respective distributions.

So..to be able to make comparisons possible, we need to compare them with their respective distributions. Unit 3 ~ Modeling Distributions of Data 1 ***Section 2.1*** Measures of Relative Standing and Density Curves (ex) Suppose that a professional soccer team has the money to sign one additional player and

More information

Ms Nurazrin Jupri. Frequency Distributions

Ms Nurazrin Jupri. Frequency Distributions Frequency Distributions Frequency Distributions After collecting data, the first task for a researcher is to organize and simplify the data so that it is possible to get a general overview of the results.

More information

The Normal Distribution

The Normal Distribution The Normal Distribution Lecture 20 Section 6.3.1 Robb T. Koether Hampden-Sydney College Wed, Sep 28, 2011 Robb T. Koether (Hampden-Sydney College) The Normal Distribution Wed, Sep 28, 2011 1 / 41 Outline

More information

CHAPTER 3: Data Description

CHAPTER 3: Data Description CHAPTER 3: Data Description You ve tabulated and made pretty pictures. Now what numbers do you use to summarize your data? Ch3: Data Description Santorico Page 68 You ll find a link on our website to a

More information

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 1.3 Homework Answers Assignment 5 1.80 If you ask a computer to generate "random numbers between 0 and 1, you uniform will

More information

CHAPTER 2 Modeling Distributions of Data

CHAPTER 2 Modeling Distributions of Data CHAPTER 2 Modeling Distributions of Data 2.2 Density Curves and Normal Distributions The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers HW 34. Sketch

More information

Chapter 3 - Displaying and Summarizing Quantitative Data

Chapter 3 - Displaying and Summarizing Quantitative Data Chapter 3 - Displaying and Summarizing Quantitative Data 3.1 Graphs for Quantitative Data (LABEL GRAPHS) August 25, 2014 Histogram (p. 44) - Graph that uses bars to represent different frequencies or relative

More information

Sections 4.3 and 4.4

Sections 4.3 and 4.4 Sections 4.3 and 4.4 Timothy Hanson Department of Statistics, University of South Carolina Stat 205: Elementary Statistics for the Biological and Life Sciences 1 / 32 4.3 Areas under normal densities Every

More information

Chapter 2: Frequency Distributions

Chapter 2: Frequency Distributions Chapter 2: Frequency Distributions Chapter Outline 2.1 Introduction to Frequency Distributions 2.2 Frequency Distribution Tables Obtaining ΣX from a Frequency Distribution Table Proportions and Percentages

More information

Organizing and Summarizing Data

Organizing and Summarizing Data 1 Organizing and Summarizing Data Key Definitions Frequency Distribution: This lists each category of data and how often they occur. : The percent of observations within the one of the categories. This

More information

The Normal Distribution

The Normal Distribution 14-4 OBJECTIVES Use the normal distribution curve. The Normal Distribution TESTING The class of 1996 was the first class to take the adjusted Scholastic Assessment Test. The test was adjusted so that the

More information

Probability and Statistics. Copyright Cengage Learning. All rights reserved.

Probability and Statistics. Copyright Cengage Learning. All rights reserved. Probability and Statistics Copyright Cengage Learning. All rights reserved. 14.6 Descriptive Statistics (Graphical) Copyright Cengage Learning. All rights reserved. Objectives Data in Categories Histograms

More information

AP Statistics Summer Assignment:

AP Statistics Summer Assignment: AP Statistics Summer Assignment: Read the following and use the information to help answer your summer assignment questions. You will be responsible for knowing all of the information contained in this

More information

Chapter 5. Normal. Normal Curve. the Normal. Curve Examples. Standard Units Standard Units Examples. for Data

Chapter 5. Normal. Normal Curve. the Normal. Curve Examples. Standard Units Standard Units Examples. for Data curve Approximation Part II Descriptive Statistics The Approximation Approximation The famous normal curve can often be used as an 'ideal' histogram, to which histograms for data can be compared. Its equation

More information

a. divided by the. 1) Always round!! a) Even if class width comes out to a, go up one.

a. divided by the. 1) Always round!! a) Even if class width comes out to a, go up one. Probability and Statistics Chapter 2 Notes I Section 2-1 A Steps to Constructing Frequency Distributions 1 Determine number of (may be given to you) a Should be between and classes 2 Find the Range a The

More information

The standard deviation 1 n

The standard deviation 1 n The standard deviation 1 SD = (xj x) n 2 The SD gives a measure of how the data are clustered around the mean. If the SD is larger, then the data are more spread out we are more likely to find data that

More information

Section 2-2 Frequency Distributions. Copyright 2010, 2007, 2004 Pearson Education, Inc

Section 2-2 Frequency Distributions. Copyright 2010, 2007, 2004 Pearson Education, Inc Section 2-2 Frequency Distributions Copyright 2010, 2007, 2004 Pearson Education, Inc. 2.1-1 Frequency Distribution Frequency Distribution (or Frequency Table) It shows how a data set is partitioned among

More information