© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D. 2/17/2014, Spring 2014 Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 4: Measures of Variability 1

Announcement Let’s switch Lecture Chapter 5 and Exam 1 2

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Calculate the range and inter-quartile range Calculate the variance and standard deviation Obtain the variance and standard deviation from a simple frequency distribution Understand the meaning of the standard deviation Calculate the coefficient of variation CHAPTER OBJECTIVES Use box plots to visualize distributions 4.6

Introduction 4.1 Measures of Central Tendency Measures of Variability 4 Summarizes what is average or typical of a distribution Summarizes how scores are scattered around the center of the distribution

Calculate the range and inter- quartile rage Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.1

6 The difference between the highest and lowest scores in a distribution Provides a crude measure of variation –Outliers affect interpretation The Range

4.1 7 The difference between the score at the first quartile and the score at the third quartile Manages the effects of extreme outliers –Sensitive to the way in which scores are concentrated around the center of the distribution The Inter-Quartile Range

IQR: Example What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4

IQR: Example What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4 1 st Quartile 3 rd Quartile

IQR: Example What is the inter-quartile range of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4

IQR: Example 3.1 What is the inter-quartile range of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4

12 IQR from Frequency Table Xfcf When you are given a frequency table instead of the raw data

13 IQR from Frequency Table X= Pos = Median Pos = X= st Quartile X= Pos = rd Quartile

IQR Advantage: Outliers 3.1 What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 1300, 11, 4

IQR Advantage: Outliers 3.1 What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4 1 st Quartile 3 rd Quartile

IQR Advantage: Outliers 3.1 What is the range and mean of the following distribution: 1, 5, 2, 9, 1300, 11, 4 vs. 1, 5, 2, 9, 13, 11, 4 Range=1300-1=1299 Range=13-1=12

Calculate the variance and standard deviation Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.2

18 We need a measure of variability that takes into account every score Deviation: the distance of any given raw score from the mean Squaring deviations eliminates the minus signs Summing the squared deviations and dividing by N gives us the average of the squared deviations The Variance

With the variance, the unit of measurement is squared It is difficult to interpret squared units We can remove the squared units by taking the square root of both sides of the equation This will give us the standard deviation The Standard Deviation “Original” formula for raw data

There is an easier way to calculate the variance and standard deviation Using raw scores The Raw-Score Formulas Formula for frequency tables

Standard Deviation: Raw Data What is the standard deviation of the following distribution: 1, 5, 2, 9, 13, 11, 4 XDev.Sq. Dev = -5.42(-5.42) 2 = = -1.42(-1.42) 2 = = -4.42(-4.42) 2 = = 2.58(2.58) 2 = = 6.58(6.58) 2 = = 4.58(4.58) 2 = = -2.42(-2.42) 2 =5.85

Obtain the variance and standard deviation from a simple frequency distribution Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.3

Example 4.3 Obtaining the variance and standard deviation from a simple frequency distribution XffXfX , , , , , ,589

24 Additional Example Find Variance and Standard Deviation using frequency table from last session

Understand the meaning of the standard deviation Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.4

26 The standard deviation converts the variance to units we can understand But, how do we interpret this new score? The standard deviation represents the average variability in a distribution –It is the average deviations from the mean The greater the variability, the larger the standard deviation Allows for a comparison between a given raw score in a set against a standardized measure The Meaning of the Standard Deviation

Calculate the coefficient of variation Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.5

28 Used to compare the variability for two or more characteristics that have been measured in different units The coefficient of variation is based on the size of the standard deviation Its value is independent of the unit of the measurement The Coefficient of Variation

29 Example 1.Find CV

Use box plots to visualize distributions Learning Objectives After this lecture, you should be able to complete the following Learning Outcomes 4.6

Figure 4.4

4.6 Figure 4.5

Box Plot: Examples Draw the box plot of the following distribution: 1, 5, 2, 9, 13, 11, 4 Draw the box plot of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4

34 Problem 12

35 Now in Excel 1. Find IQR of BMI: help/quartile-HP aspx help/quartile-HP aspx 2. Find standard deviation of BMI: us/excel-help/stdev-HP aspx us/excel-help/stdev-HP aspx 3. Find CV of BMI:

36 Homework #4 Problems (Chapter 4): Problems 20 (+boxplot) and 25

© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved Researchers can calculate the range and inter-quartile range for a crude measure of variation The variance and standard deviation provides the research with a measure of variation that takes into account every score The variance and standard deviation can also be calculated for data presented in a simple frequency distribution The standard deviation can be understood as the average of deviations from the mean The coefficient of variation is used to compare the variability for two or more characteristics that have been measured in different units CHAPTER SUMMARY Social researchers often use box plots to visualize various aspects of a distribution 4.6