# MEASURES OF VARIATION

Size: px
Start display at page:

Transcription

1 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 relatively close to, or far from the mean, so measures of variation are used to determine how far observations are from the mean. The common measures of variation are mean deviation, variance, and standard deviation. Mean Deviation The average distance each piece of data is from the mean. To calculate the mean deviation: 1. Calculate the mean of the data. 2. Subtract the mean from each piece of data, and take the absolute value of the result. 3. Add all of the results. 4. Divide by the number of values in the data set. Example 1 Consider these two sets of data: Set A: 10, 30, 39, 40, 43, 48, 70 Set B: 10, 20, 30, 40, 50, 60, 70 The mean for both sets is 40 and the range for both is = 60. To decide which set has the greater spread, calculate the mean deviation. 1. Calculate the mean: 280/7=40 2. Find the difference of each data point from the mean, then take the absolute value. Set A: Set B: = = = = = = = = = = = = = = Add all of the results from Step 2. Set A: = 82 Set B: = Divide by the number of values: Set A: 82 7! Set B: 120 7! Based on the mean deviation, Set B has the greater spread of data because the average distance of each piece of data from the mean is greater.

2 Another way to measure the spread of a set of data is called the variance. To find the variance of a set of data: 1. Calculate the mean of the data. 2. Subtract the mean from each piece of data then square the difference (this ensures that all values are positive, just as using the absolute value did for mean deviation). 3. Add all the squared differences. 4 Divide by the one less than the number of values in the data set. The symbol for variance of a sample is s 2. Example 2 Find the variance for Sets A and B in Example Calculate the mean of each set: Subtract the mean, then square the difference in each set: Set A: Set B: (40! 10) 2 = 900 (40! 10) 2 = 900 (40! 30) 2 = 100 (40! 20) 2 = 400 (40! 39) 2 = 1 (40! 30) 2 = 100 (40! 40) 2 = 0 (40! 40) 2 = 0 (40! 43) 2 = 9 (40! 50) 2 = 100 (40! 48) 2 = 64 (40! 60) 2 = 400 (40! 70) 2 = 900 (40! 70) 2 = Add all of the squared differences in each set: Set A: = 1974 Set B: = Divide by one less than the number of values in each set: Set A: 1974 (7! 1) = 329 Set B: 2800 (7! 1) = The variance (s 2 ) is 329 for Set A and s 2 = for Set B. Based on the variance, set B has a greater spread.

3 Finally, a more common measure of spread is the standard deviation. The standard deviation is the square root of the variance. The symbol for the standard deviation is s. standard deviation = s 2 = s If you know the variance (s 2 ) of a data set and want to find the standard deviation (s), take the square root of the variance. If the variance is not known, then it must first be calculated to find the standard deviation. Example 3 Example 4 Find the standard deviations for Set A and Find the standard deviation of: Set B in Example 3. 11, 19, 30, 22, 26 To find the standard deviation, take the square root of the variance. Set A: s = 329! Set B: s = ! While it is probably easier to understand and calculate the mean deviation, the standard deviation is most used because of its relationship to the normal distribution, which will be developed in the next example. 1. Mean: 108/5 = Sum of squared differences from the mean: ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 = Divide by one less than the total number of numbers: 209.2/4 = Square root: 52.3 = s = Problems 1. Find the mean deviation of the data: 5, 2, 7, 8, 1, 5, 4, 6, 10, Find the variance of the data: 5, 2, 7, 8, 1, 5, 4, 6, 10, Find the variance of the data: 24, 39, 41, 32, 56, If the variance of a data set is , find the standard deviation. 5. Find the standard deviation of the data set: 41, 58, 32, 56, 67, 49, 45, and 46.

