# Topic 9 ~ Measures of Spread

Save this PDF as:

Size: px
Start display at page:

Download "Topic 9 ~ Measures of Spread"

## Transcription

1 AP Statistics Topic 9 ~ Measures of Spread Activity 9 : Baseball Lineups The table to the right contains data on the ages of the two teams involved in game of the 200 National League Division Series. Is there a relationship between the ages of the players on the teams and the outcome of the NLDS? The Reds lost in three games. a. Identify the observational units. Also identify the explanatory and the response variable. Classify each variable as categorical or quantitative. Observational units: starting players in game of 200 NLDS Explanatory variable: age Type: quantitative Response variable: NLDS Winner Type: categorical b. Create comparative dotplots, using the axes shown here for comparing the ages between the two teams. (Be sure to label which dotplot represents which team.) Comment on how the age distributions compare. Reds Phillies Comment: The starting lineup for the Phillies tended to be older than the starting lineup for the Reds. The Phillies are more consistent in age than the Reds. c. Calculate the mean and median age of each team's lineup. Reds Mean: Median: 29 Phillies Mean: Median: 32

2 d. Which team's lineup appears to have more variability in its ages? The Red's lineup appears to have more variability in its ages. In the previous topic, you learned that the mean and median are two ways to measure the center of a distribution; you will now learn several ways to measure the spread, or variability, of a distribution. e. What is the age of the oldest player in the Phillies lineup? The youngest? What is the difference in age between the oldest and youngest player? Oldest: 38 Youngest: 30 Difference: = 8 f. Repeat part e for the Red's lineup. Oldest: 36 Youngest: 23 Difference: = 3 A very simple, but not particularly useful, measure of variability is the range, calculated as the difference between the maximum and minimum values in a data set. Another measure of variability is the interquartile range (IQR), which is the difference between the upper quartile and the lower quartile of a distribution. The lower quartile (or the 25th percentile, abbreviated Q ) is the value such that 25% of the data values are less that that value and 75% are greater than it, while the upper quartile (or the 75th percentile, abbreviated Q 3 ) is the value such that 75% of the values in the data set are less than that value and 25% are greater than it. Thus, the IQR is the range of the middle 50% of the data. g. Determine the lower and upper quartiles of the ages for the Phillies. Then find the IQR of the Phillie's ages IQR = Q3 - Q = =.5 Q = 3 Q3 = 32.5 h. Determine the lower and upper quartiles of the ages for the Reds. Then find the IQR of the Red's ages Q3 = 34.5 Q = 26.5 IQR = Q3 - Q = = 8 i. Which team has the greater age range? Which has the greater IQR? Are these values consistent with your answer to question d? The Reds have a greater range of ages, 3 versus 8 years, and a greater interquartile range of ages, 8 versus.5 years. j. Based on this analysis, summarize how the age distributions differ between the 200 Reds and Phillies (shape, center, spread). The distribution of ages for the Reds has no distinct shape, but there is a roughly symmetric cluster of ages between 23 & 30 and an evenly dispersed set of ages between 34 & 36. The median age is 29 and the IQR is 8 years. The distribution of ages for the Phillies is mound-shaped and symmetric with a potential outlier in the 38-year-old left-fielder Raul Ibanez. The median age of 32 and an IQR of.5 years. The Cincinnati team tended to be younger than the Philadelphia team, though the ages of the Red's players vary quite a bit more. 2

