HLTH 300 Biostatistics for Public Health Practice, Raul Cruz-Cano, Ph.D.

1. Fox/Levin/Forde, Elementary Statistics in Social Research, 12e • Chapter 4: Measures of Variability HLTH 300 Biostatistics for Public Health Practice,Raul Cruz-Cano, Ph.D. 2/17/2014, Spring 2014

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

3. CHAPTER OBJECTIVES 4.1 • Calculate the range and inter-quartile range 4.2 • Calculate the variance and standard deviation • Obtain the variance and standard deviation from a simple frequency distribution 4.3 4.4 • Understand the meaning of the standard deviation 4.5 • Calculate the coefficient of variation 4.6 • Use box plots to visualize distributions

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

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

6. 4.1 The Range The difference between the highest and lowest scores in a distribution • Provides a crude measure of variation • Outliers affect interpretation

7. 4.1 The Inter-Quartile Range 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

8. 3.1 IQR: Example 1 What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4 Step 1: Sort distribution from lowest to highest 1, 2, 4, 5, 9, 11, 13 Step 2: Locate the position of the median Step 3: Locate the median 1, 2, 4, 5, 9, 11, 13

9. 3.1 IQR: Example 1 What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4 Step 4: Separate the 2 halves 1, 2, 4 9, 11, 13 Step 5: Find the “median” of each half 1, 2, 4 9, 11, 13 Step 6: Calculate inter-quartile range IQR = 3rd Quartile – 1st Quartile = 11 – 2 = 9 1st Quartile 3rd Quartile

10. 3.1 IQR: Example 2 What is the inter-quartile range of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4 Step 1: Sort distribution from lowest to highest 1, 1, 2, 2, 3, 4, 4, 6 Step 2: Locate the position of the median Step 3: Locate the median 1, 1, 2, 2, 3, 4, 4, 6 Step 4: Take the halfway point between the two cases Median = 2.5

11. 3.1 IQR: Example What is the inter-quartile range of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4 Step 4: Separate the 2 halves 1, 1, 2, 2 3, 4, 4, 6 Step 5: Find the “median” of each half 1, 1, 2, 4 3, 4, 4,6 Step 6: Calculate inter-quartile range IQR = 3rd Quartile – 1st Quartile = 4 – 1.5 = 2.5

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

13. IQR from Frequency Table 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 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 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 1st Quartile Pos = 1 2 3 4 5 6 7 8 9 10 11 12 X= 23 23 24 25 26 26 26 27 27 29 30 31 3rd Quartile

14. 3.1 IQR Advantage: Outliers What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 1300, 11, 4 Step 1: Sort distribution from lowest to highest 1, 2, 4, 5, 9, 11, 1300 Step 2: Locate the position of the median Step 3: Locate the median 1, 2, 4, 5, 9, 11, 1300

15. 3.1 IQR Advantage: Outliers What is the inter-quartile range of the following distribution: 1, 5, 2, 9, 13, 11, 4 Step 4: Separate the 2 halves 1, 2, 4 9, 11, 1300 Step 5: Find the “median” of each half 1, 2, 4 9, 11, 1300 Step 6: Calculate inter-quartile range IQR = 3rd Quartile – 1st Quartile = 11 – 2 = 9 1st Quartile 3rd Quartile

16. 3.1 IQR Advantage: Outliers 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=13-1=12 Range=1300-1=1299

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

18. 4.2 The Variance 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

19. 4.2 The Standard Deviation 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 “Original” formula for raw data

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

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

23. 4.3 Example Obtaining the variance and standard deviation from a simple frequency distribution

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

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

26. 4.4 The Meaning of the Standard Deviation 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

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

28. 4.5 The Coefficient of Variation 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

29. Example Find CV

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

31. 4.6 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. Problem 12

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

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

37. CHAPTER SUMMARY • Researchers can calculate the range and inter-quartile range for a crude measure of variation 4.1 • The variance and standard deviation provides the research with a measure of variation that takes into account every score 4.2 • The variance and standard deviation can also be calculated for data presented in a simple frequency distribution 4.3 • The standard deviation can be understood as the average of deviations from the mean 4.4 • The coefficient of variation is used to compare the variability for two or more characteristics that have been measured in different units 4.5 • Social researchers often use box plots to visualize various aspects of a distribution 4.6