4 Answers 1. Mean deviation is s 2 = s 2 = s = s =

5 PREDICTIONS BASED ON NORMAL DISTRIBUTIONS A normal distribution is an important theoretical idea used in statistics to model data applicable to large populations. The graph of a normal distribution is a symmetrical bellshaped curve based on the mean and standard deviation of a sample, where the mean, median, and mode are the same. Below are the graphs of normal distributions with the same mean, but different standard deviations. The mean is located at the center of the graph. Since this is also the median, half of the values lie above the mean and the other half lie below the mean. The shape of the curve is determined by the standard deviation, or spread of the data. The normal distribution can describe many things in the world such as scores on tests like the SAT. Characteristics in nature also fall along the normal curve, such as height or weight for adult men and women. When the mean and standard deviation of a normal distribution are known, then several other values of the distribution can be calculated. It is important to be able to draw a curve and label the mean, as well as three standard deviations above and below the mean. Theoretically, almost all of the data should fall within 3 standard deviations (on each side) of the mean.

6 Example 1 Suppose the average height of women is normally distributed and has a mean of 64.5 inches and a standard deviation of 2.5 inches. Draw a normal curve out to three standard deviations on either side of the mean. Answer: Draw a bell-shaped curve with the mean of 64.5 in the center. Draw 3 equally spaced vertical lines both above and below the mean. Each vertical line represents one standard deviation. In this case the standard deviation is 2.5, so to label each vertical line, add or subtract 2.5 from the previous line to find the value. Example 2 Based on the information given, label the remaining values, state the mean and standard deviation. Numerically, the distance from 12 to 24 is 12. Since 24 is on the normal curve above is three standard deviations above 12, each standard deviation must be 4. Redraw the curve and label from left to right: 4, 8, 12, 16, 20, 24, 28 The mean of the curve is 16 with a standard deviation of 4. Problems 1. Draw and label a curve (out to three standard deviations) that has a mean of 50 and a standard deviation of Finish labeling the curve at right. State the mean and standard deviation.

8 Example 4 Based on the information in Example 3, what percentage of people taking the course are: a. Older than 36? b. Younger than 30? c. Between the ages of 36 and 42? Answers: a. In a normal distribution, half the values lie above the mean, and the other half below the mean. So 50% are older than the mean age of 36. b. We know (from the rule) that about 95% of the data lie between 30 and 42, so that must mean that the remaining 5% lie outside that interval. Since normal curves are symmetric, half of the 5% must lie below 30 and the other half above 42. So 2.5% of the people are younger than 30. c. As stated in parts (a) and (b). 50% of the data lie below the mean of 36, and 2.5% of the data lie above 42. The remaining 47.5% of the data must lie between 36 and 42. Problems 3. Sketch a normal curve that has a mean of 30 and a standard deviation of 5. Then complete the statements: 68% of the data are between and. 95% of the data are between and. 99.7% of the data are between and. 4. Using the graph you made in problem #3, answer the following questions: a. What percent of the data lie above 15? b. What percent of the data lie between 20 and 25? c. What percent of the data lie below 35? Answers 1. The curve is centered at 50, and the labels from left to right are: 5, 20, 35, 50, 65, 80, a % b. 13.5% c. 84% 2. The curve is labeled from left to right: 77, 82, 87, 92, 97, 102, 107. The mean is 92 and the standard deviation is % of the data are between 25 and % of the data are between 20 and % of the data are between 15 and 45.

### 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

### 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,

### 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

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

### 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)

### Topic 9 ~ Measures of Spread

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

### Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve

### 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

### Unit 7: Normal Curves

Unit 7: Normal Curves Summary of Video Histograms of completely unrelated data often exhibit similar shapes. To focus on the overall shape of a distribution and to avoid being distracted by the irregularities

### 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

### Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

### 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.

### 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

### 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,

### 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

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### 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

### 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

### 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

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### 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

### 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

### 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 :

### 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

### Coins, Presidents, and Justices: Normal Distributions and z-scores

activity 17.1 Coins, Presidents, and Justices: Normal Distributions and z-scores In the first part of this activity, you will generate some data that should have an approximately normal (or bell-shaped)

### 7. Normal Distributions

7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped

### 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

### 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

### The correlation coefficient

The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

### The Normal Distribution

Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution

### 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

### 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

### Frequency Distributions

Descriptive Statistics Dr. Tom Pierce Department of Psychology Radford University Descriptive statistics comprise a collection of techniques for better understanding what the people in a group look like