3 Activity 9 2: Baseball Lineups Other measures of variability examine how far the data values fall or deviate from the mean of the distribution. a. The mean age for Cincinnati's starting lineup in game one of the 200 NLDS was approximately Complete the missing entries for Votto and Rolen in the "deviation from the mean" column of the following table by calculating the differences between their ages and the mean age = = 5.33 b. Add the values in the "deviation from Mean" column. Then calculate the average deviation from the mean /9 = The un rounded values from the table appear in the table at right. Fathom calculates the sum of the deviations to be zero, as in the table below c. The sum of the deviations from the mean is always equal to zero. Verify this fact for the data set {, 5, 2}. ( )/3 = 6-6 = = = = 0 d. Given the fact that the sum of the deviations from the mean is always zero, what does that imply about using the average deviation from the mean as a measure of spread (variation) for a data set? The average deviation is a useless measure of spread since it is always going to be zero. Because a measure of spread is concerned with distances from the mean rather than direction from the mean, you could work with the absolute values of these deviations. e. Complete the missing entries in the "Absolute Deviation" column of the table below. Then calculate the average absolute deviation. Report the units of measurement for this calculation /9 = 3.63 years The measure of spread you have just calculated is the mean absolute deviation (MAD). It is certainly a reasonable measure of the amount of variation relative to the mean in a data set, but there is yet another measure of spread that has properties desirable to statisticians, as you soon shall see. f. Complete the missing entries in the "Squared Deviation" column of the table above. Then calculate the average squared deviation. Report the units of measurement for this calculation. This value is called the variance (V). g years 2 To convert back to the original units of the data set years of age take the square root of the average squared deviation. 4.2 years The measure of spread you have just calculated is the standard deviation (SD). The standard deviation is the most widely used measure of variation in statistical calculations. 3

4 The standard deviation ("baby" sigma σ) is a widely used measure of variability. To compute the standard deviation, you calculate the difference between the mean and each data value and then square the difference: (data value mean) 2. Add these squared terms, and divide the number of observational units n. The standard deviation is the square root of the result: or, more simply, The standard deviation can loosely be interpreted as the typical distance that a data value in the distribution deviates from the mean. The variance σ 2 is calculated by the formula The variance is literally the average squared deviation from the mean. σ is the Greek lowercase "sigma" and is used to represent the standard deviation (of a population). is the Greek uppercase "sigma" and is the symbol used to imply summation. μ is the Greek lowercase "myoo" and is the symbol used to represent the mean. n is the number of observational units. x is used to represent the value of a variable for a particular observational unit. Here's how to do it on the TI 83/84. standard deviation h. Calculate, with technology, the standard deviation of the ages for the Phillies' starting lineup in game of the 200 NLDS. 2.2 years i. j. Now, remove 38 year old Raul Ibanez from the Phillies lineup and calculate the standard deviation of the ages for the Phillies' starting lineup..87 Calculate the range, interquartile range (IQR), and standard deviation of the ages for the Phillies' starting lineup in game of the 200 NLDS with and without Raul Ibanez. Complete the table below years.87 years k. Which measures of spread are resistant to outliers and which are not? Explain. The IQR is least affected by the presence of Raul Ibanez since it changed the least when he is included in the Phillies lineup. 4

5 count count Activity 9-4: Placement Exam Scores Placement Score a. The distribution of placement scores appears to be roughly symmetric and mound shaped. b. μ 0.22 and σ μ - σ = = μ + σ = = 4.08 c. 46 of the 23 scores fall within one standard deviation of the mean, i.e., between 7 and 4 inclusive.this accounts for 46/ or about 69% of the scores. This is quite consistent with the 68% advertised by the Empirical Rule Placement Score d. μ - 2σ = (3.859) = μ + 2σ = (3.859) = of the 23 scores fall within two standard deviations of the mean, i.e., between 3 and 7, inclusive. This accounts for.948 or about 95% of the scores. This is exactly what the Empirical Rule advertises. e. 23 of the 23 scores fall within three standard deviations of the mean, i.e., between and 9. This accounts for 00% of the placement scores. This is quite in line with the Empirical Rule. 5

