1. © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 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

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

3. © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 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 4.1 4.2 4.3 4.4 4.5 Use box plots to visualize distributions 4.6

4. 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

5. 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. 6 The difference between the highest and lowest scores in a distribution Provides a crude measure of variation –Outliers affect interpretation The Range

7. 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

8. IQR: Example 1 3.1 What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4

9. IQR: Example 1 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

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

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

12. 12 IQR from Frequency Table Xfcf 31125 30124 29123 28022 27222 26320 25117 24116 23215 22213 21211 2039 1946 1822 When you are given a frequency table instead of the raw data

13. 13 IQR from Frequency Table X= 18 18 19 19 19 19 20 20 20 21 21 22 22 23 23 24 25 26 26 26 27 27 29 30 31 Pos = 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Median Pos = 1 2 3 4 5 6 7 8 9 10 11 12 X= 18 18 19 19 19 19 20 20 20 21 21 22 1 st Quartile X= 23 23 24 25 26 26 26 27 27 29 30 31 Pos = 1 2 3 4 5 6 7 8 9 10 11 12 3 rd Quartile

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

15. 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

16. 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

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

18. 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

19. 4.2 19 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

20. 4.2 20 There is an easier way to calculate the variance and standard deviation Using raw scores The Raw-Score Formulas Formula for frequency tables

21. Standard Deviation: Raw Data What is the standard deviation of the following distribution: 1, 5, 2, 9, 13, 11, 4 XDev.Sq. Dev 11-6.42= -5.42(-5.42) 2 =29.37 55-6.42= -1.42(-1.42) 2 =2.01 22-6.42= -4.42(-4.42) 2 =19.53 99-6.42= 2.58(2.58) 2 =6.65 1313-6.42= 6.58(6.58) 2 =43.29 1111-6.42= 4.58(4.58) 2 =20.97 44-6.42= -2.42(-2.42) 2 =5.85

22. 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

23. Example 4.3 Obtaining the variance and standard deviation from a simple frequency distribution XffXfX 2 311 961 301 900 291 841 28000 272541,458 263782,028 251 625 241 576 232461,058 22244968 21242882 203601,200 194761,444 182 36 648 57513,589

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

25. 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. 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

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

28. 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. 29 Example 1.Find CV

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

31. Figure 4.4

32. 4.6 Figure 4.5

33. 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. 34 Problem 12

35. 35 Now in Excel 1. Find IQR of BMI: http://office.microsoft.com/en-us/excel- help/quartile-HP005209226.aspx http://office.microsoft.com/en-us/excel- help/quartile-HP005209226.aspx 2. Find standard deviation of BMI: http://office.microsoft.com/en- us/excel-help/stdev-HP005209277.aspx http://office.microsoft.com/en- us/excel-help/stdev-HP005209277.aspx 3. Find CV of BMI:

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

37. © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 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 4.1 4.2 4.3 4.4 4.5 Social researchers often use box plots to visualize various aspects of a distribution 4.6