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 2/10/2014 Spring 2014 Fox/Levin/Forde, Elementary Statistics in Social Research, 12e Chapter 3: Measures of Central Tendency 1

2. Review of class intervals 2 VariableFrequency 15-1910 20-248 25-2912 30-359 Suppose we have this table What are the real interval lower and upper limits? VariableFrequency 15.5-19.510 19.5-24.58 24.5-29.512 29.5-35.59 VariableFrequency 15-19.9910 20-24.998 25-29.9912 30-34.999 Answer: It depends! You must really know the data Book VariableFrequency 15-1910 20-248 25-2912 30-359

3. Review of class intervals 3 VariableFrequency 15.5-19.510 19.5-24.58 24.5-29.512 29.5-35.59 Conclusion: It was unfair to ask you to know the real limits, i.e. to become an expert in the problem at hand (I guess that authors of the book are) Book Suppose that we are talking about Age, do you think that this are the correct limits? My guess is that the limits are more like: Supposed that we are talking about number persons that live in household, do we really have to worry about where 19.3 will fall? VariableFrequency 15-19.9910 20-24.998 25-29.9912 30-34.999 VariableFrequency 15-1910 20-248 25-2912 30-359

4. Review of class intervals 4 Jus be consistent: VariableFrequencyMidpoint 14.5-19.510(14.5+19.5)/2=17 19.5-24.58 24.5-29.512 29.5-35.59 VariableFrequencyMidpoint 15-19.9910(15+19.99)/2=17.5 20-24.998 25-29.9912 30-34.999 VariableFrequencyMidpoint 15-1910(15+19)/2=17 20-248 25-2912 30-359 17 19.5 17.5 20 In the exam I’ll clearly establish the limits

5. © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved Calculate the mode, the median, and the mean Calculate deviations Calculate the weighted mean Calculate the mode, the median, and the mean from a simple frequency distribution Understand what influences a researcher’s decision to use a specific measure of central tendency CHAPTER OBJECTIVES 3.1 3.2 3.3 3.4 3.5

6. Introduction 3.1 Measures of Central Tendency

7. 3.1 7 The most frequently occurring value in a distribution Example: 20, 21, 30, 20, 22, 20, 21, 20 –Mode = 20 Sometimes there is more than one mode –Example: 96, 91, 96, 90, 93, 90, 96, 90 –Mode = 90 and 96 This is a bimodal distribution The mode is the only measure of central tendency appropriate for nominal-level variables The Mode

8. 3.1 There is also multimodal, also the modes don’t have to be the same size

9. 3.1 9 The middlemost case in a distribution Appropriate for ordinal or interval level data How to find the median: –Cases must be ordered –If there are an odd number of cases, there will be a single middlemost case –If there are an even number of cases, there will be two middlemost cases – The halfway point between these two cases should be used as the median The Median

10. The Median: Example 1 3.1 What is the median of the following distribution: 1, 5, 2, 9, 13, 11, 4

11. The Median: Example 2 3.1 What is the median of the following distribution: 4, 3, 1, 1, 6, 2, 2, 4

12. 3.1 12 The “center of gravity” of a distribution Appropriate for interval/level data The Mean

13. 3.1 Figure 3.2

14. The Mean: Example 3.1 What is the mean of the following distribution: 4, 8, 11, 2

15. 3.2 15 The distance and direction of any raw score from the mean The sum of the deviations that fall above the mean is equal in absolute value to the sum of the deviations that fall below the mean. Deviations

16. 3.3 16 The “mean of the means” The overall mean for a number of groups The Weighted Mean

17. 3.4 17 Obtaining the Mode, Median, and Mean from a Simple Frequency Distribution XfcffX 3112531 3012430 2912329 280220 2722254 2632078 2511725 2411624 2321546 2221344 2121142 203960 194676 182236 Mo Mdn Position of the Mdn When you are given a frequency table instead of the raw data

18. Comparing the Mode, Median, and Mean 3.5 Three factors in choosing a measure of central tendency

19. 3.5 19 Level of Measurement ModeMedianMean Nominal YesNo Ordinal Yes No Interval Yes

20. 3.5 20 Symmetrical Distributions The mode, median, and mean have identical values Skewed Distributions The mode is the peak of the curve The mean is closer to the tail The median falls between the two Bimodal Distributions Both modes should be used to describe the data Shape of the Distribution

21. 3.5 Figure 3.3

22. 3.5 Figure 3.4

23. 3.5 23 Fast and Simple Research  Mode Skewed Distribution  Median Advanced Statistics Analysis  Mean Research Objective

24. 24 In class exercises

25. 25 Now in MS Excel http://office.microsoft.com/en-us/excel- help/calculate-the-median-of-a-group-of- numbers-HP003056145.aspx

26. 26 Homework: (problem 31 Chapter 2) + Chapter 3: Problem #25 and #30

27. © 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey 07458 All Rights Reserved There are three measures of central tendency: the mode, the median, and the mean How individual scores compare to the mean of the distribution can be examined by calculating deviations The mean of means, or the weighted mean, can be calculated for multiple groups The mode, the median, and the mean can also be calculated when data are presented in a simple frequency distribution Choosing which measure of central tendency to report is influenced by the level of measurement of the data, the shape of the distribution, as well as the research objective CHAPTER SUMMARY 3.1 3.2 3.3 3.4 3.5