6 Activity 9-5: SATs and ACTs a. 740 is 240 points above the mean SAT score. b. 30 is 9 points above the mean ACT score. c. No. You cannot compare these point differences because the SAT and ACT scores are not measured on the same numeric scale. d. Bobby's SAT score is 240/240 = standard deviation above the mean SAT score. e. Kathy's ACT score is 9/6 =.5 standard deviations above the mean ACT score. f. Kathy's ACT z score is which is greater than Bobby's SAT z score of =.5 g. Since Kathy's score of 30 on the ACT is.5 standard deviations above the mean score in the approximately Normal distribution of ACT scores, while Bobby's score of 540 on the SAT is only one standard deviation above the mean score in the approximately Normal distribution of SAT scores, Kathy performed better relative to the peers whose scores appear in the distribution of all ACT scores. 240 = h. z Peter = = 0.5 z Kelly = = i. Peter has the higher z score since < 0.5. j. A z score turns out to be negative when calculated for any score less than the mean score for the associated distribution. 6

7 Activity 9-6: Marriage Ages a. Husbands tend to be older than their wives by a mean of.875 years and a median of.5 years on average. b. The IQR for the distribution of husbands' ages is = 9.5 years. The IQR for the distribution of wives' ages is = 7.5 years. The standard deviation of husbands' ages is 4.26 years while the standard deviation of wives' ages is There is more variability in the distribution of husbands' ages than in the distribution of wives' ages. c. The distributions of husbands' ages and wives' ages are both skewed right. The median age of the husbands is 30.5, while the median age of the wives is only 29, indicating that the husbands tended to be older than the wives. With an interquartile range of 9.5, two more than that of the wives, there is slightly more variation in the ages of the husbands than in the ages of the wives husbands' ages wives' ages d. The mean difference (husband age minus wife age) is equal to the difference of the means, mean husband age minus mean wife age. The median difference is NOT equal to the difference of the medians. IQR = e. Neither the difference in the IQRs, nor the difference in the standard deviations, is equal to the IQR of the differences or the standard deviation of the differences. 7

### MEASURES OF VARIATION

NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

More information

### The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median

CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box

More information

### 1.3 Measuring Center & Spread, The Five Number Summary & Boxplots. Describing Quantitative Data with Numbers

1.3 Measuring Center & Spread, The Five Number Summary & Boxplots Describing Quantitative Data with Numbers 1.3 I can n Calculate and interpret measures of center (mean, median) in context. n Calculate

More information

### Center: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.)

Center: Finding the Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center

More information

### Variables. Exploratory Data Analysis

Exploratory Data Analysis Exploratory Data Analysis involves both graphical displays of data and numerical summaries of data. A common situation is for a data set to be represented as a matrix. There is

More information

### Exploratory data analysis (Chapter 2) Fall 2011

Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,

More information

### Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

More information

### Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

More information

### Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences

Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html

More information

### 3: Summary Statistics

3: Summary Statistics Notation Let s start by introducing some notation. Consider the following small data set: 4 5 30 50 8 7 4 5 The symbol n represents the sample size (n = 0). The capital letter X denotes

More information

### Lecture 2: Descriptive Statistics and Exploratory Data Analysis

Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals

More information

### Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs

Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)

More information

### Ch. 3.1 # 3, 4, 7, 30, 31, 32

Math Elementary Statistics: A Brief Version, 5/e Bluman Ch. 3. # 3, 4,, 30, 3, 3 Find (a) the mean, (b) the median, (c) the mode, and (d) the midrange. 3) High Temperatures The reported high temperatures

More information

### Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

More information

### EXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!

STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.