### 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

### 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

### WEEK #22: PDFs and CDFs, Measures of Center and Spread

WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook

### 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

### Scatter Plots with Error Bars

Chapter 165 Scatter Plots with Error Bars Introduction The procedure extends the capability of the basic scatter plot by allowing you to plot the variability in Y and X corresponding to each point. Each

### 5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

### 4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

### 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

### Statistics. Measurement. Scales of Measurement 7/18/2012

Statistics Measurement Measurement is defined as a set of rules for assigning numbers to represent objects, traits, attributes, or behaviors A variableis something that varies (eye color), a constant does

### 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.

### HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

### 6 3 The Standard Normal Distribution

290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### 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

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### Characteristics of Binomial Distributions

Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation

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

Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the mean for the given sample data. 1) Frank's Furniture employees earned the following

### 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

### 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

### 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),

### Introduction to Quantitative Methods

Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................

### 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

### Chapter 4. Probability and Probability Distributions

Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

### 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

### 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.

### 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

### 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

### 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

### The Normal distribution

The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

### Continuous Random Variables

Chapter 5 Continuous Random Variables 5.1 Continuous Random Variables 1 5.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand continuous

### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

### Midterm Review Problems

Midterm Review Problems October 19, 2013 1. Consider the following research title: Cooperation among nursery school children under two types of instruction. In this study, what is the independent variable?

### Statistics Revision Sheet Question 6 of Paper 2

Statistics Revision Sheet Question 6 of Paper The Statistics question is concerned mainly with the following terms. The Mean and the Median and are two ways of measuring the average. sumof values no. of

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade

Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements

### 8. THE NORMAL DISTRIBUTION

8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs

### 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

### Module 3: Correlation and Covariance

Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

### 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

### 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.

### 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

### 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

### 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

### CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

### 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

### REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### 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

### Content Sheet 7-1: Overview of Quality Control for Quantitative Tests

Content Sheet 7-1: Overview of Quality Control for Quantitative Tests Role in quality management system Quality Control (QC) is a component of process control, and is a major element of the quality management

### 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

### Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering

Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques

### The Standard Normal distribution

The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

### Activity 3.7 Statistical Analysis with Excel

Activity 3.7 Statistical Analysis with Excel Introduction Engineers use various tools to make their jobs easier. Spreadsheets can greatly improve the accuracy and efficiency of repetitive and common calculations;

### 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

### Lesson 20. Probability and Cumulative Distribution Functions

Lesson 20 Probability and Cumulative Distribution Functions Recall If p(x) is a density function for some characteristic of a population, then Recall If p(x) is a density function for some characteristic

### Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

### seven Statistical Analysis with Excel chapter OVERVIEW CHAPTER

seven Statistical Analysis with Excel CHAPTER chapter OVERVIEW 7.1 Introduction 7.2 Understanding Data 7.3 Relationships in Data 7.4 Distributions 7.5 Summary 7.6 Exercises 147 148 CHAPTER 7 Statistical

### 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

### Lecture 2: Discrete Distributions, Normal Distributions. Chapter 1

Lecture 2: Discrete Distributions, Normal Distributions Chapter 1 Reminders Course website: www. stat.purdue.edu/~xuanyaoh/stat350 Office Hour: Mon 3:30-4:30, Wed 4-5 Bring a calculator, and copy Tables

### 2.5 Adding and Subtracting Fractions and Mixed Numbers with Like Denominators

2.5 Adding and Subtracting Fractions and Mixed Numbers with Like Denominators Learning Objective(s) Add fractions with like denominators. 2 Subtract fractions with like denominators. Add mixed numbers

### 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

### Adding and Subtracting Positive and Negative Numbers

Adding and Subtracting Positive and Negative Numbers Absolute Value For any real number, the distance from zero on the number line is the absolute value of the number. The absolute value of any real number

### Probability. Distribution. Outline

7 The Normal Probability Distribution Outline 7.1 Properties of the Normal Distribution 7.2 The Standard Normal Distribution 7.3 Applications of the Normal Distribution 7.4 Assessing Normality 7.5 The