More information

### Lecture 1: Review and Exploratory Data Analysis (EDA)

Lecture 1: Review and Exploratory Data Analysis (EDA) Sandy Eckel seckel@jhsph.edu Department of Biostatistics, The Johns Hopkins University, Baltimore USA 21 April 2008 1 / 40 Course Information I Course

More information

### 3.2 Measures of Spread

3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread

More information

### Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

More information

### Chapter 1: Exploring Data

Chapter 1: Exploring Data Chapter 1 Review 1. As part of survey of college students a researcher is interested in the variable class standing. She records a 1 if the student is a freshman, a 2 if the student

More information

### Classify the data as either discrete or continuous. 2) An athlete runs 100 meters in 10.5 seconds. 2) A) Discrete B) Continuous

Chapter 2 Overview Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Classify as categorical or qualitative data. 1) A survey of autos parked in

More information

### Means, standard deviations and. and standard errors

CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

More information

### Exploratory Data Analysis. Psychology 3256

Exploratory Data Analysis Psychology 3256 1 Introduction If you are going to find out anything about a data set you must first understand the data Basically getting a feel for you numbers Easier to find

More information

### Interpreting Data in Normal Distributions

Interpreting Data in Normal Distributions This curve is kind of a big deal. It shows the distribution of a set of test scores, the results of rolling a die a million times, the heights of people on Earth,

More information

### Data Exploration Data Visualization

Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select

More information

### Measures of Central Tendency and Variability: Summarizing your Data for Others

Measures of Central Tendency and Variability: Summarizing your Data for Others 1 I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). 1. Mode :

More information

### DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

More information

### COMPARISON MEASURES OF CENTRAL TENDENCY & VARIABILITY EXERCISE 8/5/2013. MEASURE OF CENTRAL TENDENCY: MODE (Mo) MEASURE OF CENTRAL TENDENCY: MODE (Mo)

COMPARISON MEASURES OF CENTRAL TENDENCY & VARIABILITY Prepared by: Jess Roel Q. Pesole CENTRAL TENDENCY -what is average or typical in a distribution Commonly Measures: 1. Mode. Median 3. Mean quantified

More information

### Introduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data

A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine jutts@uci.edu 1. Statistical Methods in Water Resources by D.R. Helsel

More information

### 9.1 Measures of Center and Spread

Name Class Date 9.1 Measures of Center and Spread Essential Question: How can you describe and compare data sets? Explore Exploring Data Resource Locker Caleb and Kim have bowled three games. Their scores

More information

### DESCRIPTIVE STATISTICS & DATA PRESENTATION*

Level 1 Level 2 Level 3 Level 4 0 0 0 0 evel 1 evel 2 evel 3 Level 4 DESCRIPTIVE STATISTICS & DATA PRESENTATION* Created for Psychology 41, Research Methods by Barbara Sommer, PhD Psychology Department

More information

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Exam Name 1) A recent report stated ʺBased on a sample of 90 truck drivers, there is evidence to indicate that, on average, independent truck drivers earn more than company -hired truck drivers.ʺ Does

More information

### consider the number of math classes taken by math 150 students. how can we represent the results in one number?

ch 3: numerically summarizing data - center, spread, shape 3.1 measure of central tendency or, give me one number that represents all the data consider the number of math classes taken by math 150 students.

More information

### Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

More information

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

### Northumberland Knowledge

Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about

More information

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

More information

### Exploratory Data Analysis

Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

More information

### STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

More information

### Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts

More information

### Section 1.3 Exercises (Solutions)

Section 1.3 Exercises (s) 1.109, 1.110, 1.111, 1.114*, 1.115, 1.119*, 1.122, 1.125, 1.127*, 1.128*, 1.131*, 1.133*, 1.135*, 1.137*, 1.139*, 1.145*, 1.146-148. 1.109 Sketch some normal curves. (a) Sketch

More information

### Geostatistics Exploratory Analysis

Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa Master of Science in Geospatial Technologies Geostatistics Exploratory Analysis Carlos Alberto Felgueiras cfelgueiras@isegi.unl.pt

More information

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

More information

### Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

More information

### MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

More information

### Shape of Data Distributions

Lesson 13 Main Idea Describe a data distribution by its center, spread, and overall shape. Relate the choice of center and spread to the shape of the distribution. New Vocabulary distribution symmetric

More information

### BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

More information

### Mean = (sum of the values / the number of the value) if probabilities are equal

Population Mean Mean = (sum of the values / the number of the value) if probabilities are equal Compute the population mean Population/Sample mean: 1. Collect the data 2. sum all the values in the population/sample.

More information

### Descriptive Statistics and Measurement Scales

Descriptive Statistics 1 Descriptive Statistics and Measurement Scales Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample

More information

### Chapter 3. The Normal Distribution

Chapter 3. The Normal Distribution Topics covered in this chapter: Z-scores Normal Probabilities Normal Percentiles Z-scores Example 3.6: The standard normal table The Problem: What proportion of observations

More information

### Descriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics

Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),

More information

### Chi Square Tests. Chapter 10. 10.1 Introduction

Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

More information

### Box-and-Whisker Plots

Mathematics Box-and-Whisker Plots About this Lesson This is a foundational lesson for box-and-whisker plots (boxplots), a graphical tool used throughout statistics for displaying data. During the lesson,

More information

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

More information

### 4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: "What do the data look like?"

Data Analysis Plan The appropriate methods of data analysis are determined by your data types and variables of interest, the actual distribution of the variables, and the number of cases. Different analyses

More information

### Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion

Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

More information

### Summarizing and Displaying Categorical Data

Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency

More information

### Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

### First Midterm Exam (MATH1070 Spring 2012)

First Midterm Exam (MATH1070 Spring 2012) Instructions: This is a one hour exam. You can use a notecard. Calculators are allowed, but other electronics are prohibited. 1. [40pts] Multiple Choice Problems

More information

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

More information

### AP * Statistics Review. Descriptive Statistics

AP * Statistics Review Descriptive Statistics Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production

More information

### 2. Filling Data Gaps, Data validation & Descriptive Statistics

2. Filling Data Gaps, Data validation & Descriptive Statistics Dr. Prasad Modak Background Data collected from field may suffer from these problems Data may contain gaps ( = no readings during this period)

More information

### Final Exam Practice Problem Answers

Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

More information

### AP Statistics Solutions to Packet 2

AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

More information

### 9. Sampling Distributions

9. Sampling Distributions Prerequisites none A. Introduction B. Sampling Distribution of the Mean C. Sampling Distribution of Difference Between Means D. Sampling Distribution of Pearson's r E. Sampling

More information

### The Big Picture. Describing Data: Categorical and Quantitative Variables Population. Descriptive Statistics. Community Coalitions (n = 175)

Describing Data: Categorical and Quantitative Variables Population The Big Picture Sampling Statistical Inference Sample Exploratory Data Analysis Descriptive Statistics In order to make sense of data,

More information

### Organizing Topic: Data Analysis

Organizing Topic: Data Analysis Mathematical Goals: Students will analyze and interpret univariate data using measures of central tendency and dispersion. Students will calculate the z-scores for data.

More information

### Diagrams and Graphs of Statistical Data

Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in

More information

### Name: Date: Use the following to answer questions 2-3:

Name: Date: 1. A study is conducted on students taking a statistics class. Several variables are recorded in the survey. Identify each variable as categorical or quantitative. A) Type of car the student

More information

### AP STATISTICS REVIEW (YMS Chapters 1-8)

AP STATISTICS REVIEW (YMS Chapters 1-8) Exploring Data (Chapter 1) Categorical Data nominal scale, names e.g. male/female or eye color or breeds of dogs Quantitative Data rational scale (can +,,, with

More information

### Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

More information

### Week 1. Exploratory Data Analysis

Week 1 Exploratory Data Analysis Practicalities This course ST903 has students from both the MSc in Financial Mathematics and the MSc in Statistics. Two lectures and one seminar/tutorial per week. Exam

More information

### Foundation of Quantitative Data Analysis

Foundation of Quantitative Data Analysis Part 1: Data manipulation and descriptive statistics with SPSS/Excel HSRS #10 - October 17, 2013 Reference : A. Aczel, Complete Business Statistics. Chapters 1

More information

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

More information

### Lesson 4 Measures of Central Tendency

Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central

More information

### AP Statistics 2005 Scoring Guidelines

AP Statistics 2005 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college

More information

### Introduction; Descriptive & Univariate Statistics

Introduction; Descriptive & Univariate Statistics I. KEY COCEPTS A. Population. Definitions:. The entire set of members in a group. EXAMPLES: All U.S. citizens; all otre Dame Students. 2. All values of

More information

### Random Variables. Chapter 2. Random Variables 1

Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

More information

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

More information

### Manhattan Center for Science and Math High School Mathematics Department Curriculum

Content/Discipline Algebra 1 Semester 2: Marking Period 1 - Unit 8 Polynomials and Factoring Topic and Essential Question How do perform operations on polynomial functions How to factor different types

More information

### + Chapter 1 Exploring Data

Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1 Analyzing Categorical Data 1.2 Displaying Quantitative Data with Graphs 1.3 Describing Quantitative Data with Numbers Introduction

More information

### Thursday, November 13: 6.1 Discrete Random Variables

Thursday, November 13: 6.1 Discrete Random Variables Read 347 350 What is a random variable? Give some examples. What is a probability distribution? What is a discrete random variable? Give some examples.

More information

### 2. Here is a small part of a data set that describes the fuel economy (in miles per gallon) of 2006 model motor vehicles.

Math 1530-017 Exam 1 February 19, 2009 Name Student Number E There are five possible responses to each of the following multiple choice questions. There is only on BEST answer. Be sure to read all possible

More information

### determining relationships among the explanatory variables, and

Chapter 4 Exploratory Data Analysis A first look at the data. As mentioned in Chapter 1, exploratory data analysis or EDA is a critical first step in analyzing the data from an experiment. Here are the

More information

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

More information

### Mind on Statistics. Chapter 2

Mind on Statistics Chapter 2 Sections 2.1 2.3 1. Tallies and cross-tabulations are used to summarize which of these variable types? A. Quantitative B. Mathematical C. Continuous D. Categorical 2. The table

More information

### 5. Linear Regression

5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

More information

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

More information

### International Statistical Institute, 56th Session, 2007: Phil Everson

Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA E-mail: peverso1@swarthmore.edu 1. Introduction

More information

### THE BINOMIAL DISTRIBUTION & PROBABILITY

REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution

More information

### Problem of the Month Pick a Pocket

The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

More information

### Basics of Statistics

Basics of Statistics Jarkko Isotalo 30 20 10 Std. Dev = 486.32 Mean = 3553.8 0 N = 120.00 2400.0 2800.0 3200.0 3600.0 4000.0 4400.0 4800.0 2600.0 3000.0 3400.0 3800.0 4200.0 4600.0 5000.0 Birthweights

More information

### SPSS for Exploratory Data Analysis Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav)

Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer

More information

### Measurement with Ratios

Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical

More information

### A Correlation of. to the. South Carolina Data Analysis and Probability Standards

A Correlation of to the South Carolina Data Analysis and Probability Standards INTRODUCTION This document demonstrates how Stats in Your World 2012 meets the indicators of the South Carolina Academic Standards

More information

### Lecture 2. Summarizing the Sample

Lecture 2 Summarizing the Sample WARNING: Today s lecture may bore some of you It s (sort of) not my fault I m required to teach you about what we re going to cover today. I ll try to make it as exciting

More information

### 1 Descriptive statistics: mode, mean and median

1 Descriptive statistics: mode, mean and median Statistics and Linguistic Applications Hale February 5, 2008 It s hard to understand data if you have to look at it all. Descriptive statistics are things

More information

### Probability Distributions

Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.

More information

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

More information

### Module 4: Data Exploration

Module 4: Data Exploration Now that you have your data downloaded from the Streams Project database, the detective work can begin! Before computing any advanced statistics, we will first use descriptive

More information